% Complex Analytic and Differential Geometry, Chapter IX
% J.-P. Demailly, Universit\'e de Grenoble I, Saint Martin d'H\`eres, France

\input analgeom.mac

\def\gm{{\frak m}}
\def\red{{\rm red}}
\def\toor{\hbox{\rm T\^or}}
\def\Coker{{\rm Coker}}

\def\hotimes{\mathrel{\widehat\otimes}}


\def\stimes{\mathop{\kern0.7pt
\vrule height 0.4pt depth 0pt width 5pt\kern-5pt
\vrule height 5.4pt depth -5pt width 5pt\kern-5pt
\vrule height 5.4pt depth 0pt width 0.4pt\kern4.6pt
\vrule height 5.4pt depth 0pt width 0.4pt\kern-6.5pt
\raise0.3pt\hbox{$\times$}\kern-0.7pt}\nolimits}
\def\gge{{\scriptscriptstyle\ge}}

\catcode`@=11
\def\rightarrowfil{\m@th\mathord-\mkern-6mu%
  \cleaders\hbox{$\mkern-2mu\mathord-\mkern-2mu$}\hfill
  \mkern-6mu\mathord\rightarrow}
\catcode`@=12

\titlea{Chapter IX}{\newline Finiteness Theorems for q-Convex Spaces and
Stein Spaces}

\titleb{1.}{Topological Preliminaries}
\titlec{1.A.}{Krull Topology of $\cO_n$-Modules}
We shall use in an essential way different kind of topological results.
The first of these concern the topology of modules over a local ring
and depend on the Artin-Rees and Krull lemmas.
Let $R$ be a noetherian local ring; ``local" means that $R$ has a unique
maximal ideal $\gm$, or equivalently, that $R$ has an ideal $\gm$ such
that every element $\alpha\in R\ssm\gm$ is invertible.

\begstat{(1.1) Nakayama lemma} Let $E$ be a finitely 
generated $R$-module such that $\gm E=E$. Then $E=\{0\}$.
\endstat

\begproof{} By induction on the number of generators of $E\,$:
if $E$ is generated by $x_1\ld x_p$, the hypothesis $E=\gm E$ shows
that $x_p=\alpha_1x_1+\cdots+\alpha_px_p$ with $\alpha_j\in\gm\,$;
as $1-\alpha_p\in R\ssm\gm$ is invertible, we see that $x_p$
can be expressed in terms of $x_1\ld x_{p-1}$
if $p>1$ and that $x_1=0$ if $p=1$.\qed
\endproof

\begstat{(1.2) Artin-Rees lemma} Let $F$ be a finitely generated 
$R$-module and let $E$ be a submodule. There exists an integer $s$ such that
$$E\cap\gm^k F=\gm^{k-s}(E\cap\gm^sF)~~~~\hbox{\rm for}~~k\ge s.$$
\endstat

\begproof{} Let $R_t$ be the polynomial ring $R[\gm t]=R+\gm
t+\cdots+\gm^k t^k+\cdots$ where $t$ is an indeterminate. If $g_1\ld
g_p$ is a set of generators of the ideal $\gm$ over $R$, we see that
the ring $R_t$ is generated by $g_1t\ld g_pt$ over $R$, hence $R_t$ is
also noetherian. Now, we consider the $R_t$-modules
$$E_t=\bigoplus E\,t^k,~~~~F_t=\bigoplus~(\gm^k F)\,t^k.$$
Then $F_t$ is generated over $R_t$ by the generators of $F$ over $R$,
hence the submodule $E_t\cap F_t$ is finitely generated. Let $s$ be the 
highest exponent of $t$ in a set of generators $P_1(t)\ld P_N(t)$ of
$E_t\cap F_t$. If we identify the components of $t^k$ in the extreme
terms of the equality
$$\bigoplus~\big(E\cap\gm^kF\big)\,t^k=E_t\cap F_t=
\sum_j\Big(\bigoplus_k\gm^k\,t^k\Big)P_j(t),$$
we get
$$E\cap\gm^kF\subset\sum_{l\le s}\gm^{k-l}(E\cap\gm^lF)\subset
\gm^{k-s}(E\cap\gm^sF).$$
The opposite inclusion is clear.\qed
\endproof

\begstat{(1.3) Krull lemma} Let $F$ be a finitely generated 
$R$-module and let $E$ be a submodule. Then
\smallskip
\item{\rm a)} $\bigcap_{k\ge 0}\gm^k F=\{0\}$.
\smallskip
\item{\rm b)} $\bigcap_{k\ge 0}(E+\gm^k F)=E$.
\endstat

\begproof{} a) Put $G=\bigcap_{k\ge 0}\gm^k F\subset F$. By the Artin-Rees
lemma, there exists $s\in\bbbn$ such that $G\cap\gm^k F=
\gm^{k-s}(G\cap\gm^sF)$. Taking $k=s+1$, we find $G\subset\gm G$, hence
$\gm G=G$ and $G=\{0\}$ by the Nakayama lemma.
\medskip
\noindent b) By applying a) to the quotient module $F/E$ we get
$\bigcap\gm^k(F/E)=\{0\}$. Property b) follows.\qed
\endproof

Now assume that $R=\cO_n=\bbbc\{z_1\ld z_n\}$ and $\gm=(z_1\ld z_n)$.
Then $\cO_n/\gm^k$ is a finite dimensional vector space generated
by the monomials $z^\alpha$, $|\alpha|<k$. It follows that $E/\gm^kE$
is a finite dimensional vector space for any finitely generated
$\cO_n$-module $E$. As $\bigcap\gm^kE=\{0\}$ by 1.3~a),
there is an injection
$$E\lhra\prod_{k\in\bbbn}E/\gm^kE.\leqno(1.4)$$
We endow $E$ with the Hausdorff topology induced by the product, i.e.\
with the weakest topology that makes all projections $E\lra E/\gm^k E$
continuous for the complex vector space topology on $E/\gm^k E$. This
topology is called the {\it Krull topology} (or rather, the {\it analytic
Krull topology}; the ``algebraic'' Krull topology would be obtained by
taking the discrete topology on $E/\gm^kE$). For $E=\cO_n$, this is the
topology of simple convergence on coefficients, defined by the
collection of semi-norms $\sum c_\alpha z^\alpha\longmapsto|c_\alpha|$.
Observe that this topology is not complete: the completion of $\cO_n$
can be identified with the ring of formal power series $\bbbc[[z_1\ld
z_n]]$. In general, the completion is the inverse limit $\wh E=
\smash{\displaystyle \lim_{\longleftarrow}}~E/\gm^k E$.
Every \hbox{$\cO_n$-homomorphism} $E\lra F$ is continuous, because the 
induced finite dimensional linear maps $E/\gm^k E\lra F/\gm^k F$ are 
continuous.

\begstat{(1.5) Theorem} Let $E\subset F$ be finitely generated $\cO_n$-modules.
Then:
\smallskip
\item{\rm a)} The map $F\lra G=F/E$ is open, i.e.\ the Krull topology of
$G$ is the quotient of the Krull topology of $F\,;$
\smallskip
\item{\rm b)} $E$ is closed in $F$ and the topology induced by $F$ on
$E$ coincides with the Krull topology of $E$.\smallskip
\endstat

\begproof{} a) is an immediate consequence of the fact that the surjective
finite dimensional linear maps $F/\gm^k F\lra G/\gm^k G$ are open.
\medskip
\noindent{b)} Let $\ovl E$ be the closure of $E$ in $F$. The image of $\ovl E$
in $F/\gm^k F$ is mapped into the closure of the image of $E$. As every
subspace of a finite dimensional space is closed, the images of $E$ and
$\ovl E$ must coincide, i.e.\ $\ovl E+\gm^kF=E+\gm^k F$. Therefore
$$E\subset\ovl E\subset\bigcap~(E+\gm^k F)=E$$
thanks to 1.3~b). The topology induced by $F$ on $E$ is the
weakest that makes all projections $E\lra E/E\cap\gm^k F$ continuous
(via the injections $E/E\cap\gm^kF\lhra F/\gm^kF\,$). However, the 
Artin-Rees lemma gives
$$\gm^kE\subset E\cap\gm^kF=\gm^{k-s}(E\cap\gm^sF)\subset\gm^{k-s}E~~~~
\hbox{\rm for}~~k\ge s,$$
so the topology induced by $F$ coincides with that induced by 
$\prod E/\gm^kE$.\qed
\endproof

\titlec{1.B.}{Compact Pertubations of Linear Operators}
We now recall some basic results in the perturbation theory of linear
operators. These results will be needed in order to obtain a finiteness 
criterion for cohomology groups.

\begstat{(1.6) Definition} Let $E,F$ be Hausdorff locally convex topological 
vector spaces and $g:E\lra F$ a continuous linear operator.
\smallskip
\item{\rm a)} $g$ is said to be compact if there exists a neighborhood
$U$ of $0$ in $E$ such that the image $\ovl{g(U)}$ is compact in $F$.
\smallskip
\item{\rm b)} $g$ is said to be a monomorphism if $g$ is a topological
isomorphism of $E$ onto a closed subspace of $F$, and a quasi-monomorphism
if $\ker g$ is finite dimensional and $\wt g:E/\ker g\lra F$ a
monomorphism.
\smallskip
\item{\rm c)} $g$ is said to be an epimorphism if $g$ is surjective 
and open, and a quasi-epimorphism if $g$ is an epimorphism of $E$
onto a closed finite codimensional subspace $F'\subset F$.
\smallskip
\item{\rm d)} $g$ is said to be a quasi-isomorphism if $g$ is
simultaneously a quasi-monomorphism and a quasi-epimorphism.\smallskip
\endstat

\begstat{(1.7) Lemma} Assume that $E,F$ are Fr\'echet spaces. Then
\smallskip
\item{\rm a)} $g$ is a $($quasi-$)$ monomorphism if and only if $g(E)$ is
closed in $F$ and $g$ is injective $($resp. and $\ker g$ is 
finite dimensional$)$.
\smallskip
\item{\rm b)} $g$ is a $($quasi-$)$ epimorphism if and only if $g$ is 
surjective $($resp. $g(E)$ is finite codimensional$)$.\smallskip
\endstat

\begproof{} a) If $g(E)$ is closed, the map $\wt g:E/\ker g\lra g(E)$ is a 
continuous bijective linear map between Fr\'echet spaces, so $\wt g$ is a
topological isomorphism by Banach's theorem.
\medskip
\noindent{b)} If $g$ is surjective, Banach's theorem implies that $g$
is open, thus $g$ is an epimorphism. If $g(E)$ is finite codimensional, let 
$S$ be a supplementary subspace of $g(E)$ in $F$, $\dim S<+\infty$.
Then the map
$$G~:~~(E/\ker g)\oplus S\lra F,~~~~\wt x\oplus y\longmapsto \wt g(\wt x)+y$$
is a bijective linear map between Fr\'echet spaces, so it is a topological
isomorphism. In particular $g(E)=G\big((E/\ker g)\oplus\{0\}\big)$
is closed as an image of a closed subspace. Hence $g(E)$
is also a Fr\'echet space and $g:E\lra g(E)$ is an epimorphism.\qed
\endproof

\begstat{(1.8) Theorem} Let $h:E\lra F$ be a compact linear operator.
\smallskip
\item{\rm a)} If $g:E\lra F$ is a quasi-monomorphism, then $g+h$ is
a quasi-monomorphism.
\smallskip
\item{\rm b)} If $E,F$ are Fr\'echet spaces and if $g:E\lra F$ is a 
quasi-epimorphism, then $g+h$ is a quasi-epimorphism.\smallskip
\endstat

\begproof{} Set $f=g+h$ and let $U$ be an open convex symmetric neighbor\-hood of 
$0$ in $E$ such that $K=\ovl{h(U)}$ is compact.
\medskip
\noindent{a)} It is sufficient to show that there is a finite dimensional
subspace $E'\subset E$ such that $f_{\restriction E'}$ is a monomorphism. If
we take $E'$ equal to a supplementary subspace of $\ker g$, we see that we
may assume $g$ injective. Then $g$ is a monomorphism, so we may assume in fact 
that $E$ is a subspace of $F$ and that $g$ is the inclusion. Let $V$ be an
open convex symmetric neighborhood of $0$ in $F$ such that $U=V\cap E$.
There exists a closed finite codimensional subspace $F'\subset F$ such that
$K\cap F'\subset 2^{-1}V$ because $\bigcap_{F'}K\cap F'=\{0\}$. If we replace
$E$ by $E'=h^{-1}(F')$ and $U$ by $U'=U\cap E'$, we get
$$K':=\ovl{h(U')}\subset K\cap F'\subset 2^{-1}V.$$ 
Hence, we may assume without
loss of generality that $K\subset 2^{-1}V$. Then we show that $f=g+h$
is actually a monomorphism. If $\Omega$ is an arbitrary open
neighborhood of $0$ in $E$, we have to check that there exists a
neighborhood $W$ of $0$ in $F$ such that $f(x)\in W\Longrightarrow x\in
\Omega$. There is an integer $N$ such that $2^{-N}K\cap E\subset\Omega$.
We choose $W$ convex and so small that
$$(W+2^{-N}K)\cap E\subset\Omega~~~{\rm and}~~~2^NW+K\subset 2^{-1}V.$$
Let $x\in E$ be such that $f(x)\in W$. Then $x\in 2^nU$ for $n$ large
enough and we infer
$$x=f(x)-h(x)\in W+2^nK\subset 2^{n-1}V~~~{\rm provided~that}~~n\ge -N.$$
Thus $x\in 2^{n-1}V\cap E=2^{n-1}U$. By induction we finally get
$x\in 2^{-N}U$, so
$$x\in(W+2^{-N}K)\cap E\subset\Omega.$$
\noindent{b)} By Lemma~1.7 b), we only have to show that there is a finite
dimensional subspace $S\subset F$ such that the induced map
$$\wt f:E\lra F\lra F/S$$
is surjective. If we take $S$ equal to a supplementary subspace of $g(E)$
and replace $g,h$ by the induced maps $\wt g,\wt h:E\lra F/S$, we may 
assume that $g$ itself is surjective. Then $g$ is open, so $V=g(U)$
is a convex open neighborhood of $0$ in $F$. As $K$ is compact,
there exists a finite set of elements $b_1\ld b_N\in K$ such that 
$K\subset\bigcup(b_j+2^{-1}V)$. If we take now $S={\rm Vect}(b_1\ld b_N)$,
we obtain $\wt K\subset 2^{-1}\wt V$ where $\wt K$ is the closure of 
$\wt h(U)$ and $V=\wt g(U)$, so we may assume in addition that $K\subset 
2^{-1}V$. Then we show that $f=g+h$ is actually surjective. Let $y_0\in V$.
There exists $x_0\in U$ such that $g(x_0)=y_0$, thus
$$y_1=y_0-f(x_0)=-h(x_0)\in K\subset 2^{-1}V.$$
By induction, we construct $x_n\in 2^{-n}U$ such that $g(x_n)=y_n$ and 
$$y_{n+1}=y_n-f(x_n)=-h(x_n)\in 2^{-n}K\subset 2^{-n-1}V.$$
Hence $y_{n+1}=y_0-f(x_0+\cdots+x_n)$ tends to $0$ in $F$, but we still
have to make sure that the series $\sum x_n$ converges in $E$.
Let $U_p$ be a fundamental system of convex neighborhoods of $0$ in 
$E$ such that $U_{p+1}\subset 2^{-1}U_p$. For each $p$, $K$ is
contained in the union of the open sets $g(2^nU_p\cap 2^{-1}U)$ when
$n\in\bbbn$, equal to $g(2^{-1}U)=2^{-1}V$. There exists an increasing
sequence $N(p)$ such that $K\subset g(2^{N(p)}U_p\cap 2^{-1}U)$, thus
$$2^{1-n}K\subset g(2^{N(p)+1-n}U_p\cap 2^{-n}U).$$
As $y_n\in 2^{1-n}K$, we see that we can choose 
$x_n\in2^{N(p)+1-n}\,U_p\cap2^{-n}\,U$ for $N(p)<n\le N(p+1)\,$; then 
$$x_{N(p)+1}+\cdots+x_{N(p+1)}\in(1+2^{-1}+\cdots~~)~U_p\subset 2\,U_p.$$
As $E$ is complete, the series $x=\sum x_n$ converges towards an element 
$x$ such that $f(x)=y_0$, and $f$ is surjective.\qed
\endproof

The following important finiteness theorem due to L. Schwartz can 
be easily deduced from this.

\begstat{(1.9) Theorem} Let $(E^\bu,d)$ and $(F^\bu,\delta)$ be
complexes of Fr\'echet spaces with continuous differentials, and
$\rho^\bu:E^\bu\lra F^\bu$ a continuous complex morphism. If $\rho^q$
is compact and $H^q(\rho^\bu):H^q(E^\bu)\lra H^q(F^\bu)$ surjective, then
$H^q(F^\bu)$ is a Hausdorff finite dimensional space.
\endstat

\begproof{} Consider the operators
$$\eqalign{
&g,h~:~~Z^q(E^\bu)\oplus F^{q-1}\lra Z^q(F^\bu),\cr
&g(x\oplus y)=\rho^q(x)+\delta^{q-1}(y),~~~~h(x\oplus y)=-\rho^q(x).\cr}$$
As $Z^q(E^\bu)\subset E^q$, $Z^q(F^\bu)\subset F^q$ are closed, all our
spaces are Fr\'echet spaces. Moreover the hypotheses imply that $h$ is compact
and $g$ is surjective since $H^q(\rho^\bu)$ is surjective. Hence $g$ is an
epimorphism and $f=g+h=0\oplus\delta^{q-1}$ is a
quasi-epimorphism by 1.8 b). Therefore $B^q(F^\bu)$ is closed and finite
codimensional in $Z^q(F^\bu)$, thus $H^q(F^\bu)$ is Hausdorff and finite
dimensional.\qed
\endproof

\begstat{(1.10) Remark} \rm If $\rho^\bu:E^\bu\lra F^\bu$ is a continuous morphism
of Fr\'echet complexes and if $H^q(\rho^\bu)$ is surjective, then 
$H^q(\rho^\bu)$ is in fact open, because the above map $g$ is open. 
If $H^q(\rho^\bu)$ is bijective, it follows that $H^q(\rho^\bu)$ is 
necessarily a topological isomorphism (however $H^q(E^\bu)$ and $H^q(F^\bu)$
need not be Hausdorff).\qed
\endstat

\titlec{1.C.}{Abstract Mittag-Leffler Theorem}
We will also need the following abstract Mittag-Leffler theorem,
which is a very efficient tool in order to deal with cohomology
groups of inverse limits.

\begstat{(1.11) Proposition} Let $(E^\bu_\nu,\delta)_{\nu\in\bbbn}$ be a
sequence of Fr\'echet complexes together with morphisms
$E^\bu_{\nu+1}\lra E^\bu_\nu$. We assume that the image of
$E^\bu_{\nu+1}$ in $E^\bu_\nu$ is dense and we let $E^\bu=\displaystyle
\lim_{\displaystyle\longleftarrow}\,E^\bu_\nu$ be the inverse limit complex.
\smallskip
\item{\rm a)} If all maps $H^q(E^\bu_{\nu+1})\lra H^q(E^\bu_\nu)$,
$\nu\in\bbbn$, are surjective, then the limit $H^q(E^\bu)\lra H^q(E^\bu_0)$
is surjective.
\smallskip
\item{\rm b)} If all maps $H^q(E^\bu_{\nu+1})\lra H^q(E^\bu_\nu)$,
$\nu\in\bbbn$, have a dense range, then $H^q(E^\bu)\lra H^q(E^\bu_0)$ has
a dense range.
\smallskip
\item{\rm c)} If all maps $H^{q-1}(E^\bu_{\nu+1})\lra H^{q-1}(E^\bu_\nu)$
have a dense range and all maps\break $H^q(E^\bu_{\nu+1})\lra H^q(E^\bu_\nu)$
are injective, $\nu\in\bbbn$, then $H^q(E^\bu)\lra H^q(E^\bu_0)$ is injective.
\smallskip
\item{\rm d)} Let $\varphi^\bu:F^\bu\lra E^\bu$ be a morphism
of Fr\'echet complexes that has a dense range. If every map 
$H^q(F^\bu)\lra H^q(E^\bu_\nu)$ has a dense range, then 
$H^q(F^\bu)\lra H^q(E^\bu)$ has a dense range.\smallskip
\endstat

\begproof{} If $x$ is an element of $E^\bu$ or of $E^\bu_\mu$, $\mu\ge\nu$,
we denote by $x^\nu$ its canonical image in $E^\bu_\nu$. Let
$d_\nu$ be a translation invariant distance that defines the topology
of $E^\bu_\nu$. After replacement of $d_\nu(x,y)$ by
$$d'_\nu(x,y)=\max_{\mu\le\nu}\big\{d_\mu(x^\mu,y^\mu)\big\},~~~~
x,y\in E^\bu_\nu,$$
we may assume that all maps $E^\bu_{\nu+1}\lra E^\bu_\nu$ are Lipschitz
continuous with coefficient 1. 
\medskip
\noindent{a)} Let $x_0\in Z^q(E^\bu_0)$ represent a given cohomology class 
$\ovl x_0\in H^q(E^\bu_0)$.  We construct by induction a convergent
sequence $x_\nu\in Z^q(E^\bu_\nu)$ such that $\ovl x_\nu$ is mapped onto
$\ovl x_0$.  If $x_\nu$ is already chosen, we can find by assumption
$x_{\nu+1}\in Z^q(E^\bu_{\nu+1})$ such that $\ovl x^\nu_{\nu+1}=\ovl
x_\nu$, i.e.\ $x_{\nu+1}^\nu=x_\nu +\delta y_\nu$ for some $y_\nu\in
E^{q-1}_\nu$.  If we replace $x_{\nu+1}$ by $x_{\nu+1}-\delta
y_{\nu+1}$ where $y_{\nu+1}\in\smash{E^{q-1}_{\nu+1}}$ yields 
an approximation $y_{\nu+1}^\nu$ of $y_\nu$, we may assume that
$\max\{d_\nu(y_\nu,0),d_\nu(\delta y_\nu,0)\}\le 2^{-\nu}$.  Then
$(x_\nu)$ converges to a limit $\xi\in Z^q(E^\bu)$ and we have
$\xi^0=x_0+\delta\sum y_\nu^0$. 
\medskip
\noindent{b)} The density assumption for cohomology groups implies that the map
$$Z^q(E^\bu_{\nu+1})\times E^{q-1}_\nu\lra Z^q(E^\bu_\nu),~~~~(x_{\nu+1},
y_\nu)\longmapsto x_{\nu+1}^\nu+\delta y_\nu$$
has a dense range.  If we approximate $y_\nu$ by elements coming from
$E^{q-1}_{\nu+1}$, we see that the map $Z^q(E^\bu_{\nu+1}) \lra
Z^q(E^\bu_\nu)$ has also a dense range.  If $x_0\in Z^q(E^\bu_0)$, we
can find inductively a sequence $x_\nu\in Z^q(E^\bu_\nu)$ such that
$d_\nu(x_{\nu+1}^\nu,x_\nu)\le \varepsilon 2^{-\nu-1}$ for all $\nu$,
thus $(x_\nu)$ converges to an element $\xi\in Z^q(E^\bu)$ such that
$d_0(\xi^0,x_0)\le\varepsilon$ and $Z^q(E^\bu)\lra Z^q(E^\bu_0)$ has
a dense range.  
\medskip
\noindent{c)} Let $x\in Z^q(E^\bu)$ be such that $\ovl x^0\in H^q(E^\bu_0)$ is
zero.  By assumption, the image of $\ovl x$ in
$H^q(E^\bu_\nu)$ must be also zero, so we can write $x^\nu=dy_\nu$,
$y_\nu\in E^{q-1}_\nu$.  We have $z_\nu=y_{\nu+1}^\nu-y_\nu\in
Z^{q-1}(E^\bu_\nu)$.  Let $z_{\nu+1}\in Z^{q-1}(E^\bu_{\nu+1})$ be such
that $z_{\nu+1}^\nu$ approximates $z_\nu$.  If we replace $y_{\nu+1}$
by $y_{\nu+1}-z_{\nu+1}$, we still have $x^{\nu+1}=dy_{\nu+1}$ and we
may assume in addition that $d_\nu(y_{\nu+1}^\nu,y_\nu)\le 2^{-\nu}$. 
Then $(y_\nu)$ converges towards an element $y\in E^{q-1}$ such that
$x=dy$, thus $\ovl x=0$ and $H^q(E^\bu)\lra H^q(E^\bu_0)$ is
injective.
\medskip
\noindent{d)} For every class $\ovl y\in H^q(E^\bu)$, the hypothesis implies 
the existence of \hbox{a sequence} $x_\nu\in Z^q(F^\bu)$ such that 
$\varphi^q(\ovl x_\nu)^\nu$ converges to $\ovl y^\nu$, that is,\break
\hbox{$d_\nu(y^\nu,\varphi^q(x_\nu)^\nu+\delta z_\nu)$} tends to $0$ for some
sequence $z_\nu\in E^{q-1}_\nu$. Approximate $z_\nu$ by
$\varphi^{q-1}(w_\nu)^\nu$ for some $w_\nu\in F^{q-1}$ and
replace $x_\nu$ by $x'_\nu=x_\nu+\delta w_\nu$. Then
$\varphi^q(x'_\nu)$ converges to $y$ in $Z^q(E^\bu)$.\qed
\endproof

\titleb{2.}{q-Convex Spaces}
\titlec{2.A.}{q-Convex Functions}
The concept of $q$-convexity, first introduced in (Rothstein~1955)
and further developed by (Andreotti-Grauert~1962), generalizes the 
concepts of pseudoconvexity already considered in chapters 1 and 8.
Let $M$ be a complex manifold, $\dim_\bbbc M=n$. A function $v\in C^2(M,\bbbr)$
is said to be strongly (resp. weakly) $q$-convex at a point $x\in M$
if $id'd''v(x)$ has at least $(n-q+1)$ strictly positive (resp. nonnegative)
eigenvalues, or equivalently if there exists a $(n-q+1)$-dimensional subspace
$F\subset T_xM$ on which the complex Hessian $H_x v$ is positive definite
(resp. semi-positive). Weak 1-convexity is thus equivalent to
plurisubharmonicity. Some authors use different conventions for
the number of positive eigenvalues in $q$-convexity. The reason why
we introduce the number $n-q+1$ instead of $q$ is mainly due to the 
following result:

\begstat{(2.1) Proposition} If $v\in C^2(M,\bbbr)$ is strongly $($weakly$)$
$q$-convex and if $Y$ is a submanifold of $M$, then 
$v_{\restriction Y}$ is strongly $($weakly$)$ $q$-convex.
\endstat

\begproof{} Let $d=\dim Y$. For every $x\in Y$, there exists $F\subset T_xM$
with $\dim F=n-q+1$ such that $Hv$ is (semi-) positive on $F$.
Then $G=F\cap T_xY$ has dimension $\ge (n-q+1)-(n-d)=d-q+1$,
and $H(v_{\restriction Y})$ is (semi-) positive on $G\subset T_xY$. Hence
$v_{\restriction Y}$ is strongly (weakly) $q$-convex at $x$.\qed
\endproof

\begstat{(2.2) Proposition} Let $v_j\in C^2(M,\bbbr)$ be a weakly $($strongly$)$
$q_j$-convex function, $1\le j\le s$, and $\chi\in C^2(\bbbr^s,\bbbr)$ 
a convex function that is increasing $($strictly increasing$)$ in all variables.
Then $v=\chi(v_1\ld v_s)$ is weakly $($strongly$)$ $q$-convex with 
$q-1=\sum(q_j-1)$. In particular $v_1+\cdots+v_s$ is weakly
$($strongly$)$ $q$-convex.
\endstat

\begproof{} A simple computation gives
$$Hv=\sum_j{\partial\chi\over\partial t_j}(v_1\ld v_s)\,Hv_j+
\sum_{j,k}{\partial^2\chi\over\partial t_j\partial t_k}(v_1\ld v_s)\,
d'v_j\otimes\ovl{d'v_k},\leqno(2.3)$$
and the second sum defines a semi-positive hermitian form.
In every tangent space $T_xM$ there exists a subspace $F_j$ of codimension 
$q_j-1$ on which $Hv_j$ is semi-positive (positive definite). Then 
$F=\bigcap F_j$ has codimension $\le q-1$ and $Hv$ is semi-positive 
(positive definite) on $F$.\qed
\endproof

The above result cannot be improved, as shown by the trivial example
$$v_1(z)=-2|z_1|^2+|z_2|^2+|z_3|^2,~~~~v_2(z)=|z_1|^2-2|z_2|^2+
|z_3|^2~~~\hbox{\rm on}~~\bbbc^3,$$
in which case $q_1=q_2=2$ but $v_1+v_2$ is only 3-convex.
However, formula (2.3) implies the following result.

\begstat{(2.4) Proposition} Let $v_j\in C^2(M,\bbbr)$, $1\le j\le s$, be such
that every convex linear combination $\sum\alpha_jv_j$, $\alpha_j\ge 0$,
$\sum\alpha_j=1$, is weakly $($strongly$)$ $q$-convex. If 
$\chi\in C^2(\bbbr^s,\bbbr)$ is a convex function that is increasing 
$($strictly increasing$)$ in all variables, then $\chi(v_1\ld v_s)$ is weakly 
$($strongly$)$ $q$-convex.\qed
\endstat

The invariance property of Prop.~2.1 immediately suggests the 
definition of $q$-convexity on complex spaces or analytic schemes:

\begstat{(2.5) Definition} Let $(X,\cO_X)$ be an analytic scheme.
A function $v$ on $X$ is said to be strongly $($resp. weakly$)$ 
$q$-convex of class $C^k$ on $X$ if $X$ can be covered by patches
$G:U\buildo\raise-1.5pt\hbox{$\scriptstyle\simeq$}\over\lra A$, 
$A\subset\Omega\subset\bbbc^N$ such that for each
patch there exists a function $\wt v$ on $\Omega$ with $\wt v_{\restriction A}
\circ G=v_{\restriction U}$,
which is strongly $($resp. weakly$)$ $q$-convex of class $C^k$.
\endstat

The notion of $q$-convexity on a patch $U$ does not depend on the
way $U$ is embedded in $\bbbc^N$, as shown by the following lemma.

\begstat{(2.6) Lemma} Let $G:U\lra A\subset\Omega\subset\bbbc^N$ and
$G':U'\lra A'\subset\Omega'\subset\bbbc^{N'}$ be two patches of $X$. 
Let $\wt v$ be a strongly $($weakly$)$ $q$-convex function on $\Omega$
and $v=\wt v_{\restriction A}\circ G$. For every $x\in U\cap U'$ there 
exists a strongly $($weakly$)$ \hbox{$q$-convex}
 function $\wt v'$ on a neighborhood $W'\subset\Omega'$ of $G'(x)$
such that $\wt v'_{\restriction A'\cap W'}\circ G'$ coincides with $v$ on
$G^{\prime-1}(W')$.
\endstat

\begproof{} The isomorphisms
$$\eqalign{
&G'\circ G^{-1}:A\supset G(U\cap U')\lra G'(U\cap U')\subset A'\cr
&G\circ G^{\prime-1}:A'\supset G'(U\cap U')\lra G(U\cap U')\subset A\cr}$$
are restrictions of holomorphic maps $H:W\lra\Omega'$, $H':W'\lra\Omega$
defined on neighborhoods $W\ni G(x)$, $W'\ni G'(x)\,$; we can shrink
$W'$ so that $H'(W')\subset W$. If we compose the automorphism
$(z,z')\longmapsto(z,z'-H(z))$ of $W\times\bbbc^{N'}$ with the function
$v(z)+|z'|^2$ we see that the function $\varphi(z,z')=\wt v(z)+|z'-H(z)|^2$ 
is strongly (weakly) $q$-convex on $W\times\Omega'$. Now, $W'$
can be embedded in $W\times\Omega'$ via the map $z'\longmapsto
\big(H'(z'),z'\big)$, so that the composite function
$$\wt v'(z')=\varphi\big(H'(z'),z'\big)=\wt v\big(H'(z')\big)+
|z'-H\circ H'(z')|^2$$
is strongly (weakly) $q$-convex on $W'$ by Prop.~2.1. 
Since $H\circ G=G'$ and $H'\circ G'=G$ on $G^{\prime-1}(W')$, we have
$\wt v'\circ G'=\wt v\circ G=v$ on $G^{\prime-1}(W')$ and the lemma 
follows.\qed
\endproof

A consequence of this lemma is that Prop.~2.2 is still valid for
an analytic scheme $X$ (all the extensions $\wt v_j$ near a given point
$x\in X$ can be obtained with respect to the same local embedding).

\begstat{(2.7) Definition} An analytic scheme $(X,\cO_X)$ is said to be 
strongly
$($resp. weakly$)$ $q$-convex if $X$ has a $\ci$ exhaustion function
$\psi$ which is strongly $($resp. weakly$)$ $q$-convex outside an
exceptional compact set $K\subset X$. We say that $X$ is strongly 
$q$-complete if $\psi$ can be chosen so that $K=\emptyset$. 
By convention, a compact scheme $X$ is said to be strongly
$0$-complete, with exceptional compact set $K=X$.
\endstat

We consider the sublevel sets
$$X_c=\{x\in X~;~\psi(x)<c\},~~~~c\in\bbbr.\leqno(2.8)$$
If $K\subset X_c$, we may select a convex increasing function $\chi$
such that $\chi=0$ on $]-\infty,c]$ and $\chi'>0$ on $]c,+\infty[$.
Then $\chi\circ\psi=0$ on $X_c$, so that $\chi\circ\psi$ is weakly
$q$-convex everywhere in virtue of (2.3). In the weakly
$q$-convex case, we may therefore always assume $K=\emptyset$. 
The following properties are almost immediate consequences of the 
definition:

\begstat{(2.9) Theorem} \smallskip
\item{\rm a)} A scheme $X$ is strongly $($weakly$)$ 
$q$-convex if and only if the reduced space $X_\red$ is strongly 
$($weakly$)$ $q$-convex.
\smallskip
\item{\rm b)} If $X$ is strongly $($weakly$)$ $q$-convex, every closed
analytic subset $Y$ of $X_\red$ is strongly $($weakly$)$ $q$-convex.
\smallskip
\item{\rm c)} If $X$ is strongly $($weakly$)$ $q$-convex, every sublevel
set $X_c$ containing the exceptional compact set $K$ is strongly 
$($weakly$)$ $q$-convex.
\smallskip
\item{\rm d)} If $U_j$ is a weakly $q_j$-convex open subset of $X$,
$1\le j\le s$, the intersection $U=U_1\cap\ldots\cap U_s$ is
weakly $q$-convex with $q-1=\sum(q_j-1)$~$;$ $U$ is strongly
$q$-convex $($resp. $q$-complete$)$ as soon as one of the sets $U_j$ is 
strongly $q_j$-convex $($resp. $q_j$-complete$)$.\smallskip
\endstat

\begproof{} a) is clear, since Def.~2.5 does not involve the structure
sheaf $\cO_X$. In cases b) and c), let $\psi$ be an exhaustion of the
required type on $X$. Then $\psi_{\restriction Y}$ and $1/(c-\psi)$ are
exhaustions on $Y$ and $X_c$ respectively (this is so only if $Y$ is 
closed). Moreover, these functions are strongly (weakly) $q$-convex
on $Y\ssm(K\cap Y)$ and $X_c\ssm K$, thanks to 
Prop.~2.1 and 2.2. For property d), note that a sum
$\psi=\psi_1+\cdots+\psi_s$ of exhaustion functions on the sets $U_j$ is an
exhaustion on $U$, choose the $\psi_j$'s weakly $q_j$-convex 
everywhere, and apply Prop.~2.2.\qed
\endproof

\begstat{(2.10) Corollary} Any finite intersection $U=U_1\cap\ldots\cap U_s$
of weakly $1$-convex open subsets is weakly $1$-convex. 
The set $U$ is strongly $1$-convex $($resp. $1$-complete$)$ as soon
as one of the sets $U_j$ is strongly $1$-convex $($resp. 
$1$-complete$)$.
\endstat

\titlec{2.B.}{Neighborhoods of q-complete subspaces}
We prove now a rather useful result asserting the existence of
$q$-complete neighborhoods for $q$-complete subvarieties. The case 
$q=1$ goes back to (Siu~1976), who used a much more complicated method.
The first step is an approximation-extension theorem for strongly
$q$-convex functions.

\begstat{(2.11) Proposition} Let $Y$ be an analytic set in a complex space
$X$ and $\psi$ a strongly $q$-convex $\ci$ function on $Y$. For every
continuous function $\delta>0$ on $Y$, there exists a strongly $q$-convex
$\ci$ function $\varphi$ on a neighborhood $V$ of $Y$ such that
$\psi\le\varphi_{\restriction Y}<\psi+\delta$.
\endstat

\begproof{} Let $Z_k$ be a stratication of $Y$ as given by Prop.~II.5.6,
i.e.\ $Z_k$ is an increasing sequence of analytic subsets of $Y$ such that
$Y=\bigcup Z_k$ and $Z_k\ssm Z_{k-1}$ is a smooth $k$-dimensional
manifold (possibly empty for some $k$'s). We shall prove by induction on
$k$ the following statement:

{\it There exists a $\ci$ function $\varphi_k$ on $X$ which is strongly
$q$-convex along $Y$ and on a closed neighborhood $\ovl V_k$ of $Z_k$
in $X$, such that $\psi\le\varphi_{k\restriction Y}<\psi+\delta$.}

We first observe that any smooth extension $\varphi_{-1}$ of $\psi$ to
$X$ satisfies the requirements with $Z_{-1}=V_{-1}=\emptyset$.  Assume that
$V_{k-1}$ and $\varphi_{k-1}$ have been constructed.  Then $Z_k\ssm
V_{k-1}\subset Z_k\ssm Z_{k-1}$ is contained in $Z_{k,\reg}$. 
The closed set $Z_k\ssm V_{k-1}$ has a locally finite covering 
$(A_\lambda)$ in $X$ by open coordinate patches
$A_\lambda\subset\Omega_\lambda\subset\bbbc^{N_\lambda}$ in which
$Z_k$ is given by equations $z'_\lambda=(z_{\lambda,k+1}\ld
z_{\lambda,N_\lambda})=0$. Let $\theta_\lambda$ be $\ci$ functions
with compact support in $A_\lambda$ such that $0\le\theta_\lambda
\le 1$ and $\sum\theta_\lambda=1$ on $Z_k\ssm V_{k-1}$.  We set
$$\varphi_k(x)=\varphi_{k-1}(x)+\sum~\theta_\lambda(x)\,\varepsilon_\lambda^3
\,\log(1+\varepsilon_\lambda^{-4}|z'_\lambda|^2)~~~~\hbox{\rm on}~~X.$$ 
For $\varepsilon_\lambda>0$ small enough, we will have
$\psi\le\varphi_{k-1\restriction Y}\le\varphi_{k\restriction Y}<
\psi+\delta$. Now, we check that $\varphi_k$ is still strongly
$q$-convex along $Y$ and near any $x_0\in\ovl V_{k-1}$,
and that $\varphi_k$ becomes strongly $q$-convex near any
$x_0\in Z_k\ssm V_{k-1}$. We may assume that $x_0\in\Supp
\theta_\mu$ for some $\mu$, otherwise $\varphi_k$ coincides with
$\varphi_{k-1}$ in a neighborhood of $x_0$. Select $\mu$ and a 
small neighborhood $W\compact\Omega_\mu$ of $x_0$ such that
\medskip
\item{a)} if $x_0\in Z_k\ssm V_{k-1}$, then
$\theta_\mu(x_0)>0$ and $A_\mu\cap W\compact\{\theta_\mu>0\}\,$;
\smallskip
\item{b)} if $x_0\in A_\lambda$ for some $\lambda$ (there is only a finite
set $I$ of such $\lambda$'s), then\break
$A_\mu\cap W\compact A_\lambda$ and $z_{\lambda\restriction A_\mu\cap W}$
has a holomorphic extension $\wt z_\lambda$ to $\ovl W\,$;
\smallskip
\item{c)} if $x_0\in\ovl V_{k-1}$, then $\varphi_{k-1
\restriction A_\mu\cap W}$ has a strongly $q$-convex extension 
$\wt\varphi_{k-1}$ to $\ovl W\,;$
\smallskip
\item{d)} if $x_0\in Y\ssm\ovl V_{k-1}$, then $\varphi_{k-1
\restriction Y\cap W}$ has a strongly $q$-convex extension 
$\wt\varphi_{k-1}$ to $\ovl W\,.$
\medskip
\noindent{}Otherwise take an arbitrary smooth extension 
$\wt\varphi_{k-1}$ of $\varphi_{k-1\restriction A_\mu\cap W}$ to $\ovl W$
and let $\smash{\wt\theta_\lambda}$ be an extension of $\theta_{\lambda
\restriction A_\mu\cap W}$ to $\ovl W$. Then
$$\wt\varphi_k=\wt\varphi_{k-1}+\sum\wt\theta_\lambda\,\varepsilon_\lambda^3\,
\log(1+\varepsilon_\lambda^{-4}|{\wt z}^{\,\prime}_\lambda|^2)$$
is an extension of $\varphi_{k\restriction A_\mu\cap W}$ to $\ovl W$,
resp. of $\varphi_{k\restriction Y\cap W}$ to $\ovl W$ in case d).
As the function $\log(1+\varepsilon_\lambda^{-4}
|{\wt z}^{\,\prime}_\lambda|^2)$ is plurisubharmonic and as its first
derivative $\langle{\wt z}^{\,\prime}_\lambda,
d{\wt z}^{\,\prime}_\lambda\rangle\,(\varepsilon_\lambda^4+
|{\wt z}^{\,\prime}_\lambda|^2)^{-1}$ is bounded by
$O(\varepsilon_\lambda^{-2})$, we see that
$$id'd''\wt\varphi_k\ge id'd''\wt\varphi_{k-1}-
O({\scriptstyle\sum}\varepsilon_\lambda).$$ 
Therefore, for $\varepsilon_\lambda$ small enough, $\wt\varphi_k$
remains $q$-convex on $\ovl W$ in cases c) and d). Since
all functions ${\wt z}^{\,\prime}_\lambda$ vanish along $Z_k\cap W$,
we have
$$id'd''\wt\varphi_k\ge id'd''\wt\varphi_{k-1}
+\sum_{\lambda\in I}\theta_\lambda\,\varepsilon_\lambda^{-1}\,id'd''
|{\wt z}^{\,\prime}_\lambda|^2\ge id'd''\wt\varphi_{k-1}
+\theta_\mu\,\varepsilon_\mu^{-1}\,id'd''|z'_\mu|^2$$
at every point of $Z_k\cap W$.
Moreover $id'd''\wt\varphi_{k-1}$ has at most $(q-1)$-negative
eigenvalues on $TZ_k$ since $Z_k\subset Y$, whereas
$id'd''|z'_\mu|^2$ is positive definite in the normal directions to
$Z_k$ in $\Omega_\mu$.  In case a), we thus find that $\wt\varphi_k$ 
is strongly $q$-convex on $\smash{\ovl W}$ for $\varepsilon_\mu$ small 
enough; we also observe that only finitely many conditions are
required on each $\varepsilon_\lambda$ if we choose a locally finite
covering of $\bigcup\Supp\theta_\lambda$ by neighborhoods
$W$ as above. Therefore, for $\varepsilon_\lambda$ small enough,
$\varphi_k$ is strongly $q$-convex on a
neighborhood $\smash{\ovl V'_k}$ of $Z_k\ssm V_{k-1}$.  The
function $\varphi_k$ and the set $V_k=V'_k\cup V_{k-1}$ satisfy the
requirements at order $k$.  It is clear that we can choose the sequence
$\varphi_k$ stationary on every compact subset of $X\,$; the limit
$\varphi$ and the open set $V=\bigcup V_k$ fulfill the proposition.\qed
\endproof

The second step is the existence of almost plurisubharmonic functions
having poles along a prescribed analytic set.  By an almost
plurisubharmonic function on a manifold, we mean a function that is
locally equal to the sum of a plurisubharmonic function and of a smooth
function, or equivalently, a function whose complex Hessian has bounded
negative part.  On a complex space, we require that our function can be
locally extended as an almost plurisubharmonic function in the ambient
space of an embedding. 

\begstat{(2.12) Lemma} Let $Y$ be an analytic subvariety in a complex 
space $X$. There is an almost plurisubharmonic function $v$ on $X$
such that $v=-\infty$ on $Y$ with logarithmic poles and 
$v\in C^\infty(X\ssm Y)$.
\endstat

\begproof{} Since $\cI_Y\subset\cO_X$ is a coherent subsheaf, there is a
locally finite covering of $X$ by patches $A_\lambda$ isomorphic to
analytic sets in balls $B(0,r_\lambda)\subset\bbbc^{N_\lambda}$, such
that $\cI_Y$ admits a system of generators $g_\lambda=(g_{\lambda,j})$
on a neighborhood of each set $\ovl A_\lambda$.  We set
$$\eqalign{
&v_\lambda(z)=\log|g_\lambda(z)|^2-{1\over r^2_\lambda
-|z-z_\lambda|^2}~~~~\hbox{\rm on}~~A_\lambda,\cr
&v(z)=M_{(1\ld 1)}\big(\ldots\,,v_\lambda(z),\,\ldots\big)~~~~\hbox{\rm
for}~~\lambda~~\hbox{\rm such that}~~A_\lambda\ni z,\cr}$$ 
where $M_\eta$ is the regularized max function defined in I-3.37.
As the generators $(g_{\lambda,j})$ can be expressed in terms of
one another on a neighborhood of $\ovl A_\lambda\cap \ovl A_\mu$, we see
that the quotient $|g_\lambda|/|g_\mu|$ remains bounded on this set. 
Therefore none of the values $v_\lambda(z)$ for $A_\lambda\ni z$ and
$z$ near $\partial A_\lambda$ contributes to the value of $v$,
since $1/(r_\lambda^2-|z-z_\lambda|^2)$ tends to $+\infty$ on
$\partial A_\lambda$. It follows that $v$ is
smooth on $X\ssm Y\,$; as each $v_\lambda$ is almost 
plurisubharmonic on $A_\lambda$, we also see that $v$ is almost 
plurisubharmonic on $X$.\qed
\endproof

\begstat{(2.13) Theorem} Let $X$ be a complex space and $Y$ a strongly 
$q$-complete analytic subset. Then $Y$ has a fundamental family of 
strongly $q$-complete neighborhoods $V$ in $X$.
\endstat

\begproof{} By Prop.~2.11 applied to a strongly
$q$-convex exhaustion of $Y$ and $\delta=1$, there exists a strongly
$q$-convex function $\varphi$ on a neighborhood $W_0$ of $Y$ such that
$\varphi_{\restriction Y}$ is an exhaustion.  Let $W_1$ be a
neighborhood of $Y$ such that $\ovl W_1\subset W_0$ and such that
$\varphi_{\restriction\ovl W_1}$ is an exhaustion.  We are going to show
that every neighborhood $W\subset W_1$ of $Y$ contains a strongly
$q$-complete neighborhood $V$.  If $v$ is the function given by
Lemma~2.12, we set
$$\wt v=v+\chi\circ\varphi~~~~\hbox{\rm on}~~\ovl W$$
where $\chi:\bbbr\to\bbbr$ is a smooth convex increasing function.  If
$\chi$ grows fast enough, we get $\wt v>0$ on $\partial W$ and the
$(q-1)$-codimensional subspace on which $id'd''\varphi$ is positive 
definite (in some ambient space) is also positive definite for
$id'd''\wt v$ provided that $\chi'$ be large enough to compensate the
bounded negative part of $id'd''v$. Then $\wt v$ is
strongly $q$-convex.  Let $\theta$ be a smooth convex increasing
function on $]-\infty,0[$ such that $\theta(t)=0$ for $t<-3$ and
$\theta(t)=-1/t$ on $]-1,0[$.  The open set $V=\{z\in W\,;\,\wt v(z)<0\}$
is a neighborhood of $Y$ and $\smash{\wt\psi}=\varphi+\theta\circ\wt v$ 
is a strongly $q$-convex exhaustion of~$V$.\qed
\endproof

\titlec{2.C.}{Runge Open Subsets}
In order to extend the classical Runge theorem into an approximation
result for sheaf cohomology groups, we need the concept of
a $q$-Runge open subset.

\begstat{(2.14) Definition} An open subset $U$ of a complex space $X$
is said to be $q$-Runge $($resp. $q$-Runge complete$)$ in $X$ if 
for every compact subset $L\subset U$
there exists a smooth exhaustion function $\psi$ on $X$ and a sublevel set
$X_b$ of $\psi$ such that $L\subset X_b\compact U$ and
$\psi$ is strongly $q$-convex on $X\ssm\ovl X_b$
$($resp. on the whole space $X)$.
\endstat

\begstat{(2.15) Example} \rm If $X$ is strongly $q$-complete and if $\psi$
is a strongly $q$-convex exhaustion function of $X$, then every 
sublevel set $X_c$ of $\psi$ is $q$-Runge complete
in $X\,$: every compact set $L\subset X_c$ satisfies
$L\subset X_b\compact X_c$ for some $b<c$. More generally, if
$X$ is strongly $q$-convex and if $\psi$ is strongly $q$-convex
on $X\ssm K$, every sublevel set $X_c$ containing $K$
is $q$-Runge in $X$.
\endstat

Later on, we shall need the following technical result.

\begstat{(2.16) Proposition} Let $Y$ be an analytic subset of a complex
space $X$. If $U$ is a $q$-Runge complete open subset of $Y$
and $L$ a compact subset, there exist a neighborhood $V$ of $Y$ in
$X$ and a strongly $q$-convex exhaustion $\wt\psi$ on $V$
such that $U=Y\cap V$ and $L\subset Y\cap V_b\compact U$ for some 
sublevel set $V_b$ of $\wt\psi$.
\endstat

\begproof{} Let $\psi$ be a strongly $q$-convex exhaustion on $Y$ with
$L\subset\{\psi<b\}\compact U$ as in Def.~2.14.  Then
$L\subset\{\psi<b-\delta\}$ for some number $\delta>0$ and Lemma~2.11
gives a strongly $q$-convex function $\varphi$ on a neighborhood $W_0$
of $Y$ so that $\psi\le\varphi_{\restriction Y}<\psi+\delta$.  The
neighborhood $V$ and the function $\smash{\wt\psi}=
\varphi+\theta\circ\wt v$ constructed in the proof of Th.~2.13 are
the desired ones: we have $\psi\le\smash{\wt\psi_{\restriction Y}}
=\varphi_{\restriction Y}<\psi+\delta$, thus 
$$L\subset Y\cap V_{b-\delta}\subset\{\psi<b\}\compact U.\eqno{\square}$$
\endproof

\titleb{3.}{q-Convexity Properties in Top Degrees}
It is obvious by definition that a $n$-dimensional complex manifold $M$ is
strongly $q$-complete for $q\ge n+1$ (an arbitrary smooth function is
then strongly $q$-convex~!). If $M$ is connected
and non compact, (Greene and Wu~1975) have shown that $M$ is strongly 
$n$-complete, i.e.\ there is a smooth exhaustion function $\psi$ on $M$
such that $id'd''\psi$ has at least one positive eigenvalue everywhere.
We need the following lemmas.

\begstat{(3.1) Lemma} Let $\psi$ be a strongly $q$-convex function 
on $M$ and $\varepsilon>0$ a given number.
There exists a hermitian metric $\omega$ on $M$
such that the eigenvalues $\gamma_1\le\ldots\le\gamma_n$ of the Hessian
form $id'd''\psi$ with respect to $\omega$ satisfy $\gamma_1\ge-\varepsilon$ 
and $\gamma_q=\ldots=\gamma_n=1$.
\endstat

\begproof{} Let $\omega_0$ be a fixed hermitian metric, $A_0\in\ci(\End TM)$
the hermitian endomorphism associated to the hermitian form $id'd''\psi$ with
respect to $\omega_0$, and $\gamma^0_1\le\ldots\le\gamma^0_n$ the 
eigenvalues of $A_0$ (or $id'd''\psi$). 
We can choose a function $\eta\in\ci(M,\bbbr)$ such that
$0<\eta(x)\le\gamma^0_q(x)$ at each point $x\in M$. Select a positive 
function $\theta\in\ci(\bbbr,\bbbr)$ such that
$$\theta(t)\ge |t|/\varepsilon~~\hbox{\rm for}~~t\le 0,~~~
\theta(t)\ge t~~\hbox{\rm for}~~t\ge 0,~~~\theta(t)=t~~\hbox{\rm for}~~t\ge 1.$$
We let $\omega$ be the hermitian metric defined by the hermitian
endomorphism
$$A(x)=\eta(x)\,\theta[(\eta(x))^{-1}A_0(x)]$$
where $\theta[\eta^{-1}A_0]\in\ci(\End TM)$ is defined as in 
Lemma~VII-6.2. By construction, the eigenvalues of $A(x)$ are 
$\alpha_j(x)=\eta(x)\theta\big(\gamma^0_j(x)/\eta(x)\big)>0$
and we have
$$\cmalign{
&\alpha_j(x)\ge|\gamma^0_j(x)|/\varepsilon~~~~
&\hbox{\rm for}~~\gamma^0_j(x)\le 0,\cr
&\alpha_j(x)\ge\gamma^0_j(x)~~~~&\hbox{\rm for}~~\gamma^0_j(x)\ge 0,\cr
&\alpha_j(x)=\gamma^0_j(x)~~~~&\hbox{\rm for}~~j\ge q~~~
\big(\hbox{\rm then}~\gamma^0_j(x)\ge\eta(x)\big).\cr}$$
The eigenvalues of $id'd''\psi$ with respect to $\omega$ are
$\gamma_j(x)=\gamma^0_j(x)/\alpha_j(x)$ and they have the required
properties.\qed
\endproof

On a hermitian manifold $(M,\omega)$, we consider the Laplace operator 
$\Delta_\omega$ defined by
$$\Delta_\omega v=\hbox{\rm Trace}_\omega(id'd''v)=\sum_{1\le j,k\le n}
\omega^{jk}(z){\partial^2v\over\partial z_j\partial\ovl z_k}\leqno(3.2)$$
where $(\omega^{jk})$ is the conjugate of the inverse matrix of 
$(\omega_{jk})$. Note that $\Delta_\omega$ may differ from the usual
Laplace-Beltrami operator if $\omega$ is not K\"ahler. We say that $v$
is strongly $\omega$-subharmonic if $\Delta_\omega v>0$.  This property
implies clearly that $v$ is strongly $n$-convex; however, as
$$\eqalign{
\Delta_\omega\chi(v_1\ld v_s)=\sum_j&{\partial\chi\over\partial t_j}(v_1\ld
v_s)\,\Delta_\omega v_j\cr
&{}+\sum_{j,k}{\partial^2\chi\over\partial t_j\partial
t_k}(v_1\ld v_s)\,\langle d'v_j,d'v_k\rangle_\omega,\cr}$$
subharmonicity has the advantage of being preserved by all convex increasing 
transformations. Conversely, if $\psi$ is strongly $n$-convex and $\omega$
chosen as in Lemma~3.1 with $\varepsilon$ small enough, we get
$\Delta_\omega\psi\ge 1-(n-1)\varepsilon>0$, thus $\psi$ is strongly
subharmonic for a suitable metric $\omega$. 

\begstat{(3.3) Lemma} Let $U,W\subset M$ be open sets such that for every
connected component $U_s$ of $U$ there is a connected component
$W_{t(s)}$ of $W$ such that $W_{t(s)}\cap U_s\ne\emptyset$ and
$W_{t(s)}\ssm\smash{\ovl U_s}\ne\emptyset$.  Then there exists a 
function $v\in\ci(M,\bbbr)$, $v\ge 0$, with support contained in 
$\smash{\ovl U\cup\ovl W}$, such that $v$ is strongly
$\omega$-subharmonic and $>0$ on $U$.
\endstat

\begproof{} We first prove that the result is true when $U,W$ are small cylinders
with the same radius and axis. Let $a_0\in M$ be a given
point and $z_1\ld z_n$ holomorphic coordinates centered at $a_0$. 
We set $\Re z_j=x_{2j-1}$, $\Im z_j=x_{2j}$,
$x'=(x_2\ld x_{2n})$ and $\omega=\sum\wt\omega_{jk}(x)dx_j\otimes dx_k$.
Let $U$ be the cylinder $|x_1|<r$,
$|x'|<r$, and $W$ the cylinder $r-\varepsilon<x_1<r+\varepsilon$,
$|x'|<r$. There are constants $c,C>0$ such that
$$\sum\wt\omega^{jk}(x)\xi_j\xi_k\ge c|\xi|^2~~~\hbox{\rm and}~~~
\sum|\wt\omega^{jk}(x)|\le C~~~\hbox{\rm on}~~\ovl U.$$
Let $\chi\in\ci(\bbbr,\bbbr)$ be a nonnegative
function equal to $0$ on $]-\infty,-r]\cup[r+\varepsilon,+\infty[$ and 
strictly convex on $]-r,r]$. We take explicitly $\chi(x_1)=(x_1+r)
\exp(-1/(x_1+r)^2\big)$ on $]-r,r]$ and
$$v(x)=\chi(x_1)\exp\big(1/(|x'|^2-r^2)\big)~~~\hbox{\rm on}~~U\cup W,~~~
v=0~~~\hbox{\rm on}~~M\ssm(U\cup W).$$
We have $v\in\ci(M,\bbbr)$, $v>0$ on $U$, and a simple computation gives
$$\eqalign{
&{\Delta_\omega v(x)\over v(x)}=
\wt\omega^{11}(x)\big(4(x_1+r)^{-5}-2(x_1+r)^{-3}\big)\cr
&~~+\sum_{j>1}\wt
\omega^{1j}(x)\big(1+2(x_1+r)^{-2}\big)(-2x_j)(r^2-|x'|^2)^{-2}\cr 
&~~+\sum_{j,k>1}\wt\omega^{jk}(x)\Big( x_jx_k\big(4-8(r^2-|x'|^2)
\big)-2(r^2-|x'|^2)^2\delta_{jk}\Big)(r^2-|x'|^2)^{-4}.\cr}$$
For $r$ small, we get
$$\eqalign{
{\Delta_\omega v(x)\over v(x)}\ge 2c(x_1+r)^{-5}
&-C_1(x_1+r)^{-2}|x'|(r^2-|x'|^2)^{-2}\cr
&+(2c|x'|^2-C_2r^4)(r^2-|x'|^2)^{-4}\cr}$$
with constants $C_1,C_2$ independent of $r$.  
The negative term is bounded by $C_3(x_1+r)^{-4}+c|x'|^2(
r^2-|x'|^2)^{-4}$, hence
$$\Delta_\omega v/v(x)\ge c(x_1+r)^{-5}+(c|x'|^2-C_2r^4)(r^2-|x'|^2)^{-4}.$$
The last term is negative only when $|x'|<C_4r^2$, in which case
it is bounded by $C_5r^{-4}<c(x_1+r)^{-5}$. Hence
$v$ is strongly $\omega$-subharmonic on $U$.

Next, assume that $U$ and $W$ are connected. Then $U\cup W$ is connected. 
Fix a point $a\in W\ssm\ovl U$. If $z_0\in U$ is given, we choose a 
path $\Gamma\subset U\cup W$ from $z_0$ to $a$
which is piecewise linear with respect to holomorphic coordinate
patches. Then we can find a finite sequence of
cylinders $(U_j,W_j)$ of the type described above, $1\le j\le N$,
whose axes are segments contained in $\Gamma$, such that 
$$U_j\cup W_j\subset U\cup W,~~~\ovl W_j\subset U_{j+1}~~~\hbox{\rm and}~~~
z_0\in U_0,~~~a\in W_N\subset W\ssm\ovl U.$$
For each such pair, we have a function $v_j\in\ci(M)$ with support in
$\ovl U_j\cup\ovl W_j$, $v_j\ge 0$, strongly $\omega$-subharmonic and $>0$ on
$U_j$. By induction, we can find constants $C_j>0$ such that
$v_0+C_1v_1+\cdots+C_j v_j$ is strongly $\omega$-subharmonic on
$U_0\cup\ldots\cup U_j$ and $\omega$-subharmonic on $M\ssm\ovl W_j$. 
Then
$$w_{z_0}=v_0+C_1v_1+\ldots+C_Nv_N\ge 0$$
is $\omega$-subharmonic on $U$ and strongly $\omega$-subharmonic
$>0$ on a neighborhood $\Omega_0$ of the given point $z_0$. Select
a denumerable covering of $U$ by such neighborhoods $\Omega_p$ and set
$v(z)=\sum\varepsilon_pw_{z_p}(z)$ where $\varepsilon_p$ is a sequence
converging sufficiently fast to $0$ so that $v\in\ci(M,\bbbr)$. Then $v$
has the required properties.

In the general case, we find for each pair $(U_s,W_{t(s)})$ a function
$v_s$ with support in $\smash{\ovl U_s\cup\ovl W_{t(s)}}$, strongly
$\omega$-subharmonic and $>0$ on $U_s$.  Any convergent series
$v=\sum\varepsilon_sv_s$ yields a function with the desired
properties.\qed
\endproof

\begstat{(3.4) Lemma} Let $X$ be a connected, locally connected and
locally compact topological space. If $U$ is a relatively compact
open subset of $X$, we let $\wt U$ be the union of $U$ with all compact
connected components of $X\ssm U$. Then $\wt U$ is open and
relatively compact in $X$, and $X\ssm\smash{\wt U}$ has only finitely
many connected components, all non compact.
\endstat

\begproof{} A rather easy exercise of general topology. Intuitively, $\wt U$
is obtained by ``filling the holes" of $U$ in $X$.\qed
\endproof

\begstat{(3.5) Theorem {\rm(Greene-Wu~1975)}} Every $n$-dimensional
connected non compact complex manifold $M$ has a strongly subharmonic 
exhaustion function with respect to any hermitian metric $\omega$. 
In particular, $M$ is strongly $n$-complete.
\endstat

\begproof{} Let $\varphi\in\ci(M,\bbbr)$ be an arbitrary exhaustion function. 
There exists a sequence of connected smoothly bounded open sets
$\Omega'_\nu\compact M$ such that $\smash{\ovl\Omega'_\nu}\subset\Omega'_{\nu+1}$
and $M=\bigcup\Omega'_\nu$.  Let $\Omega_\nu=\smash{\wt\Omega'_\nu}$ be
the relatively compact open set given by Lemma~3.4.  Then
$\smash{\ovl\Omega_\nu}\subset\Omega_{\nu+1}$, $M=\bigcup\Omega_\nu$ 
and $M\ssm\Omega_\nu$ has no compact connected component.  We set
$$U_1=\Omega_2,~~~~U_\nu=\Omega_{\nu+1}\ssm\ovl{\Omega_{\nu-2}}
~~~\hbox{\rm for}~~\nu\ge 2.$$ 
Then $\partial U_\nu=\partial\Omega_{\nu+1}\cup\partial\Omega_{\nu-2}\,$;
any connected component $U_{\nu,s}$ of $U_\nu$ has its boundary
$\partial U_{\nu,s}\not\subset\partial\Omega_{\nu-2}$, otherwise $\ovl
U_{\nu,s}$ would be open and closed in $M\ssm\Omega_{\nu-2}$,
hence $\ovl U_{\nu,s}$ would be a compact component of
$M\ssm\Omega_{\nu-2}$.  Therefore $\partial U_{\nu,s}$ intersects
$\partial\Omega_{\nu+1}\subset U_{\nu+1}$. If $\partial U_{\nu+1,t(s)}$
is a connected component of $U_{\nu+1}$ containing a point of
$\partial U_{\nu,s}$, then $U_{\nu+1,t(s)}\cap U_{\nu,s}\ne
\emptyset$ and $U_{\nu+1,t(s)}\ssm\ovl U_{\nu,s}\ne\emptyset$.  
Lemma 7 implies that there is a nonnegative function
$v_\nu\in\ci(M,\bbbr)$ with support in $U_\nu\cup U_{\nu+1}$, which is
strongly $\omega$-subharmonic on $U_\nu$.  An induction yields
constants $C_\nu$ such that
$$\psi_\nu=\varphi+C_1v_1+\cdots+C_\nu v_\nu$$ 
is strongly $\omega$-subharmonic on $\ovl{\Omega_\nu}\subset
U_0\cup \ldots\cup U_\nu$, thus $\psi=\varphi+\sum C_\nu v_\nu$ is a
strongly $\omega$-subharmonic exhaustion function on
$M$.\qed
\endproof

By an induction on the dimension, the above result can be generalized 
to an arbi\-trary complex space (or analytic scheme), as was first 
shown by T.~Ohsawa.

\begstat{(3.6) Theorem {\rm(Ohsawa~1984)}} Let $X$ be a complex 
space of maximal dimension $n$. 
\smallskip
\item{\rm a)} $X$ is always strongly $(n+1)$-complete.
\smallskip
\item{\rm b)} If $X$ has no compact irreducible component of 
dimension $n$, then $X$ is strongly $n$-complete.
\smallskip
\item{\rm c)} If $X$ has only finitely many irreducible 
components of dimension $n$, then $X$ is strongly $n$-convex.\smallskip
\endstat

\begproof{} We prove a) and b) by induction on $n=\dim X$. For $n=0$, 
property b) is void and a) is obvious (any function can then be
considered as strongly $1$-convex). Assume that a) has been proved
in dimension $\le n-1$. Let $X'$ be the union
of $X_\sing$ and of the irreducible components of $X$ of dimension
at most $n-1$, and $M=X\ssm X'$ the $n$-dimensional part of
$X_\reg$. As $\dim X'\le n-1$, the induction hypothesis shows
that $X'$ is strongly \hbox{$n$-complete}. By Th.~2.13,
there exists a strongly $n$-convex exhaustion function $\varphi'$
on a neighborhood $V'$ of $X'$. Take a closed neighborhood
$\ovl V\subset V'$ and an arbitrary exhaustion $\varphi$ on $X$ that extends 
$\varphi'_{\smash{\restriction\ovl V}}$. Since every function on a
$n$-dimensional manifold is strongly $(n+1)$-convex, we conclude
that $X$ is at worst $(n+1)$-complete, as stated in a).
\medskip
In case b), the hypothesis means that the connected components $M_j$ 
of $M=X\ssm X'$ have non compact closure $\ovl M_j$ in $X$.
On the other hand, Lemma~3.1 shows that there exists a hermitian metric
$\omega$ on $M$ such that $\varphi_{\restriction M\cap V}$ is
strongly $\omega$-subharmonic. Consider the open sets 
$U_{j,\nu}\subset M_j$ provided by Lemma~3.7 below. By the arguments 
already used in Th.~3.5, we can find a strongly $\omega$-subharmonic
exhaustion $\psi=\varphi+\sum_{j,\nu}C_{j,\nu}v_{j,\nu}$ on $X$,
with $v_{j,\nu}$ strongly $\omega$-subharmonic on
$U_{j,\nu}$, $\Supp v_{j,\nu}\subset U_{j,\nu}\cup U_{j,\nu+1}$ and
$C_{j,\nu}$ large. Then $\psi$ is strongly $n$-convex on $X$.
\endproof

\begstat{(3.7) Lemma} For each $j$, there exists a sequence of open sets 
$U_{j,\nu}\compact M_j$, $\nu\in\bbbn$, such that
\smallskip
\item{\rm a)} $M_j\ssm V'\subset\bigcup_\nu U_{j,\nu}$~ and
$(U_{j,\nu})$ is locally finite in $\ovl M_j\,;$
\smallskip
\item{\rm b)} for every connected component $U_{j,\nu,s}$ of $U_{j,\nu}$
there is a connected component $U_{j,\nu+1,t(s)}$ of $U_{j,\nu+1}$
such that $U_{j,\nu+1,t(s)}\cap U_{j,\nu,s}\ne\emptyset$ and
$U_{j,\nu+1,t(s)}\ssm\smash{\ovl U_{j,\nu,s}}\ne\emptyset$.\smallskip
\endstat

\begproof{} By Lemma~3.4 applied to the space $\ovl M_j$, there exists a sequence
of relatively compact connected open sets $\Omega_{j,\nu}$ in $\ovl M_j$ such 
that $\ovl M_j\ssm\Omega_{j,\nu}$ has no compact connected component, 
$\ovl\Omega_{j,\nu}\subset\Omega_{j,\nu+1}$ and
$\ovl M_j=\bigcup\Omega_{j,\nu}$. We define a compact set 
$K_{j,\nu}\subset M_j$ and an open set $W_{j,\nu}\subset\ovl M_j$
containing $K_{j,\nu}$ by
$$K_{j,\nu}=(\ovl\Omega_{j,\nu}\ssm\Omega_{j,\nu-1})\ssm V',
~~~~W_{j,\nu}=\Omega_{j,\nu+1}\ssm\ovl\Omega_{j,\nu-2}.$$
By induction on $\nu$, we construct an open set
$U_{j,\nu}\compact W_{j,\nu}\ssm X'\subset M_j$ and a finite set 
$F_{j,\nu}\subset \partial U_{j,\nu}\ssm\ovl\Omega_{j,\nu}$.
We let $F_{j,-1}=\emptyset$. 
If these sets are already constructed for $\nu-1$, the
compact set $K_{j,\nu}\cup F_{j,\nu-1}$ is contained in the open
set $W_{j,\nu}$, thus contained in a finite union of connected components 
$W_{j,\nu,s}$. We can write $K_{j,\nu}\cup F_{j,\nu-1}=
\bigcup L_{j,\nu,s}$ where $L_{j,\nu,s}$ is contained in 
$W_{j,\nu,s}\ssm X'\subset M_j$. The open set $W_{j,\nu,s}\ssm X'$
is connected and non contained in $\ovl\Omega_{j,\nu}\cup
L_{j,\nu,s}$, otherwise its closure $\ovl W_{j,\nu,s}$ would have no
boundary point $\in\partial\Omega_{j,\nu+1}$, thus would be open and 
compact in $\ovl M_j\ssm\Omega_{j,\nu-2}$, contradiction. 
We select a point $a_s\in (W_{j,\nu,s}\ssm X')\ssm(
\ovl\Omega_{j,\nu}\cup L_{j,\nu,s})$ and a smoothly bounded connected 
open set $U_{j,\nu,s}\compact W_{j,\nu,s}\ssm X'$ containing
$L_{j,\nu,s}$ with $a_s\in\partial U_{j,\nu,s}$. 
Finally, we set $U_{j,\nu}=\bigcup_s U_{j,\nu,s}$
and let $F_{j,\nu}$ be the set of all points $a_s$. By construction, we
have $U_{j,\nu}\supset K_{j,\nu}\cup F_{j,\nu-1}$, thus 
$\bigcup U_{j,\nu}\supset\bigcup K_{j,\nu}=M_j\ssm V'$, and $\partial
U_{j,\nu,s}\ni a_s$ with $a_s\in F_{j,\nu}\subset U_{j,\nu+1}$.
Property b) follows.\qed
\endproof

\begproof{of Theorem 3.6~c) (end).} Let $Y\subset X$ be the
union of $X_\sing$ with all irreducible components of $X$ that are
non compact or of dimension $<n$.  Then $\dim Y\le n-1$, so $Y$ is
$n$-convex and Th.~2.13 implies that there is an exhaustion
function $\psi\in\ci(X,\bbbr)$ such that $\psi$ is strongly $n$-convex on
a neighborhood $V$ of $Y$.  Then the complement $K=X\ssm V$ is
compact and $\psi$ is strongly $n$-convex on $X\ssm K$.\qed
\endproof

\begstat{(3.8) Proposition} Let $M$ be a connected non compact
$n$-dimensional complex manifold and $U$ an open subset of $M$.
Then $U$ is $n$-Runge complete in $M$ if and only if $M\ssm U$ has no
compact connected component.\qed
\endstat

\begproof{} First observe that a strongly $n$-convex function cannot have
any local maximum, so it satisfies the maximum principle.  If
$M\ssm U$ has a compact connected component $T$, then $T$ has a
compact neighborhood $L$ in $M$ such that $\partial L\subset U$.  We
have $\max_L\psi=\max_{\partial L}\psi$ for every strongly $n$-convex
function, thus $\partial L\subset M_b$ implies $L\subset M_b\,$; thus we
cannot find a sublevel set $M_b$ such that $\partial L\subset M_b\compact U$,
and $U$ is not $n$-Runge in $M$. 

On the other hand, assume that $M\ssm U$ has no compact connected
component and let $L$ be a compact subset of $U$. Let $\omega$ be
any hermitian metric on $M$ and $\varphi$ a strongly $\omega$-subharmonic 
exhaustion function on $M$. Set $b=1+\sup_L\varphi$ and 
$$P=\{x\in M\ssm U\,;\,\varphi(x)\le b\}.$$
As $M\ssm U$ has no compact connected component, all its components
$T_\alpha$ contain a point $y_\alpha$ in 
$$W=\{x\in X\,;\,\varphi(x)>b+1\}.$$
For every point $x\in P$ with $x\in T_\alpha$, there exists a connected open 
set $V_x\compact M\ssm L$ containing $x$ such that $\partial V_x\ni
y_\alpha$ ($M\ssm L$ is a neighborhood of $M\ssm U$ and we can
consider a tubular neighborhood of a path from $x$ to $y_\alpha$ in
$M\ssm L$). The
compact set $P$ can be covered by a finite number of open sets $V_{x_j}$.
Then Lemma~3.3 yields functions $v_j$ with support in 
$\smash{\ovl V_{x_j}\cup\ovl W}$ which are strongly 
$\omega$-subharmonic on $V_{x_j}$.  Let $\chi$ be a convex increasing
function such that $\chi(t)=0$ on $]-\infty,b]$ and $\chi'(t)>0$ on
$]b,+\infty[$.  Consider the function
$$\psi=\varphi+\sum C_jv_j+\chi\circ\varphi.$$
First, choose $C_j$ large enough so that $\psi\ge b$ on $P$. Then choose
$\chi$ increasing fast enough so that $\psi$ is strongly $\omega$-subharmonic
on $\ovl W$. Then $\psi$ is a strongly $n$-convex exhaustion function 
on $M$, and as $\psi\ge\varphi$ on $M$ and $\psi=\varphi$ on $L$, we see that
$$L\subset\{x\in M\,;\,\psi(x)<b\}\subset U.$$
This proves that $U$ is $n$-Runge complete in $M$.\qed
\endproof

\titleb{4.}{Andreotti-Grauert Finiteness Theorems}
\titlec{4.A.}{Case of Vector Bundles over Manifolds}
The crucial point in the proof of the Andreotti-Grauert theorems 
is the following special case, which is easily obtained by the
methods of chapter 8.

\begstat{(4.1) Proposition} Let $M$ be a strongly $q$-complete 
manifold with $q\ge 1$, and $E$ a holomorphic vector bundle over 
$M$. Then:
\smallskip
\item{\rm a)} $H^k\big(M,\cO(E)\big)=0$~~for $k\ge q$.
\smallskip
\item{\rm b)} Let $U$ be a $q$-Runge complete open subset of $M$.
Every $d''$-closed form 
$h\in\ci_{0,q-1}(U,E)$ can be approximated uniformly with
all derivatives on every compact subset of $U$ by a sequence of
global $d''$-closed forms $\smash{\wt h_\nu}\in\ci_{0,q-1}(M,E)$.\smallskip
\endstat

\begproof{} We replace $E$ by 
$\wt E=\Lambda^nTM\otimes E\,$; then we can work with forms of bidegree
$(n,k)$ instead of $(0,k)$. Let $\psi$ be a strongly $q$-convex
exhaustion function on $M$ and $\omega$ the metric given by
Lemma~3.1. Select a function $\rho\in\ci(M,\bbbr)$ which increases rapidly 
at infinity so that the hermitian metric $\wt\omega=e^\rho\omega$ is
complete on $M$. Denote by $E_\chi$ the bundle $E$ endowed with the
hermitian metric obtained by multiplication of a fixed metric of $E$ 
by the weight $\exp(-\rho\circ\psi)$ where $\chi\in\ci(\bbbr,\bbbr)$ is a convex
increasing function. We apply Th.~VIII-4.5 for the bundle $E_\chi$ over the
complete hermitian manifold $(M,\wt\omega)$. Then
$$ic(E_\chi)=ic(E)+id'd''(\chi\circ\psi)\otimes\Id_E
\ge_{\rm Nak}ic(E)+\chi'\circ\psi~id'd''\psi\otimes\Id_E.$$
The eigenvalues of $id'd''\psi$ with respect to $\wt\omega$ are
$e^{-\rho}\gamma_j$, so Lemma~VII-7.2 and Prop.~VI-8.3 yield
$$\eqalign{
[ic(E_\chi),\Lambda]+T_{\wt\omega}&\ge[ic(E),\Lambda]+T_{\wt\omega}+
\chi'\circ\psi~[id'd''\psi,\Lambda]\otimes\Id_E\cr
&\ge[ic(E),\Lambda]+T_{\wt\omega}+\chi'\circ\psi~e^{-\rho}(\gamma_1+\cdots+
\gamma_k)\otimes\Id_E\cr}$$
when this curvature tensor acts on $(n,k)$-forms. For $k\ge q$, we have
$$\gamma_1+\cdots+\gamma_k\ge 1-(q-1)\varepsilon>0~~~~\hbox{\rm if}~~
\varepsilon\le 1/q.$$
We choose $\chi_0$ increasing fast enough so that all the eigenvalues 
of the above curvature tensor are $\ge 1$ when $\chi=\chi_0$. 
Then for every $g\in\ci_{n,k}(M,E)$ with $D''g=0$
the equation $D''f=g$ can be solved with an estimate
$$\int_M|f|^2e^{-\chi\circ\psi}dV\le\int_M|g|^2e^{-\chi\circ\psi}dV,
$$
where $\chi=\chi_0+\chi_1$ and where $\chi_1$ is a convex increasing function
chosen so that the integral of $g$ converges. This gives a).
In order to prove b), let $h\in\ci_{n,q-1}(U,E)$ be such that $D''h=0$
and let $L$ be an arbitrary compact subset of $U$.
Thanks to Def.~2.14, we can choose $\psi$ such that there is
a sublevel set $M_b$ with $L\subset M_b\compact U$.
Select $b_0<b$ so that $L\subset M_{b_0}$, and let
$\theta\in\ci(\bbbr,\bbbr)$ be a convex increasing function such that
$\theta=0$ on $]-\infty,b_0[$ and $\theta\ge 1$ on $]b,+\infty[$.
Let $\eta\in\cD(U)$ be a cut-off function such that $\eta=1$
on $M_b$. We solve the equation $D''f=g$ for $g=D''(\eta h)$
with the weight $\chi=\chi_0+\nu \theta\circ\psi$ and let $\nu$ tend to
infinity. As $g$ has compact support in $U\ssm M_b$ and
$\chi\circ\psi\ge\chi_0\circ\psi+\nu$ on this set, we find a solution 
$f_\nu$ such that
$$\int_{M_{b_0}}|f_\nu|^2e^{-\chi_0\circ\psi}dV\le
\int_M|f_\nu|^2e^{-\chi\circ\psi}dV\le 
\int_{U\ssm M_b}|g|^2e^{-\chi\circ\psi}dV\le Ce^{-\nu},$$
thus $f_\nu$ converges to $0$ in $L^2(M_{b_0})$ and $h_\nu=\eta h-f_\nu
\in\ci_{n,q-1}(M,E)$ is a $D''$-closed form converging to $h$ 
in $L^2(M_{b_0})$. However, if we choose
the minimal solution such that $\delta''_\chi f_\nu=0$ as in Rem.~VIII-4.6,
we get $\Delta''_\chi f_\nu=\delta''_\chi g$ on $M$ and in particular
$\Delta''_{\chi_0}f_\nu=0$ on $M_{b_0}$. G\aa rding's inequality
VI-3.3 applied to the elliptic operator $\Delta''_{\chi_0}$
shows that $f_\nu$ converges to $0$ with all derivatives on $L$,
hence $h_\nu$ converges to $h$ on $L$. Now, replace $L$ by an
exhaustion $L_\nu$ of $U$ by compact sets; some diagonal subsequence 
$h_\nu$ converges to $h$ in $\ci_{n,q-1}(U,E)$.\qed
\endproof

\titlec{4.B.}{A Local Vanishing Result for Sheaves}
Let $(X,\cO_X)$ be an analytic scheme and $\cS$ a coherent sheaf of
$\cO_X$-modules.  We wish to extend Prop.~4.1 to the cohomology
groups $H^k(X,\cS)$.  The first step is to show that the result holds on 
small open sets, and this is done by means of local resolutions of $\cS$. 

For a given point $x\in X$, we choose a patch $(A,\cO_\Omega/\cJ)$ of $X$
containing $x$, where $A$ is an analytic subset of $\Omega\subset\bbbc^N$
and $\cJ$ a sheaf of ideals with zero set $A$. Let $i_A:A\lra\Omega$ be 
the inclusion. Then $(i_A)_\star\cS$ is a coherent $\cO_\Omega$-module 
supported on $A$. In particular there is a neighborhood
$W_0\subset\Omega$ of $x$ and a surjective sheaf morphism 
$$\cO^{p_0}\lra(i_A)_\star\cS~~~\hbox{\rm on}~~W_0,~~~~(u_1\ld u_{p_0})
\longmapsto\sum_{1\le j\le p_0}u_jG_j$$
where $G_1\ld G_{p_0}\in\cS(A\cap W_0)$ are generators of $(i_A)_\star\cS$ 
on $W_0$. If we repeat the procedure inductively for the kernel of the above
surjective morphism, we get a {\it homological free resolution} of 
$(i_A)_\star\cS\,$:
$$\cO^{p_l}\lra\cdots\lra\cO^{p_1}\lra\cO^{p_0}\lra(i_A)_\star\cS\lra 0~~~
\hbox{\rm on}~~W_l\leqno(4.3)$$
of arbitrary large length $l$, on neighborhoods $W_l\subset
W_{l-1}\subset \ldots\subset W_0$.  In particular, after replacing $\Omega$ 
by $W_{2N}$ and $A$ by $A\cap W_{2N}$, we may assume that $(i_A)_\star\cS$
has a resolution of length $2N$ on $\Omega$. In this case, we shall say
that $A\subset\Omega$ is a $\cS$-{\it distinguished patch} of $X$.

\begstat{(4.4) Lemma} Let $A\subset\Omega$ be a $\cS$-distinguished patch
of $X$ and $U$ a strongly $q$-convex open subset of $A$. Then
$$H^k(U,\cS)=0~~~~\hbox{\it for}~~k\ge q.$$
\endstat

\begproof{} Theorem 2.13 shows that there exists a strongly $q$-convex
open set $V\subset\Omega$ such that $U=A\cap V$.  Let us denote by
$\cZ^l$ the kernel of $\cO^{p_l}\lra\cO^{p_{l-1}}$ for $l\ge 1$ and
$\cZ^0=\ker\big(\cO^{p_0}\lra(i_A)_\star\cS\big)$.  There are exact
sequences $$\eqalign{ &0\lra\cZ^0\lra\cO^{p_0}\lra(i_A)_\star\cS\lra 0,\cr
&0\lra\cZ^l\lra\cO^{p_l}\lra\cZ^{l-1}\lra 0,~~~~1\le l\le 2N.\cr}$$
For $k\ge q$, Prop.~4.1~a) gives $H^k(V,\cO^{p_l})=0$, 
therefore we get
$$H^k(U,\cS)\simeq H^k\big(V,(i_A)_\star\cS\big) \simeq
H^{k+1}(V,\cZ^0)\simeq\ldots\simeq H^{k+2N+1}(V,\cZ^{2N}),$$ 
and the last group vanishes because $\hbox{\rm topdim}\,V\le\dim_\bbbr
V=2N$.\qed
\endproof

\titlec{4.C.}{Topological Structure on Spaces of Sections and on
Cohomology Groups}
Let $V\subset\Omega$ be a strongly $1$-complete open set
relatively to a $\cS$-distinguished patch $A\subset\Omega$
and let $U=A\cap V$.  By the proof of Lemma~4.4, we have 
$$H^1(V,\cZ^0)\simeq H^{2N+1}(V,\cZ^{2N})=0,$$
hence we get an exact sequence
$$0\lra\cZ^0(V)\lra\cO^{p_0}(V)\lra\cS(U)\lra 0.\leqno(4.5)$$
We are going to show that the Fr\'echet space structure on $\cO^{p_0}(V)$
induces a natural Fr\'echet space structure on the groups of sections of
$\cS$ over any open subset. We first note that $\cZ^0(V)$ is closed in
$\cO^{p_0}(V)$. Indeed, let $f_\nu\in\cZ^0(V)$ be a sequence converging 
to a limit $f\in\cO^{p_0}(V)$ uniformly on compact subsets of $V$.
For every $x\in V$, the germs $(f_\nu)_x$ converge to $f_x$ with
respect to the topology defined by (1.4) on $\cO^{p_0}$. As $\cZ^0_x$
is closed in $\cO_x^{p_0}$ in view of Th.~1.5~b), we get $f_x\in\cZ^0_x$
for all $x\in V$, thus $f\in\cZ^0(V)$. 

\begstat{(4.6) Proposition} The quotient topology on $\cS(U)$ is 
independent of the choices made above.
\endstat

\begproof{} For a smaller set $U'=A\cap V'$ where $V'$ is a strongly $1$-convex 
open subset of $V$, the restriction map $\cO^{p_0}(V)\lra\cO^{p_0}(V')$ 
is continuous, thus $\cS(U)\lra\cS(U')$ is continuous.  If $(V_\alpha)$ is a 
countable covering of $V$ by such sets and $U_\alpha=A\cap V_\alpha$, we 
get an injection of $\cS(U)$ onto the closed subspace of the product
$\prod\cS(U_\alpha)$ consisting of families which are compatible in the
intersections.  Therefore, the Fr\'echet topology induced by the product
coincides with the original topology of $\cS(U)$.  If we choose other
generators $H_1\ld H_{q_0}$ for $(i_A)_\star\cS$, the germs $H_{j,x}$
can be expressed in terms of the $G_{j,x}\,$'s, thus we get a
commutative diagram 
$$\cmalign{ 
&\cO^{p_0}(V)&\buildo G\over\lra&\cS(U)&\lra 0\cr 
&~~~\big\downarrow&&~~\big|\big|&\cr
&\cO^{q_0}(V)&\buildo H\over\lra&\cS(U)&\lra 0\cr}$$ 
provided that $U$ and $V$ are small enough.  If we express the
generators $G_j$ in terms of the $H_j$'s, we find a similar diagram with
opposite vertical arrows and we conclude easily that the topology
obtained in both cases is the same.  Finally, it remains to show that
the topology of $\cS(U)$ is independent of the embedding
$A\subset\Omega$ near a given point $x\in X$.  We compare the given
embedding with the Zariski embedding $(A,x)\subset\Omega'$ of minimal
dimension $d$.  After shrinking $A$ and changing coordinates, we may
assume $\Omega=\Omega'\times\bbbc^{N-d}$ and that the embedding
$i_A:A\lra\Omega$ is the composite of $i'_A:A\lra\Omega'$ and of the 
inclusion $j:\Omega'\lra\Omega'\times\{0\}\subset\Omega$. For
$V'\subset\Omega'$ sufficient small and $U'=A\cap V'$, we have a
surjective map $G':\cO^{p_0}(V')\lra\cS(U')$ obtained by choosing generators
$G'_j$ of $(i'_A)^\star\cS$ on a neighborhood of $x$ in $\Omega'$. 
Then we consider the open set $V=V'\times\bbbc^{N-d}\subset\Omega$
and the surjective map onto $\cS(U')$ equal to the composite 
$$\cO^{p_0}(V)\buildo j^\star\over\lra
\cO^{p_0}(V')\buildo G'\over\lra\cS(U).$$ 
This map corresponds to a choice of generators $G_j\in(i_A)^\star\cS(V)$ 
equal to the functions $G'_j$, considered as functions independent of 
the last variables $z_{d+1}\ld z_N$.  Since $j^\star$ is open, it is
obvious that the quotient topology on $\cS(U')$ is the same for both
embeddings.\qed
\endproof

Now, there is a natural topology on the cohomology groups $H^k(X,\cS)$.
In fact, let $(U_\alpha)$ be a countable covering of $X$
by strongly \hbox{$1$-complete} open sets, such that each
$U_\alpha$ is contained in a $\cS$-distinguished patch.  Since the
intersections $U_{\alpha_0\ldots\alpha_k}$ are again strongly
1-complete, the covering $\cU$ is acyclic by Lemma~4.4 
and Leray's theorem shows that $H^k(X,\cS)$ is isomorphic to 
$\check H^q(\cU,\cS)$. 
We consider the product topology on the spaces of \v Cech cochains
$C^k(\cU,\cS)=\prod\cS(U_{\alpha_0\ldots\alpha_k})$ and the quotient
topology on $\check H^k(\cU,\cS)$.  It is clear that $\check
H^0(\cU,\cS)$ is a Fr\'echet space; however the higher cohomology
groups $\check H^k(\cU,\cS)$ need not be Hausdorff because the
coboundary groups may be non closed in the cocycle groups.  
The resulting topology on $H^k(X,\cS)$ is independent of the choice 
of the covering: in fact we only have to check that the bijective
continuous map $\check H^k(\cU,\cS)\lra\check H^k(\cU',\cS)$ 
is a topological isomorphism if $\cU'$ is a refinement of
$\cU$, and this follows from Rem.~1.10 applied to the morphism 
of \v Cech complexes $C^\bu(\cU,\cS)\lra C^\bu(\cU',\cS)$.

Finally, observe that when $\cS$ is the locally free sheaf associated
to a holomorphic vector bundle $E$ on a smooth manifold $X$, the topology 
on $H^k\big(X,\cO(E)\big)$ is the same as the topology associated to the
Fr\'echet space structure on the Dolbeault complex 
$\big(\ci_{0,\bu}(X,E),d''\big)\,$: by the analogue of formula (IV-6.11) 
we have a bijective continuous map
$$\eqalign{
\check H^k\big(\cU,\cO(E)\big)&\lra H^k\big(\ci_{0,\bu}(X,E)\big)\cr
\hfill \{(c_{\alpha_0\ldots\alpha_k})\}&\longmapsto
f(z)=\sum_{\alpha_0\ld\alpha_q}c_{\alpha_0\ldots\alpha_q}(z)\,\theta_{\alpha_q}
\,d''\theta_{\alpha_0}\wedge\ldots\wedge d''\theta_{\alpha_{q-1}}\cr}$$
where $(\theta_\alpha)$ is a partition of unity subordinate to $\cU$.
As in Rem.~1.10, the continuity of the inverse follows by the open 
mapping theorem applied to the surjective map
$$Z^k\big(C^\bu(\cU,\cO(E))\big)\oplus\ci_{0,k-1}(X,E)
\lra Z^k\big(\ci_{0,\bu}(X,E)\big).$$
We shall need a few simple additional results.

\begstat{(4.7) Proposition} The following properties hold:
\smallskip
\item{\rm a)} For every $x\in X$, the map $\cS(X)\lra\cS_x$ is 
continuous with respect to the topology of $\cS_x$ defined by $(1.4)$.
\smallskip
\item{\rm b)} If $\cS'$ is a coherent analytic subsheaf of $\cS$, 
the space of global sections $\cS'(X)$ is closed in $\cS(X)$.
\smallskip
\item{\rm c)} If $U'\subset U$ are open in $X$,
the restriction maps $H^k(U,\cS)\lra H^k(U',\cS)$ are continuous.
\smallskip
\item{\rm d)} If $U'$ is relatively compact in $U$, the
restriction operator $\cS(U)\lra\cS(U')$ is compact.
\smallskip
\item{\rm e)} Let $\cS\lra\cS'$ be a morphism of coherent sheaves
over $X$. Then the induced maps $H^k(X,\cS)\lra H^k(X,\cS')$ are 
continuous.\smallskip
\endstat

\begproof{} a) Let $V\subset\Omega$ be a strongly $1$-convex open
neighborhood of $x$ relatively to a $\cS$-distinguished patch
$A\subset\Omega$. The map $\cO^{p_0}(V)\lra\cO^{p_0}_x$ is continuous, 
and the same is true for $\cO^{p_0}_x\lra\cS_x$ by \S 1. Therefore 
the composite $\cO^{p_0}(V)\lra\cS_x$ and its factorization $\cS(U)\lra\cS_x$ 
are continuous.
\medskip
\noindent{b)} is a consequence of the above property a) and of the fact 
that each stalk $\cS'_x$ is closed in $\cS_x$ (cf.\ 1.5 b)).
\medskip
\noindent{c)} The restriction map $\cS(U)\lra\cS(U')$ is continuous, and the
case of higher cohomology groups follows immediately.
\medskip
\noindent{d)} Assume first that $U=A\cap V$ and $U'=A\cap V'$, where $A\subset
\Omega$ is a \hbox{$\cS$-distinguished} patch and $V'\compact V$ are strongly 
$1$-convex open subsets of $\Omega$.  The operator
$\cO^{p_0}(V)\lra\cO^{p_0}(V')$ is compact by Montel's theorem, thus
$\cS(U)\lra\cS(U')$ is also compact. In the general case, select
a finite family of strongly $1$-convex sets 
$U'_\alpha\compact U_\alpha\subset U$ such that $(U'_\alpha)$
covers $\ovl U'$ and $U_\alpha$ is contained in some distinguished patch.
There is a commutative diagram
$$\cmalign{
&~~~\cS(U)~~\rightarrowfil~~&\cS(U')\cr
&~~~~~~\big\downarrow&~~~\big\downarrow\cr
&\prod\cS(U_\alpha)\lra\prod\cS(U'_\alpha)\lra\prod\cS&(U'\cap U'_\alpha)\cr}$$
where the right vertical arrow is a monomorphism and where the first arrow in 
the bottom line is compact. Thus $\cS(U)\lra\cS(U')$ is compact.
\medskip
\noindent{e)} It is enough to check that $\cS(U)\lra\cS'(U)$ is continuous, and
for this we may assume that $U=A\cap V$ where $V$ is a small neighborhood of
a given point $x$. Let $G_1\ld G_{p_0}$ be generators of $\cS_x$,
$G'_1\ld G'_{p_0}$ their images in $\cS'_x$. Complete these elements
in order to obtain a system of generators $(G'_1\ld G'_{q_0})$ of $\cS'_x$.
For $V$ small enough, the map $\cS(U)\lra\cS'(U)$ is induced by the
inclusion $\cO^{p_0}(V)\lra\cO^{p_0}(V)\times\{0\}\subset\cO^{q_0}(V)$,
hence continuous.\qed
\endproof

\titlec{4.D.}{Cartan-Serre Finiteness Theorem}
The above results enable us to prove a finiteness theorem for 
cohomology groups over compact analytic schemes.

\begstat{(4.8) Theorem {\rm(Cartan-Serre)}} Let $\cS$ be a coherent 
analytic sheaf over an analytic scheme $(X,\cO_X)$. If $X$ is compact, all
cohomology groups $H^k(X,\cS)$ are finite dimensional $($and Hausdorff~$)$.
\endstat

\begproof{} There exist finitely many strongly $1$-complete 
open sets $U'_\alpha\compact U_\alpha$ such that each 
$U_\alpha$ is contained in some $\cS$-distinguished patch and such that 
$\bigcup U'_\alpha=X$.  By Prop.~4.7~d), the restriction map 
on \v Cech cochains
$$C^\bu(\cU,\cS)\lra C^\bu(\cU',\cS)$$
defines a compact morphism of complexes of Fr\'echet spaces.
As the coverings $\cU=(U_\alpha)$ and $\cU'=(U'_\alpha)$ are acyclic 
by 4.4, the induced map 
$$\check H^k(\cU,\cS)\lra\check H^k(\cU',\cS)$$
is an isomorphism, both spaces being isomorphic to $H^k(X,\cS)$.
We conclude by Schwartz' theorem 1.9.\qed
\endproof

\titlec{4.E.}{Local Approximation Theorem}
We show that a local analogue of the approximation result 4.1 b) holds
for a sheaf $\cS$ over an analytic scheme $(X,\cO_X)$.

\begstat{(4.9) Lemma} Let $A\subset\Omega$ be a $\cS$-distinguished patch of
$X$, and $U'\subset U\subset A$ open subsets such that
$U'$ is $q$-Runge complete in $U$. Then the restriction map
$$H^{q-1}(U,\cS)\lra H^{q-1}(U',\cS)$$
has a dense range.
\endstat

\begproof{} Let $L$ be an arbitrary compact subset of $U'$.
Proposition 2.16 applied with $Y=U$ embedded in some neighborhood in
$\Omega$ shows that there is a neighborhood $V$ of $U$ in $\Omega$ such
that $A\cap V=U$ and a strongly $q$-convex function $\psi$ on $V$ such that 
$L\subset U_b\compact U'$ for some $U_b=A\cap V_b$. The proof of 
Lemma~4.4 gives $H^q(V,\cZ^0)=H^q(V_b,\cZ^0)=0$ and the cohomology
exact sequences of $0\to\cZ^0\to\cO^{p_0}\to i_A^\star\cS\to 0$ 
over $V$ and $V_b$ yield
a commutative diagram of continuous maps
$$\cmalign{
&H^{q-1}\big(V,\cO^{p_0}\big)&\lra&H^{q-1}\big(V,i_A^\star\cS\big)&=
H^{q-1}\big(&U,\cS)\cr
&~~~~~~~~\big\downarrow&&~~~~~~~~\big\downarrow&&\big\downarrow\cr
&H^{q-1}\big(V_b,\cO^{p_0}\big)&\lra&H^{q-1}\big(V_b,i_A^\star\cS\big)&=
H^{q-1}\big(&U_b,\cS)\cr}$$
where the horizontal arrows are surjective. Since $V_b$ is $q$-Runge
complete in $V$, the left vertical arrow has a dense range 
by Prop.~4.1 b). As $U'$ is the union of an increasing sequence
of sets $U_{b_\nu}$, we only have to show that the range remains dense
in the inverse limit $H^{q-1}(U',\cS)$. For that, we apply
Property 1.11~d) on a suitable covering of $U$.
Let $\cW$ be a countable basis of the topology of $U$, consisting of
strongly $1$-convex open subsets contained in $\cS$-distinguished 
patches. We let $\cW'$ (resp. $\cW_\nu$) be the subfamily of 
$W\in\cW$ such that $W\compact U'$ (resp. $W\compact U_{b_\nu}$). Then
$\cW,~\cW',~\cW_\nu$ are acyclic coverings of $U,~U',~U_{b_\nu}$
and each restriction map $C^\bu(\cW,\cS)\lra C^\bu(\cW_\nu,\cS)$
is surjective. Property 1.11~d) can thus be applied and the
lemma follows.\qed
\endproof

\titlec{4.F.}{Statement and Proof of the Andreotti-Grauert Theorem}
\begstat{(4.10) Theorem {\rm(Andreotti-Grauert~1962)}} Let $\cS$ be a 
coherent analytic sheaf over a strongly $q$-convex analytic scheme 
$(X,\cO_X)$. Then
\smallskip
\item{\rm a)} $H^k(X,\cS)$ is Hausdorff and finite dimensional
for $k\ge q$.
\smallskip
\noindent{}Moreover, let $U$ be a $q$-Runge open subset of $X$, $q\ge 1$. Then
\smallskip
\item{\rm b)} the restriction map $H^k(X,\cS)\to H^k(U,\cS)$ 
is an \hbox{isomorphism for $k\ge q\,;$}
\smallskip
\item{\rm c)} the restriction map $H^{q-1}(X,\cS)\to
H^{q-1}(U,\cS)$ has a dense range.
\endstat

The compact case $q=0$ of 4.10~a) is precisely the Cartan-Serre
finiteness theorem.  For $q\ge 1$, the special case when $X$ is
strongly $q$-complete and $U=\emptyset$ yields the following very
important consequence. 

\begstat{(4.11) Corollary} If $X$ is strongly $q$-complete, then 
$$H^k(X,\cS)=0~~~~\hbox{\it for}~~k\ge q.$$
\endstat

Assume that $q\ge 1$ and let $\psi$ be a smooth exhaustion on $X$
that is strongly $q$-convex on $X\ssm K$. We first consider 
sublevel sets $X_d\supset X_c\supset K$, $d>c$,
and verify assertions 4.10 b), c) for all restriction maps 
$$H^k(X_d,\cS)\lra H^k(X_c,\cS),~~~~k\ge q-1.$$
The basic idea, already contained in (Andreotti-Grauert~1962), is 
to deform $X_c$ into
$X_d$ through a sequence of strongly $q$-convex open sets $(G_j)$
such that $G_{j+1}$ is obtained from $G_j$ by making a small bump.

\begstat{(4.12) Lemma} There exist a sequence of strongly $q$-convex
open sets\break $G_0\subset\ldots\subset G_s$ and a sequence
of strongly $q$-complete open sets $U_0\ld U_{s-1}$ in $X$ such that 
\smallskip
\item{\rm a)} $G_0=X_c$,~~~$G_s=X_d$,~~~$G_{j+1}=G_j\cup U_j$~~for~
$0\le j\le s-1\,;$
\smallskip
\item{\rm b)} $G_j=\{x\in X\,;\,\psi_j(x)<c_j\}$ where $\psi_j$ is an
exhaustion function on $X$ that is strongly $q$-convex on $X\ssm K\,;$
\smallskip
\item{\rm c)} $U_j$ is contained in a
$\cS$-distinguished patch $A_j\subset\Omega_j$ of $X\,;$
\smallskip
\item{\rm d)} $G_j\cap U_j$ is strongly $q$-complete and
$q$-Runge complete in $U_j$.
\endstat

\begproof{} There exists a finite covering of the compact set 
$\ovl X_d\ssm X_c$ by
$\cS$-distinguished patches $A_j\subset\Omega_j$, $0\le j<s$, where
$\Omega_j\subset\bbbc^{N_j}$ is a euclidean ball and $K\cap A_j=\emptyset$. 
Let $\theta_j\in\cD(X)$ be a family of functions such that 
$\Supp\theta_j\subset A_j$, $\theta_j\ge 0$,
$\sum\theta_j\le 1$ and $\sum\theta_j=1$ on a neighborhood of 
$\ovl X_d\ssm X_c$. We can find $\varepsilon_0>0$ so small that 
$$\psi_j=\psi-\varepsilon\sum_{0\le k<j}\theta_k$$
is still strongly $q$-convex on $X\ssm K$ for $0\le j\le s$
and $\varepsilon\le\varepsilon_0$.  We have
$\psi_0=\psi$ and $\psi_s=\psi-\varepsilon$ on $\ovl X_d\ssm X_c$, thus
$$G_j=\{x\in X\,;\,\psi_j(x)<c\},~~~~0\le j\le s$$
is an increasing sequence of strongly $q$-convex open sets such that
$G_0=X_c$, $G_s=X_{c+\varepsilon}$. Moreover, as $\psi_{j+1}-\psi_j=
-\varepsilon\theta_j$ has support in $A_j$, we have
$$G_{j+1}=G_j\cup U_j~~~~\hbox{\rm where}~~U_j=G_{j+1}\cap A_j.$$
It follows that conditions a), b), c) are satisfied with $c+\varepsilon$
instead of $d$. Finally, the functions
$$\varphi_j=1/(c-\psi_{j+1})+1/(r_j^2-|z-z_j|^2),~~~~
\wt\varphi_j=1/(c-\psi_j)+1/(r_j^2-|z-z_j|^2)$$
are strongly $q$-convex exhaustions on $U_j$ and $G_j\cap U_j=G_j\cap A_j$.
Let $L$ be an arbitrary compact subset of $G_j\cap U_j$ and
$a=\sup_L\psi_j<c$. Select $b\in]a,c[$ and set
$$\psi_{j,\eta}=\psi_j+\eta\varphi_j~~~~\hbox{\rm on}~~U_j,~~~~\eta>0.$$
Then $\psi_{j,\eta}$ is an exhaustion of $U_j$. As $\varphi_j$ is 
bounded below, we have
$$L\subset\{\psi_{j,\eta}<b\}\compact\{\psi_j<c\}\cap U_j=G_j\cap U_j$$
for $\eta$ small enough. Moreover
$$(1-\alpha)\psi_j+\alpha\psi_{j+1}=\psi-\varepsilon\sum_{0\le k<j}
\theta_k-\alpha\varepsilon\,\theta_j$$ 
is strongly $q$-convex for all $\alpha\in[0,1]$ and $\varepsilon\le
\varepsilon_0$ small enough, so Prop.~2.4 implies that $\psi_{j,\eta}$
is strongly $q$-convex. By definition, $G_j\cap U_j$ is thus 
\hbox{$q$-Runge} complete in $U_j$, and Lemma~4.12 
is proved with $X_{c+\varepsilon}$ instead of $X_d$.  In order to 
achieve the proof, we consider an increasing sequence 
$c=c_0<c_1<\ldots<c_N=d$ with $c_{k+1}-c_k\le\varepsilon_0$ and perform
the same construction for each pair $X_{c_k}\subset X_{c_{k+1}}$, with
$c$ replaced by $c_k$ and $\varepsilon=c_{k+1}-c_k$.\qed
\endproof

\begstat{(4.13) Proposition} For every sublevel set $X_c\supset K$, the
group $H^k(X_c,\cS)$ is Hausdorff and finite dimensional when $k\ge q$.
Moreover, for $d>c$, the restriction map
$$H^k(X_d,\cS)\lra H^k(X_c,\cS)$$
is an isomorphism when $k\ge q$ and has a dense range when $k=q-1$.
\endstat

\begproof{} Thanks to Lemma~4.12, we are led to consider the restriction maps
$$H^k(G_{j+1},\cS)\lra H^k(G_j,\cS).\leqno(4.14)$$
Let us apply the Mayer-Vietoris exact sequence IV-3.11 to $G_{j+1}=G_j\cup U_j$.
For $k\ge q$ we have $H^k(U_j,\cS)=H^k(G_j\cap U_j,\cS)=0$ by Lemma~4.4.
Hence we get an exact sequence
$$\cmalign{
H^{q-1}(G_{j+1},\cS)&\lra H^{q-1}(G_j,\cS)\oplus H^{q-1}(U_j,\cS)&\lra 
H^{q-1}(G_j\cap U_j,\cS)\lra\cr
H^k(G_{j+1},\cS)&\lra~~~~~~~~~~~~~H^k(G_j,\cS)&\lra~~~0~~~\lra\cdots,~~~~
k\ge q.\cr}$$
In this sequence, all the arrows are induced by restriction maps, so
they define continuous linear operators. We already infer that
the map (4.14) is bijective for $k>q$ and surjective for $k=q$.
There exist a $\cS$-acyclic covering $\cV=(V_\alpha)$ of $X_d$ and 
a finite family $\cV'=(V'_{\alpha_1}\ld V'_{\alpha_p})$ of open sets 
such that $V'_{\alpha_j}\compact V_{\alpha_j}$ and $\bigcup V'_{\alpha_j}
\supset\smash{\ovl X_c}$. Let $\cW$ be a locally finite $\cS$-acyclic covering 
of $X_c$ which refines $\cV'\cap X_c=(V'_{\alpha_j}\cap X_c)$.
The refinement map
$$C^\bu(\cV,\cS)\lra C^\bu(\cV'\cap X_c,\cS)\lra C^\bu(\cW,\cS)$$
is compact because the first arrow is, and it induces a surjective map
$$H^k(X_d,\cS)\lra H^k(X_c,\cS)~~~~\hbox{\rm for}~~k\ge q.$$
By Schwartz' theorem 1.9, we conclude that $H^k(X_c,\cS)$ is Hausdorff and 
finite dimensional for $k\ge q$. This is equally true for $H^q(G_j,\cS)$ 
because $G_j$ is also a global sublevel set $\{x\in X\,;\,\psi_j(x)<c_j\}$
containing $K$. Now, the Mayer-Vietoris exact sequence implies that the
composite
$$H^{q-1}(U_j,\cS)\lra H^{q-1}(G_j\cap U_j,\cS)\buildo\partial\over
\lra H^q(G_{j+1},\cS)$$
is equal to zero. However, the first arrow has a dense range by
Lemma~4.9. As the target space is Hausdorff, the second arrow must
be zero; we obtain therefore the injectivity of
$H^q(G_{j+1},\cS)\lra H^q(G_j,\cS)$ and an exact sequence
$$\cmalign{
H^{q-1}(G_{j+1},\cS)\lra H^{q-1}(G_j,\cS)&\oplus H^{q-1}(U_j,\cS)
&\lra H^{q-1}(G_j\cap U_j,\cS)\lra 0\cr
\hfill g&\oplus u&\longmapsto u_{\restriction G_j\cap U_j}-
g_{\restriction G_j\cap U_j}.\cr}$$
The argument used in Rem.~1.10 shows that the surjective arrow is open.
Let $g\in H^{q-1}(G_j,\cS)$ be given. By Lemma~4.9, we can
approximate $g_{\restriction G_j\cap U_j}$ by a sequence
$u_{\nu\restriction G_j\cap U_j}$, $u_\nu\in H^{q-1}(U_j,\cS)$.
Then $w_\nu=u_{\nu\restriction G_j\cap U_j}-g_{\restriction G_j\cap U_j}$
tends to zero. As the second map in the exact sequence is open, we can
find a sequence
$$g'_\nu\oplus u'_\nu\in H^{q-1}(G_j,\cS)\oplus H^{q-1}(U_j,\cS)$$
converging to zero which is mapped on $w_\nu$. Then
$(g-g'_\nu)\oplus(u_\nu-u'_\nu)$ is mapped on zero, and there exists
a sequence $f_\nu\in H^{q-1}(G_{j+1},\cS)$ which coincides
with $g-g'_\nu$ on $G_j$ and with $u_\nu-u'_\nu$ on $U_j$. In particular
$f_{\nu\restriction G_j}$ converges to $g$ and we have shown that
$$H^{q-1}(G_{j+1},\cS)\lra H^{q-1}(G_j,\cS)$$
has a dense range.\qed
\endproof

\begproof{of Andreotti-Grauert's Theorem 4.10.} Let $\cW$
be a countable basis of the topology of $X$
consisting of strongly $1$-convex open sets $W_\alpha$
contained in $\cS$-distinguished patches of $X$. 
Let $L\subset U$ be an arbitrary compact subset. Select
a smooth exhaustion function $\psi$ on $X$ such that $\psi$ is
strongly $q$-convex on $X\ssm\ovl X_b$ and
$L\subset X_b\compact U$ for some sublevel set $X_b$ of $\psi\,$;
choose $c>b$ such that $X_c\compact U$.
For every $d\in\bbbr$, we denote by $\cW_d\subset\cW$
the collection of sets $W_\alpha\in\cW$ such that $W_\alpha\subset X_d$.
Then $\cW_d$ is a $\cS$-acyclic covering of $X_d$. We consider the 
sequence of \v Cech complexes
$$E^\bu_\nu=C^\bu(\cW_{c+\nu},\cS),~~~~\nu\in\bbbn$$
together with the surjective projection maps $E^\bu_{\nu+1}\lra E^\bu_\nu$, 
and their inverse limit $E^\bu=C^\bu(\cW,\cS)$. Then we have
$H^k(E^\bu)=H^k(X,\cS)$ and $H^k(E^\bu_\nu)=H^k(X_{c+\nu},\cS)$.
Propositions 1.11 (a,b,c) and 4.13 imply that $H^k(X,\cS)\lra
H^k(X_c,\cS)$ is bijective for $k\ge q$ and has a dense range for
$k=q-1$. It already follows that $H^k(X,\cS)$ is Hausdorff for
$k\ge q$. Now, take an increasing sequence of open sets
$X_{c_\nu}$ equal to sublevel sets of a sequence of exhaustions 
$\psi_\nu$, such that $U=\bigcup X_{c_\nu}$. Then all groups
$H^k(X_{c_\nu},\cS)$ are in bijection with $H^k(X,\cS)$ for $k\ge q$,
and the image of $H^{q-1}(X_{c_{\nu+1}},\cS)$ in $H^{q-1}(X_{c_\nu},\cS)$
is dense because it contains the image of $H^{q-1}(X,\cS)$.
Proposition 1.11 (a,b,c) again shows that 
$H^k(U,\cS)\lra H^k(X_{c_0},\cS)$ is bijective
for $k\ge q$, and d) shows that $H^{q-1}(X,\cS)\lra H^{q-1}(U,\cS)$
has a dense range. The theorem follows.\qed
\endproof

A combination of Andreotti-Grauert's theorem with Th.~3.6 
yields the following important consequence.

\begstat{(4.15) Corollary} Let $\cS$ be a coherent sheaf over an analytic 
scheme $(X,\cO_X)$ with $\dim X\le n$.
\smallskip
\item{\rm a)} We have~~$H^k(X,\cS)=0$~~for all $k\ge n+1\,;$
\smallskip
\item{\rm b)} If $X$ has no compact irreducible component of dimension 
$n$, then we have $H^n(X,\cS)=0$.
\smallskip
\item{\rm c)} If $X$ has only finitely many $n$-dimensional compact
irreducible components, then $H^n(X,\cS)$ is finite dimensional.
\qed\smallskip
\endstat

The special case of 4.15 b) when $X$ is smooth and $\cS$ locally free
has been first proved by (Malgrange~1955), and the general case is due to
(Siu~1969). Another consequence is the following
approximation theorem for coherent sheaves over manifolds, which
results from Prop.~3.8.

\begstat{(4.16) Proposition} Let $\cS$ be a coherent sheaf 
over a non compact connected complex manifold $M$ with $\dim M=n$.
Let $U\subset M$ be an open subset such that the complement
$M\ssm U$ has no compact connected component. Then the
restriction map $H^{n-1}(M,\cS)\lra H^{n-1}(U,\cS)$ has a dense range.\qed
\endstat

\titleb{5.}{Grauert's Direct Image Theorem}
The goal of this section is to prove the following fundamental
result on direct images of coherent analytic sheaves, due to
(Grauert~1960).
\begstat{(5.1) Direct image theorem} Let $X$, $Y$ be complex
analytic schemes and let $F:X\to Y$ be a proper analytic morphism.
If $\cS$ is a coherent $\cO_X$-module, the direct
images $R^qF_\star\cS$ are coherent $\cO_Y$-modules.
\endstat

We give below a beautiful proof due to (Kiehl-Verdier~1971), which is 
much simpler than Grauert's original proof; this proof rests
on rather deep properties of nuclear modules over nuclear Fr\'echet
algebras. We first introduce the basic concept of topological tensor 
product. Our presentation owes much to the seminar lectures by
(Douady-Verdier~1973).

\titlec{5.A.}{Topological Tensor Products and Nuclear Spaces}
The algebra of holomorphic functions on a product space $X\times Y$
is a completion $\cO(X)\hotimes\cO(Y)$ of the algebraic tensor
product $\cO(X)\otimes\cO(Y)$. We are going to describe the
construction and the basic properties of the required topological 
tensor products $\hotimes$.

Let $E$, $F$ be (real or complex) vector spaces equipped with semi-norms
$p$ and $q$, respectively. Then $E\otimes F$ can be equipped with any
one of the two natural semi-norms $p\otimes_\pi q$, 
$p\otimes_\varepsilon q$ defined by
$$\eqalign{
p\otimes_\pi q(t)&=\inf\Big\{\sum_{1\le j\le N}p(x_j)\,q(y_j)\,;~
t=\!\!\sum_{1\le j\le N}x_j\otimes y_j\,,~x_j\in E\,,~y_j\in F~\Big\},\cr
p\otimes_\varepsilon q(t)&=\sup_{||\xi||_p\le 1,\,||\eta||_q\le 1}
\big|\xi\otimes\eta(t)\big|,~~~~\xi\in E',~~\eta\in F'\,;\cr}$$
the inequalities in the last line mean that $\xi$, $\eta$ satisfy $|\xi(x)|
\le p(x)$ and $|\eta(y)|\le q(y)$ for all $x\in E$, $y\in F$. Then clearly
$p\otimes_\varepsilon q\le p\otimes_\pi q$, for
$$p\otimes_\varepsilon q\Big(\sum x_j\otimes y_j\Big)\le
\sum p\otimes_\varepsilon q(x_j\otimes y_j)\le\sum p(x_j)\,q(y_j).$$
Given $x\in E$, $y\in F$, the Hahn-Banach theorem implies that there exist
$\xi$,~$\eta$ such that $||\xi||_p=||\eta||_q=1$ with
$\xi(x)=p(x)$ and $\eta(y)=q(y)$, hence 
$p\otimes_\varepsilon q(x\otimes y)\ge p(x)\,q(y)$. On the other hand
$p\otimes_\pi q(x\otimes y)\le p(x)\,q(y)$, thus
$$p\otimes_\varepsilon q(x\otimes y)=p\otimes_\pi q(x\otimes y)=p(x)\,q(y).$$

\begstat{(5.2) Definition} Let $E$, $F$ be locally convex topological vector
spaces. The topological tensor product $E\hotimes_\pi F$ $($resp.
$E\hotimes_\varepsilon F\,)$ is the Hausdorff completion of $E\otimes F$,
equipped with the family of semi-norms $p\otimes_\pi q$ $($resp. 
$p\otimes_\varepsilon q)$ associated to fundamental families of 
semi-norms on $E$ and $F$.
\endstat

Since we may also write
$$p\otimes_\pi q(t)=\inf\Big\{\sum|\lambda_j|\,;~
t=\sum\lambda_j\,x_j\otimes y_j\,,~p(x_j)\le 1\,,~q(y_j)\le 1~\Big\}$$
where the $\lambda_j$'s are scalars, we see that the closed unit ball
$B(p\hotimes_\pi q)$ in $E\hotimes_\pi F$ is the closed convex hull
of $B(p)\otimes B(q)$. From this, we easily infer that the topological
dual space $(E\hotimes_\pi F)'$ is isomorphic to the space of continuous
bilinear forms on $E\times F$. Another simple consequence of this 
interpretation of $B(p\hotimes_\pi q)$ is example a) below.

\begstat{\bf(5.3) Examples} \smallskip\rm
\noindent{a)} For all discrete spaces $I$, $J$, there is an isometry 
$$\ell^1(I)\hotimes_\pi\ell^1(J)\simeq\ell^1(I\times J).$$
\noindent{b)} For Banach spaces $(E,p)$, $(F,q)$, the closed unit
ball in $E\hotimes_\varepsilon F$ is dual to the unit ball
$B(p'\hotimes_\pi q')$ of $E'\hotimes_\pi F'$ through the natural
pairing extending the algebraic pairing of $E\otimes F$ and $E'\otimes F'$.
If $c_0(I)$ denotes the space of bounded sequences on $I$ converging to 
zero at infinity, we have $c_0(I)'=\ell^1(I)$, hence
by duality $c_0(I)\hotimes_\varepsilon c_0(J)$ is isometric to
$c_0(I\times J)$.
\medskip
\noindent{c)} If $X$, $Y$ are compact topological spaces and if $C(X)$, $C(Y)$
are their algebras of continuous functions with the sup norm, then
$$C(X)\hotimes_\varepsilon C(Y)\simeq C(X\times Y).$$
Indeed, $C(X)'$ is the space of finite Borel measures equipped with the 
mass norm. Thus for $f\in C(X)\otimes C(Y)$, the $\otimes_\varepsilon$-seminorm
is given by
$$||f||_\varepsilon=\sup_{||\mu||\le 1,\,||\nu||\le 1}\mu\otimes\nu(f)=
\sup_{X\times Y}|f|\,;$$
the last equality is obtained by taking Dirac measures $\delta_x$, $\delta_y$
for $\mu$, $\nu$ (the inequality $\le$ is obvious). Now $C(X)\otimes C(Y)$ is 
dense in $C(X\times Y)$ by the Stone-Weierstrass theorem, hence its completion
is $C(X\times Y)$, as desired.\qed
\endstat

Let $f:E_1\to E_2$ and $g:F_1\to F_2$ be continuous morphisms. For all
semi-norms $p_2$, $q_2$ on $E_2$, $F_2$, there exist semi-norms
$p_1$, $q_1$ on $E_1$, $F_1$ and constants $||f||=||f||_{p_1,p_2}$,
$||g||=||g||_{q_1,q_2}$ such that
$p_2\circ f\le ||f||\,p_1$ and $q_2\circ g\le ||g||\,q_1$. Then we find
$$(p_2\otimes_\pi q_2)\circ(f\otimes g)\le ||f||\,||g||\,p_1\otimes_\pi q_1$$
and a similar formula with $p_j\otimes_\varepsilon q_j$. It follows that
there are well defined continuous maps
\medskip\noindent
$\cmalign{
(5.4')\hfill\qquad
&f\hotimes_\pi g&{}:E_1\hotimes_\pi F_1&{}\lra E_2\hotimes_\pi F_2,\cr
(5.4'')\hfill\qquad
&f\hotimes_\varepsilon g&{}:E_1\hotimes_\varepsilon F_1&{}\lra
E_2\hotimes_\varepsilon F_2.\cr}$
\medskip\noindent
Another simple fact is that $\hotimes_\pi$ preserves open morphisms:

\begstat{(5.5) Proposition} If $f:E_1\to E_2$ and $g:F_1\to F_2$ are
epimorphisms, then $f\hotimes_\pi g:E_1\hotimes_\pi F_1\lra 
E_2\hotimes_\pi F_2$ is an epimorphism.
\endstat

\begproof{} Recall that when $E$ is locally convex complete and $F$ Hausdorff,
a morphism $u:E\to F$ is open if and only if $\ovl{u(V)}$ is a neighborhood
of $0$ for every neighborhood of $0$ (this can be checked essentially by the
same proof as 1.8~b)). Here, for any semi-norms $p$, $q$ on
$E_1$, $F_1$ the closure of $f\hotimes_\pi g\big(B(p\hotimes_\pi q)\big)$
contains the closed convex hull of $f\big(B(p)\big)\otimes g\big(B(q)\big)$
in which $f\big(B(p)\big)$ and $g\big(B(q)\big)$ are neighborhoods
of~$0$, so it is a neighborhood of $0$ in $E\hotimes_\pi F$.\qed
\endproof

If $E_1\subset E_2$ is a closed subspace, every continuous semi-norm
$p_1$ on $E_1$ is the restriction of a continuous semi-norm on $E_2$, and
every linear form $\xi_1\in E_1'$ such that $||\xi_1||_{p_1}\le 1$ can
be extended to a linear form $\xi_2\in E_2$ such that
$||\xi_2||_{p_2}=||\xi_1||_{p_1}$ (Hahn-Banach theorem); similar
properties hold for a closed subspace $F_1\subset F_2$. We infer that
$$(p_2\otimes_\varepsilon q_2)_{\restriction E_1\otimes F_1}=
p_1\otimes_\varepsilon q_1\,,$$
thus $E_1\hotimes_\varepsilon F_1$ is a closed subspace of
$E_2\hotimes_\varepsilon F_2$. In other words:

\begstat{(5.6) Proposition} If $f:E_1\to E_2$ and $g:F_1\to F_2$ are
monomorphisms, then $f\hotimes_\varepsilon g:E_1\hotimes_\varepsilon F_1\lra 
E_2\hotimes_\varepsilon F_2$ is a monomorphism.\qed
\endstat

Unfortunately, 5.5 fails for $\hotimes_\varepsilon$ and 5.6 fails for
$\hotimes_\pi$, even with Fr\'echet or Banach spaces. It follows that 
neither $\hotimes_\pi$ nor $\hotimes_\varepsilon$ are exact functors 
in the category of Fr\'echet spaces. In order to circumvent this difficulty, 
it is necessary to work in a suitable subcategory.

\begstat{(5.7) Definition} A morphism $f:E\to F$ of complete locally convex
spaces is said to be nuclear if $f$ can be written as
$$f(x)=\sum\lambda_j\,\xi_j(x)\,y_j$$
where $(\lambda_j)$ is a sequence of scalars with $\sum|\lambda_j|<+\infty$,
$\xi_j\in E'$ an equi\-con\-tinuous sequence of linear forms and $y_j\in F$ a
bounded sequence.
\endstat

When $E$ and $F$ are Banach spaces, the space of nuclear morphisms is
isomorphic to $E'\hotimes_\pi F$ and the nuclear norm $||f||_\nu$ is
defined to be the norm in this space, namely
$$||f||_\nu=\inf\Big\{\sum|\lambda_j|\,;~f=\sum\lambda_j\,\xi_j\otimes y_j,~
||\xi_j||\le 1,~||y_j||\le 1~\Big\}.\leqno(5.8)$$
For general spaces $E$, $F$, the equicontinuity of $(\xi_j)$ means that
there is a semi-norm $p$ on $E$ and a constant $C$ such that
$|\xi_j(x)|\le C\,p(x)$ for all $j$.  Then the definition shows that
$f:E\to F$ is nuclear if and only if $f$ can be factorized as $E\to
E_1\to F_1\to F$ where $E_1\to F_1$ is a nuclear morphism of Banach
spaces: indeed we need only take $E_1$ be equal to the Hausdorff
completion $\wh E_p$ of $(E,p)$ and let $F_1$ be the subspace of $F$
generated by the closed balanced convex hull of~$\{y_j\}$ ($=$ unit
ball in $F_1)\,$; moreover, if $u:S\to E$ and $v:F\to T$ are
continuous, the nuclearity of $f$ implies the nuclearity of $v\circ
f\circ u\,$; its nuclear decomposition is then $v\circ f\circ u=
\sum\lambda_j\,(\xi_j\circ u)\otimes v(y_j)$.

\begstat{(5.9) Remark} \rm Every nuclear morphism is compact: indeed, we may assume
in Def.~5.7 that $(y_j)$ converges to $0$ and $\sum|\lambda_j|\le 1$,
otherwise we replace $y_j$ by $\varepsilon_j y_j$ with $\varepsilon_j$
converging to zero such that $\sum|\lambda_j/\varepsilon_j|\le 1\,$;
then, if $U\subset F$ is  a neighborhood of $0$ such that
$|\xi_j(U)|\le 1$ for all $j$, the image $f(U)$ is contained in the
closed convex hull of the compact set $\{y_j\}\cup\{0\}$, which is
compact.
\endstat

\begstat{(5.10) Proposition} If $E,F,G$ are Banach spaces and if
$f:E\to F$ is nuclear, there is a continuous morphism
$$f\hotimes\Id_G:E\hotimes_\varepsilon G\lra F\hotimes_\pi G$$
extending $f\otimes\Id_G$, such that
$||f\hotimes\Id_G||\le||f||_\nu$.
\endstat

\begproof{} If $f=\sum\lambda_j\,\xi_j\otimes y_j$ as in (5.8), then
for any $t\in E\otimes G$ we have
$$(f\otimes\Id_G)(t)=\sum\lambda_j\big(\xi_j\otimes\Id_G(t)\big)
\otimes y_j$$
where $(\xi_j\otimes\Id_G)(t)\in G$ has norm
$$||(\xi_j\otimes\Id_G)(t)||=
\sup_{\eta\in G',\,||\eta||\le 1}\big|\eta\big(\xi_j\otimes\Id_G(t)\big)\big|
=\sup_\eta\big|\xi_j\otimes\eta(t)\big|\le ||t||_\varepsilon.$$
Therefore $||f\otimes\Id_G(t)||_\pi\le\sum|\lambda_j|\,||t||_\varepsilon$,
and the infimum over all decompositions of $f$ yields
$$||f\otimes\Id_G(t)||_\pi\le||f||_\nu||t||_\varepsilon.$$
Proposition 5.10 follows.\qed
\endproof

If $E$ is a Fr\'echet space and $(p_j)$ an increasing sequence of
semi-norms on $E$ defining the topology of $E$, we have
$$E=\lim_{{\displaystyle\longleftarrow}}\wh E_{p_j},$$
where $\wh E_{p_j}$ is the Hausdorff completion of $(E,p_j)$ and
$\wh E_{p_{j+1}}\to\wh E_{p_j}$ the canonical morphism. Here
$\wh E_{p_j}$ is a Banach space for the induced norm $\wh p_j$.

\begstat{(5.11) Definition} A Fr\'echet space $E$ is said to be nuclear if the
topology of $E$ can be defined by an increasing sequence of semi-norms
$p_j$ such that each canonical morphism
$$\wh E_{p_{j+1}}\lra\wh E_{p_j}$$
of Banach spaces is nuclear.
\endstat

If $E,F$ are arbitrary locally convex spaces, we always have a
continuous morphism $E\hotimes_\pi F\to E\hotimes_\varepsilon F$, because
$p\otimes_\varepsilon q\le p\otimes_\pi q$. If $E$, say, is nuclear,
this morphism yields in fact an isomorphism $E\hotimes_\varepsilon F
\simeq E\hotimes_\pi F\,$: indeed, by Prop.~5.10, we have
$p_j\hotimes_\pi q\le C_j\,p_{j+1}\hotimes_\varepsilon q$ where $C_j$
is the nuclear norm of $\wh E_{p_{j+1}}\to\wh E_{p_j}$. Hence, when
$E$ or $F$ is nuclear, we will identify $E\hotimes_\pi F$ and
$E\hotimes_\varepsilon F$ and omit $\varepsilon$ or $\pi$ in the
notation $E\hotimes F$.

\begstat{(5.12) Example} \rm Let $D=\prod D(0,R_j)$ be a polydisk in $\bbbc^n$.
For any $t\in{}]0,1[$, we equip $\cO(D)$ with the semi-norm
$$p_t(f)=\sup_{tD}|f|.$$
The completion of $\big(\cO(D),p_t\big)$ is the Banach space $E_t$ of
holomorphic functions on $tD$ which are continuous up to the boundary.
We claim that for $t'<t<1$ the restriction map
$$\rho_{t,t'}:E_{t'}\lra E_t$$
is nuclear. In fact, for $f\in\cO(D)$, we have $f(z)=\sum a_\alpha
z^\alpha$ where $a_\alpha=a_\alpha(f)$ satisfies the Cauchy inequalities
$|a_\alpha(f)|\le p_{t'}(f)/(t'R)^{\alpha}$ for all $\alpha\in\bbbn^n$. 
The formula $f=\sum a_\alpha(f)\,e_\alpha$ with $e_\alpha(z)=z^\alpha$ 
shows that
$$||\rho_{t,t'}||_\nu\le\sum||a_\alpha||_{p_{t'}}||e_\alpha||_{p_t}\le
\sum(t'R)^{-\alpha}(tR)^\alpha=(1-t/t')^{-n}<+\infty.$$
We infer that $\cO(D)$ is a nuclear Fr\'echet space. It is also in a
natural way a fully nuclear Fr\'echet algebra (see Def.~5.39 below).\qed
\endstat

\begstat{(5.13) Proposition} Let $E$ be a nuclear space. A morphism
$f:E\to F$ is nuclear if and only if $f$ admits a factorization
$E\to M\to F$ through a Banach space $M$.
\endstat

\begproof{} By definition, a nuclear map $f:E\to F$ always has a factorization
through a Banach space (even if $E$ is not nuclear). Conversely, if $E$ is
nuclear, any continuous linear map $E\to M$ into a Banach space $M$ is
continuous for some semi-norm $p_j$ on $E$, so this map has a factorization
$$E\to\wh E_{p_{j+1}}\to\wh E_{p_j}\to M$$
in which the second arrow is nuclear. Hence any map $E\to M\to F$ is
nuclear.\qed
\endproof

\begstat{(5.14) Proposition} \smallskip
\item{\rm a)} If $E$, $F$ are nuclear spaces, then $E\hotimes F$ is nuclear.
\smallskip
\item{\rm b)} Any closed subspace or quotient space of a nuclear space
is nuclear.
\smallskip
\item{\rm c)} Any countable product of nuclear spaces is nuclear.
\smallskip
\item{\rm d)} Any countable inverse limit of nuclear spaces is nuclear.
\vskip0pt
\endstat

\begproof{} a) If $f:E_1\to F_1$ and $g:E_2\to F_2$ are nuclear morphisms of
Banach spaces, it is easy to check that $f\hotimes_\pi g$ and
$f\hotimes_\varepsilon g$ are nuclear with
$||f\hotimes_? g||_\nu\le||f||_\nu||g||_\nu$ in both cases. Property a)
follows by applying this to the canonical morphisms
$\wh E_{p_{j+1}}\to\wh E_{p_j}$ and $\wh F_{q_{j+1}}\to\wh F_{q_j}$.
\medskip
\noindent{c)} Let $E_k$, $k\in\bbbn$, be nuclear spaces and $F=\prod E_k$.
If $(p^k_j)$ is an increasing family of semi-norms on $E_k$ as in
Def.~5.11, then the topology of $F$ is defined by the family of
semi-norms
$$q_j(x)=\max_{0\le k\le j}p^k_j(x_k),~~~~x=(x_k)\in F.$$
Then $\wh F_{q_j}=\bigoplus_{0\le k\le j}\wh E_{k,p^k_j}$ and
$$\big(\wh F_{q_{j+1}}\to\wh F_{q_j}\big)=\bigoplus_{0\le k\le j}
\big(\wh E_{k,p^k_{j+1}}\to\wh E_{k,p^k_j}\big)\oplus
\big(\wh E_{j+1,p^{j+1}_{j+1}}\to\{0\}\big)$$
is easily seen to be nuclear.
\medskip
\noindent{b)} If $F\subset E$ is closed, then $\wh F_{p_j}$ can be
identified to a closed subspace of~$\wh E_{p_j}$, the map $\wh F_{p_{j+1}}
\to\wh F_{p_j}$ is the restriction of $\wh E_{p_{j+1}}\to\wh E_{p_j}$
and we have \hbox{$\wh{E/F}_{p_j}\simeq\wh E_{p_j}/\wh F_{p_j}$.}
It is not true in general that the restriction or quotient of a nuclear
morphism is nuclear, but this is true for a binuclear \hbox{
$=(\hbox{nuclear}\circ\hbox{nuclear})$} morphism, as shown by Lemma~5.15~b)
below. Hence $\wh F_{p_{2j+2}}\to\wh E_{p_{2j}}$ and $\wh{E/F}_{p_{2j+2}}
\to\wh{E/F}_{p_{2j}}$ are nuclear, so $(p_{2j})$ is a fundamental family
of semi-norms on $F$ or $E/F$, as required in Def.~5.11.
\medskip
\noindent{d)} follows immediately from b) and c), since
$\smash{\displaystyle\lim_{\displaystyle\longleftarrow}}E_k$ is a closed 
subspace of $\prod E_k$.\qed
\endproof

\begstat{(5.15) Lemma} Let $E$, $F$, $G$ be Banach spaces.\smallskip
\item{\rm a)} If $f:E\to F$ is nuclear, then $f$ can be factorized
through a Hilbert space $H$ as a morphism $E\to H\to F$.
\smallskip
\item{\rm b)} Let $g:F\to G$ be another nuclear morphism. If
$\Im(g\circ f)$ is contained in a closed subspace $T$ of $G$, then
$g\circ f:E\to T$ is nuclear. If $\ker(g\circ f)$ contains a closed
subspace $S$ of $E$, the induced map $(g\circ f)^\sim:E/S\to G$ is
nuclear.\smallskip
\endstat

\begproof{} a) Write $f=\sum_{j\in I}\xi_j\otimes y_j\in E'\hotimes_\pi F$ with
$\sum||\xi_j||\,||y_j||<+\infty$. Without loss of generality, we may
suppose $||\xi_j||=||y_j||$. Then $f$ is the composition
$$E\lra\ell^2(I)\lra F,~~~~x\longmapsto\big(\xi_j(x)\big),~~
(\lambda_j)\longmapsto\sum\lambda_jy_j.$$
b) Decompose $g$ into $g=v\circ u$ as in a) and write $g\circ f$ 
as the composition
$$E\buildo f\over\lra F\buildo u\over\lra H\buildo v\over\lra G$$
where $H$ is a Hilbert space. If $\Im(g\circ f)\subset T$ and if
$T\subset G$ is closed, then $H_1=v^{-1}(T)$ is a closed subspace of
$H$ containing $\Im(u\circ f)$. Therefore $g\circ f:E\to T$ is the composition
$$E\buildo f\over\lra F\buildo u\over\lra H\buildo{\rm pr}^\perp\over\lra
H_1\buildo v_{\restriction H_1}\over\lra T$$
where $f$ is nuclear and $g\circ f:E\lra T$ is nuclear. Similar
proof for $(g\circ f)^\sim:E/S\to G$ by using decompositions 
$f=v\circ u:E\to H\to F$ and
$$(g\circ f)^\sim:E/S\buildo\wt u\over\lra H/H_1\simeq H_1^\perp
\buildo v_{\restriction H_1^\perp}\over\lra F\buildo g\over\lra G$$
where $H_1=\ovl{u(S)}$ satisfies $H_1\subset\ker(g\circ v)\subset H$.\qed
\endproof

\begstat{(5.16) Corollary} Let $E$ be a nuclear space and let $E\to F$ be a
nuclear morphism.
\smallskip
\item{\rm a)} If $f(E)$ is contained in a closed subspace $T$ of $F$, then
the morphism \hbox{$f_1:E\to T$} induced by $f$ is nuclear.
\smallskip
\item{\rm b)} If $\ker f$ contains a closed subspace $S$ of $E$, then
$\wt f:E/S\to F$ is nuclear.
\endstat

\begproof{} Let $E\buildo u\over\lra M\buildo v\over\lra F$ be a factorization 
of $f$ through a Banach space~$M$. In case a), resp. b), $M_1=v^{-1}(T)$
is a closed subspace of $M$, resp. $M/\ovl{u(S)}$ is a Banach space,
and we have factorizations
$$f_1:E\buildo u_1\over\lra M_1\buildo v_1\over\lra T,~~~~
\wt f:E/S\buildo\wt u\over\lra M/\ovl{u(S)}\buildo\wt v\over\lra F$$
where $u_1$, $\wt u$ are induced by $u$ and $v_1$, $\wt v$ by $v$. Hence
$f_1$ and $\wt f$ are nuclear.\qed
\endproof

\begstat{(5.17) Proposition} Let $0\to E_1\to E_2\to E_3\to 0$ be an
exact sequence of Fr\'echet spaces and let $F$ be a Fr\'echet space.
If $E_2$ or $F$ is nuclear, there is an exact sequence
$$0\lra E_1\hotimes F\lra E_2\hotimes F\lra E_3\hotimes F\lra 0.$$
\endstat

\begproof{} If $E_2$ is nuclear, then so are $E_1$ and $E_3$ by Prop.~5.14~b).
Hence $E_1\hotimes F\to E_2\hotimes F$ is a monomorphism and 
$E_2\hotimes F\to E_3\hotimes F$ an epi\-mor\-phism by Prop.~5.6 and 5.5.
It only remains to show that
$$\Im\big(E_1\hotimes F\lra E_2\hotimes F\big)=
\ker\big(E_2\hotimes F\lra E_3\hotimes F\big)$$
and for this, we need only show that the left hand side is dense in the
right hand side (we already know it is closed). Let
$\varphi\in(E_2\hotimes F)'$ be a linear form, viewed as a continuous
bilinear form on $E_2\times F$. If $\varphi$ vanishes on the image
of $E_1\hotimes F$, then $\varphi$ induces a continuous bilinear form
on $E_3\times F$ by passing to the quotient. Hence $\varphi$ must
vanish on the kernel of $E_2\hotimes F\to E_3\hotimes F$, and
our density statement follows by the Hahn-Banach theorem.\qed
\endproof

\titlec{5.B.}{K\"unneth Formula for Coherent Sheaves}
As an application of the above general concepts, we now show how 
topological tensor products can be used to compute holomorphic 
functions and cohomology of coherent sheaves on product spaces.

\begstat{(5.18) Proposition} Let $\cF$ be a coherent analytic
sheaf on a complex analytic scheme $(X,\cO_X)$. Then $\cF(X)$ is
a nuclear space.
\endstat

\begproof{} Let $A\subset\Omega\subset\bbbc^N$ be an open patch of $X$ such that
the image sheaf $(i_A)_\star\cF_{\restriction A}$ on $\Omega$ has a
resolution
$$\cO_\Omega^{p_1}\lra\cO_\Omega^{p_0}\lra(i_A)_\star\cF_{\restriction A}\lra0$$
and let $D\compact\Omega$ be a polydisk. As $D$ is Stein, we get an exact
sequence
$$\cO^{p_1}(D)\lra\cO^{p_0}(D)\lra\cF(A\cap D)\lra 0.\leqno(5.19)$$
Hence $\cF(A\cap D)$ is a quotient of the nuclear space $\cO^{p_0}(D)$
and so $\cF(A\cap D)$ is nuclear by (5.14~b). Let $(U_\alpha)$ be a 
countable covering of $X$ by open sets of the form $A\cap D$. Then $\cF(X)$ 
is a closed subspace of $\prod\cF(U_\alpha)$, thus $\cF(X)$ is nuclear
by \hbox{(5.14~b,c)}.\qed
\endproof

\begstat{(5.20) Proposition} Let $\cF$, $\cG$ be coherent
sheaves on complex analytic schemes $X$, $Y$ respectively.
Then there is a canonical isomorphism
$$\cF\stimes\cG(X\times Y)\simeq\cF(X)\hotimes\cG(Y).$$
\endstat

\begproof{} We show the proposition in several steps of increasing generality.
\medskip
\item{a)}{\it $X=D\subset\bbbc^n$, $Y=D'\subset\bbbc^p$ are
polydisks, $\cF=\cO_X$, $\cG=\cO_Y$.}\smallskip
Let $p_t(f)=\sup_{tD}|f|$, $p'_t(f)=\sup_{tD'}|f|$ and
$q_t(f)=\sup_{t(D\times D')}|f|$ be the semi-norms
defining the topology of $\cO(D)$, $\cO(D')$ and $\cO(D\times D')$, 
respectively.
Then $\wh E_{p_t}$ is a closed subspace of the space $C(t\ovl D)$ of 
continuous functions on $t\ovl D$ with the sup norm, and we have
$p_t\otimes_\varepsilon p'_t=q_t$ by example (5.3~c). Now, 
$\cO(D)\otimes\cO(D')$ is dense in $\cO(D\times D')$, hence its
completion with respect to the family $(q_t)$ is
$\cO(D)\hotimes_\varepsilon\cO(D')=\cO(D\times D')$.
\medskip
\item{b)}{\it $X$ is embedded in a polydisk $D\subset\bbbc^n$,
$X=A\cap D\buildo i\over\lhra D$,\hfill\break
$i_\star\cF$ is the cokernel of a morphism 
$\cO_D^{p_1}\lra\cO_D^{p_0}$,\hfill\break
$Y=D'\subset\bbbc^p$ is a polydisk and $\cG=\cO_Y$.}\smallskip
\noindent{}By taking the external tensor product with $\cO_Y$, we get an exact 
sequence
$$\cO_{D\times Y}^{p_1}\lra\cO_{D\times Y}^{p_0}\lra i_\star\cF\stimes
\cO_Y\lra 0.\leqno(5.21)$$
Then we find a commutative diagram
$$\cmalign{
\cO^{p_1}(D)&\hotimes\cO(Y)&\lra\cO^{p_0}(D)&\hotimes\cO(Y)&\lra
\hfill\cF(X)&\hotimes\cO(Y)&\lra 0\cr
&~\big\downarrow\simeq&&~\big\downarrow\simeq&&~\big\downarrow&\cr
\hfill\cO^{p_1}(D&\times Y)&\lra\hfill\cO^{p_0}(D&\times Y)&\lra
\cF\stimes\cO_Y&(X\times Y)&\lra 0\cr}$$
in which the first line is exact as the image of (5.19) by the exact functor
$\bu\hotimes\cO(Y)$, and the second line is exact because the exact sequence 
of sheaves (5.21) gives an exact sequence of spaces of sections on the Stein 
space $D\times Y\,$; note that $i_\star\cF\stimes\cO_Y(D\times Y)=
\cF\stimes\cO_Y(X\times Y)$. As the first two vertical arrows are isomorphisms
by a), the third one is also an isomorphism.
\medskip
\item{c)}{\it $X$, $\cF$ are as in {\rm b)},\hfill\break
$Y$ is embedded in a polydisk $D'\subset\bbbc^p$,
$\smash{Y=A'\cap D'\buildo j\over\lhra D'}$\hfill\break
and $j_\star\cG$ is the cokernel of $\cO_{D'}^{q_1}\lra\cO_{D'}^{q_0}$.}
\smallskip
\noindent{}Taking the external tensor product with $\cF$, we get an exact sequence
$$\cF\stimes\cO_{D'}^{q_1}\lra\cF\stimes\cO_{D'}^{q_0}\lra\cF\stimes 
j_\star\cG\lra 0$$
and with the same arguments as above we obtain a commutative diagram
$$\cmalign{
\hfill\cF(X)&\hotimes\cO^{q_1}(D')&\lra\hfill\cF(X)&\hotimes\cO^{q_0}(D')
&\lra\hfill\cF(X)&\hotimes\cG(Y)&\lra 0\cr
&~\big\downarrow\simeq&&~\big\downarrow\simeq&&~\big\downarrow&\cr
\hfill\cF\stimes\cO^{q_1}_{D'}&(X\times D')&\lra\hfill\cF\stimes
\cO^{q_0}_{D'}&(X\times D')&\lra\cF\stimes\cG&(X\times Y)&\lra 0.\cr}$$
\item{d)}{\it $X$, $\cF$ are as in {\rm b),c)} and $Y$, $\cG$ are
arbitrary.}\smallskip
Then $Y$ can be covered by open sets $U_\alpha=A_\alpha\cap D_\alpha$ 
embedded in polydisks $D_\alpha$, on which the image of $\cG$ admits a two-step
resolution. We have $\cF\stimes\cG(X\times U_\alpha)\simeq
\cF(X)\hotimes\cG(U_\alpha)$ by c), and the same is true over the
intersections $X\times U_{\alpha\beta}$ because $U_{\alpha\beta}=U_\alpha
\cap U_\beta$ can be embedded by the cross product embedding
$j_\alpha\times j_\beta:U_{\alpha\beta}\to D_\alpha\times D_\beta$.
We have an exact sequence
$$0\lra\cG(Y)\lra\prod_\alpha\cG(U_\alpha)\lra\prod_{\alpha,\beta}
\cG(U_{\alpha\beta})$$
where the last arrow is $(c_\alpha)\mapsto(c_\beta-c_\alpha)$, and a
commutative diagram with exact lines
$$\cmalign{
&0\lra\hfill\cF(X)&\hotimes\cG(Y)&\lra\hfill\prod\cF(X)&\hotimes\cG(U_\alpha)
&\lra\prod\hfill\cF(X)&\hotimes\cG(U_{\alpha\beta})\cr
&&~\big\downarrow&&~\big\downarrow\simeq&&~\big\downarrow\simeq\cr
&0\lra\cF\stimes\cG(&X\times Y)&\lra\prod\cF\stimes\cG(&X\times U_\alpha)
&\lra\prod\cF\stimes\cG(&X\times U_{\alpha\beta}).\cr}$$
Therefore the first vertical arrow is an isomorphism.
\medskip
\item{e)}{\it $X$, $\cF$, $Y$, $\cG$ are arbitrary.}\smallskip
This case is treated exactly in the same way as d) by reversing the 
roles of $\cF$, $\cG$ and by using d) to get the isomorphism in the last
two vertical arrows.\qed
\endproof

\begstat{(5.22) Corollary} Let $\cF$, $\cG$ be coherent sheaves over complex
analytic schemes $X$, $Y$ and let $\pi:X\times Y\to X$ be the projection.
Suppose that $H^\bu(Y,\cG)$ is Hausdorff.
\smallskip
\item{\rm a)} If $X$ is Stein, then
$H^q(X\times Y,\cF\stimes\cG)\simeq\cF(X)\hotimes H^q(Y,\cG)$.
\smallskip
\item{\rm b)} In general, for every open set $U\subset X$,
$$\big(R^q\pi_\star(\cF\stimes\cG)\big)(U)=\cF(U)\hotimes H^q(Y,\cG).$$
\smallskip
\item{\rm c)} If $H^q(Y,\cG)$ is finite dimensional, then
$$R^q\pi_\star(\cF\stimes\cG)=\cF\otimes H^q(Y,\cG).$$
\endstat

\begproof{} a) Let $\cV=(V_\alpha)$ be a countable Stein covering of $Y$. By 
the Leray theorem, $H^\bu(Y,\cG)$ is equal to the cohomology of the \v Cech 
complex $C^\bu(\cV,\cG)$. Similarly $X\times\cV=(X\times V_\alpha)$ is a
Stein covering of $X\times Y$ and we have
$$H^q(X\times Y,\cF\stimes\cG)=H^q\big(C^\bu(X\times\cV,\cF\stimes\cG)\big).$$
However, Prop.~5.20 shows that $C^\bu(X\times\cV,\cF\stimes\cG)=
\cF(X)\hotimes C^\bu(\cV,\cG)$. Our assumption that 
$C^\bu(\cV,\cG)$ has Hausdorff cohomology implies that the cocycle and
coboundary groups are (nuclear) Fr\'echet spaces, and that each cohomology 
group can be computed by means of short exact sequences in this category. By 
Prop.~5.17, we thus get the desired equality
$$H^q\big(C^\bu(X\times\cV,\cF\stimes\cG)\big)=
\cF(X)\hotimes H^q\big(C^\bu(\cV,\cG)\big).$$
\noindent{b)} The presheaf $U\mapsto\cF(U)\hotimes H^q(Y,\cG)$ is in fact
a sheaf, because the tensor product with the nuclear space $H^q(Y,\cG)$ 
preserves the exactness of all sequences
$$0\lra\cF(U)\lra\prod\cF(U_\alpha)\lra\prod\cF(U_{\alpha\beta})$$
associated to arbitrary coverings $(U_\alpha)$ of $U$. Property b)
thus follows from a) and from the fact that $R^q\pi_\star(\cF\stimes\cG)$ is
the sheaf associated to the presheaf $U\mapsto H^q(U\times Y,\cF\stimes\cG)$.
\medskip
\noindent{c)} is an immediate consequence of b), since the finite
dimensionality of $H^q(Y,\cG)$ implies that this space is Hausdorff.\qed
\endproof

\begstat{(5.23) K\"unneth formula} Let $\cF$, $\cG$ be coherent 
sheaves over complex analytic schemes $X$, $Y$ and suppose that the 
cohomology spaces $H^\bu(X,\cF)$ and $H^\bu(Y,\cG)$ are Hausdorff. 
Then there is an isomorphism
$$\eqalign{
\bigoplus_{p+q=k}H^p(X,\cF)\hotimes H^q(Y,\cG)&\buildo\simeq\over\lra
H^k(X\times Y,\cF\stimes\cG)\cr
\bigoplus\alpha_p\otimes\beta_q&\buildo~\over\longmapsto
\sum\alpha_p\smallsmile\beta_q.\cr}$$
\endstat

\begproof{} Consider the Leray spectral sequence associated to the coherent
sheaf $\cS=\cF\stimes\cG$ and to the projection $\pi:X\times Y\to X$. By 
Cor.~5.22~b) and a use of \v Cech cohomology, we find 
$$E_2^{p,q}=H^p(X,R^q\pi_\star\cF\stimes\cG)=H^p(X,\cF)\hotimes H^q(Y,\cG).$$
It remains to show that the Leray spectral sequence degenerates in
$E_2$. For this, we argue as in the proof of Th.~IV-15.9. In that proof,
we defined a morphism of the double complex $C^{p,q}=\cF^{[p]}(X)\otimes
\cG^{[q]}(Y)$ into the double complex that defines the Leray
spectral sequence (in IV-15.9, we only considered the sheaf theoretic
external tensor product $\cF\stimes\cG$, but there is an obvious morphism
of that one into the analytic tensor product). We get a morphism of
spectral sequences which induces at the $E_2$-level the obvious morphism
$$H^p(X,\cF)\otimes H^q(Y,\cG)\lra H^p(X,\cF)\hotimes H^q(Y,\cG).$$
It follows that the Leray spectral sequence $E^{p,q}_r$ is
obtained for $r\ge 2$ by taking the completion of the spectral
sequence of $C^{\bu,\bu}$. Since this spectral sequence degenerates
in $E_2$ by the algebraic K\"unneth theorem, the Leray spectral
sequence also satisfies $d_r=0$ for $r\ge 2$.\qed
\endproof

\begstat{(5.24) Remark} \rm If $X$ or $Y$ is compact, the K\"unneth formula holds
with $\otimes$ instead of $\hotimes$, and the assumption that both
cohomology spaces are Hausdorff is unnecessary. The proof is exactly
the same, except that we use (5.22~c) instead of (5.22~b).
\endstat

\titlec{5.C.}{Modules over Nuclear Fr\'echet Algebras}
Throughout this subsection, we work in the category of nuclear
Fr\'echet spaces. Recall that a topological algebra (commutative, with
unit element~$1$) is an algebra $A$ together with a topological vector
space structure such that the multiplication $A\times A\to A$ is
continuous. $A$ is said to be a Fr\'echet (resp. nuclear) algebra if
it is Fr\'echet (resp. nuclear) as a topological vector space.

\begstat{(5.25) Definition} A $($Fr\'echet, resp. nuclear$)$ $A$-module $E$ is a
$($Fr\'echet, resp. nuclear$)$ space $E$ with a $A$-module structure
such that the multiplication $A\times E\to E$ is continuous. The
module $E$ is said to be nuclearly free if $E$ is of the form $A\hotimes V$
where $V$ is a nuclear Fr\'echet space.
\endstat

Assume that $A$ is nuclear and let $E$ be a nuclear $A$-module.
A {\it nuclearly free resolution} $L_\bu$ of $E$ is an exact sequence
of $A$-modules and continuous $A$-linear morphisms
$$\cdots\lra L_q\buildo d_q\over\lra L_{q-1}\lra\cdots\lra L_0\lra E\lra 0
\leqno(5.26)$$
in which each $L_q$ is a nuclearly free $A$-module. Such a resolution is
said to be {\it direct} if each map $d_q$ is direct, i.e.\ if $\Im d_q$
has a topological supplementary space in $L_{q-1}$ (as a vector space 
over $\bbbr$ or $\bbbc$, not necessarily as a $A$-module).

\begstat{(5.27) Proposition} Every nuclear $A$-module $E$ admits a 
direct nuclearly free resolution.
\endstat

\begproof{} We define the ``standard resolution" of $E$ to be
$$L_q=A\hotimes\ldots\hotimes A\hotimes E$$ 
where $A$ is repeated \hbox{$(q+1)$} times; the $A$-module structure of 
$L_q$ is chosen to be the one given by the first factor and we set 
$d_0(a_0\otimes x)=a_0x$,
$$\eqalign{
d_q(a_0\otimes\ldots\otimes a_q\otimes x)=
\sum_{0\le i<q}&(-1)^ia_0\otimes\ldots\otimes a_ia_{i+1}\otimes\ldots
\otimes a_q\otimes x\cr
{}+{}&(-1)^qa_0\otimes\ldots\otimes a_{q-1}\otimes a_qx.\cr}$$
Then there is a homotopy operator $h_q:L_q\to L_{q+1}$ given by 
$h_q(t)=1\otimes t$ for all $q$ ($h_q$, however, is not $A$-linear). 
This implies easily that $L_\bu$ is a direct nuclearly free resolution.\qed
\endproof

If $E$ and $F$ are two nuclear $A$-modules, we define $E\hotimes_A F$
to be
$$\leqalignno{
&E\hotimes_A F=\hbox{\rm coker}\big(E\hotimes A\hotimes F\buildo d\over\lra
E\hotimes F\big)~~~~\hbox{\rm where}&(5.28)\cr
&d(x\otimes a\otimes y)=ax\otimes y-x\otimes ay.\cr}$$
Then $E\hotimes_A F$ is a $A$-module which it is not necessarily
Hausdorff. If \hbox{$E\hotimes_A F$} is Hausdorff, it is in fact
a nuclear $A$-module by Prop.~5.14. If $E$ is nuclearly free, say 
$E=A\hotimes V\simeq V\hotimes A$, we have $E\hotimes_A F=V\hotimes F$ 
(which is thus Hausdorff): indeed, there is an exact sequence
$$\eqalign{
&V\hotimes A\hotimes A\hotimes F\lra V\hotimes A\hotimes F\lra V\hotimes F
\lra 0,\cr
&v\otimes a_0\otimes a_1\otimes x\longmapsto v\otimes a_0a_1\otimes x-
v\otimes a_0\otimes a_1x,~~~~v\otimes a\otimes x\longmapsto v\otimes ax,\cr}$$
obtained by tensoring the standard resolution of $F$ with $V\hotimes\,$;
observe that the tensor product $\hotimes$ with a nuclear space preserves
exact sequences thanks to Prop.~5.17. We further define $\toor_q^A(E,F)$ to be
$$\toor_q^A(E,F)=H_q(E\hotimes_A L_\bu),\leqno(5.29)$$
where $L_\bu$ is the standard resolution of $F$. There is in fact an 
isomorphism
$$\eqalign{
E\hotimes_A L_\bu&\buildo\simeq\over\lra 
E\hotimes A\hotimes\cdots\hotimes A\hotimes F\cr
x\otimes_A(a_0\otimes a_1\otimes\ldots\otimes a_q\otimes y)
&\buildo\over\longmapsto a_0x\otimes a_1\otimes\ldots\otimes a_q\otimes y\cr}$$
where $A$ is repeated $q$ times in the target space. In this
isomorphism, the differential becomes
$$\eqalign{
&d_q(x\otimes a_1\otimes\ldots\otimes a_q\otimes y)=
a_1x\otimes a_2\otimes\ldots\otimes a_q\otimes y\cr
&\qquad\qquad\qquad{}+\sum_{1\le i<q}(-1)^i
x\otimes a_1\otimes\ldots\otimes a_ia_{i+1}\otimes\ldots\otimes a_q\otimes y\cr
&\qquad\qquad\qquad{}+(-1)^qx\otimes a_1\otimes\ldots\otimes a_{q-1}
\otimes a_qy.\cr}$$
In particular, we get $\toor_0^A(E,F)=E\hotimes_A F$. Moreover, if we
exchange the roles of $E$ and $F$, we obtain a complex which is
isomorphic to the above one up to the sign of $d_q$, hence
$\toor_q^A(E,F)\simeq\toor_q^A(F,E)$. If $E=A\hotimes V$ is nuclearly
free, the complex $E\hotimes_A L_\bu=V\hotimes L_\bu$ is exact, thus
$$\hbox{\rm$E$ or $F$ nuclearly free}\Longrightarrow
\toor_q^A(E,F)=0~~\hbox{\rm for $q\ge 1$.}$$

\begstat{(5.30) Proposition} For any exact sequence
$0\to E_1\to E_2\to E_3\to 0$ of nuclear $A$-modules and any nuclear
$A$-module $F$, there is an $($algebraic$)$ exact sequence
$$\cmalign{
\hfill\cdots\toor_q^A(E_1,F)&\lra\toor_q^A(E_2,F)&\lra\toor_q^A(E_3,F)&\lra
\toor_{q-1}^A(E_1,F)\cdots\cr
\lra~~E_1\hotimes_A F~~&\lra~~E_2\hotimes_A F&\lra~~E_3\hotimes_A F
&\lra 0.\cr}$$
\endstat

\begproof{} As the standard resolution $L_\bu\to F$ is nuclearly free,
$L_q=A\hotimes V_q$ say, then $E_j\hotimes_A L_\bu=E_j\hotimes V_\bu$
for $j=1,2,3$, so we have a short exact sequence of complexes
$$0\lra E_1\hotimes_A L_\bu\lra E_2\hotimes_A L_\bu\lra E_3\hotimes_A
L_\bu\lra 0.\eqno{\square}$$
\endproof

\begstat{(5.31) Corollary} For any nuclearly free $($possibly non direct$)$
resolution $L_\bu$ of $F$, there is a canonical isomorphism
$$\toor_q^A(E,F)\simeq H_q(E\hotimes_A L_\bu).$$
\endstat

\begproof{} Set $B_q=\Im(L_{q+1}\to L_q)$ for all $q\ge 0$ and $B_{-1}=F$. 
Then apply (5.30) to the short exact sequences $0\to B_q\to L_q\to
B_{q-1}\to 0$ and the fact that $L_q$ is nuclearly free to get
$$\toor_k^A(E,B_{q-1})\simeq\cases{
\toor_{k-1}^A(E,B_q)&for $k>1$,\cr
\ker(E\hotimes_A B_q\to E\hotimes_A L_q)&for $k=1$.\cr}$$
Hence we obtain inductively
$$\eqalign{
\toor_q^A(E,F)&=\toor_q^A(E,B_{-1})\simeq\ldots\simeq\toor_1^A(E,B_{q-2})\cr
&\simeq\ker(E\hotimes_A B_{q-1}\to E\hotimes_A L_{q-1})\cr}$$
and a commutative diagram
$$\eqalign{
&E\hotimes_A L_{q+1}\lra E\hotimes_A L_q\lra E\hotimes_A B_{q-1}\lra 0\cr
&\qquad\searrow\qquad\qquad\nearrow\cr
&~~~~\qquad E\hotimes_A B_q\cr}$$
in which the horizontal line is exact (thanks to the surjectivity of
the left oblique arrow and the exactness of the sequence with
$E\hotimes_A B_q$ as first term). Therefore
$\ker(E\hotimes_A B_{q-1}\to E\hotimes_A L_{q-1})$
can be interpreted as the kernel of $E\hotimes_A L_q\to E\hotimes_A L_{q-1}$
modulo the image of $E\hotimes_A L_{q+1}\to E\hotimes_A L_q$,\break
and this is is precisely the definition of $H_q(E\hotimes_A L_\bu)$.\qed
\endproof

Now, we are ready to introduce the crucial concept of transversality.

\begstat{(5.32) Definition} We say that two nuclear $A$-modules $E$, $F$ are
transverse if $E\hotimes_A F$ is Hausdorff and if $\toor_q^A(E,F)=0$
for $q\ge 1$.
\endstat

For example, a nuclearly free $A$-module $E=A\hotimes V$ is transverse to
any nuclear $A$-module~$F$. Before proving further general properties, we 
give a fundamental example.

\begstat{(5.33) Proposition} Let $X$, $Y$ be Stein spaces and let
$U'\subset U\compact X$, $V\compact Y$ be Stein open subsets. If $\cF$ is
a coherent sheaf over $X\times Y$, then $\cO(U')$ and $\cF(U\times V)$ are
transverse over $\cO(U)$. Moreover
$$\cO(U')\hotimes_{\cO(U)}\cF(U\times V)=\cF(U'\times V).$$
\endstat

\begproof{} Let $\cL_\bu\to\cF$ be a free resolution of $\cF$ over $U\times V\,$;
such a resolution exists by Cartan's theorem~A. Then $\cL_\bu(U\times V)$
is a resolution of \hbox{$\cF(U\times V)$} which is nuclearly free over
$\cO(U)$, for $\cO(U\times V)=\cO(U)\hotimes\cO(V)\,$; in particular,
we get
$$\eqalign{
\cO(U')\hotimes_{\cO(U)}\cO(U\times V)&=\cO(U')\hotimes\cO(V)=\cO(U'\times V),\cr
\cO(U')\hotimes_{\cO(U)}\cL_\bu(U\times V)&=\cL_\bu(U'\times V).\cr}$$
But $\cL_\bu(U'\times V)$ is a resolution of $\cF(U'\times V)$, so
$$\toor_q^{\cO(U)}\big(\cO(U'),\cF(U\times V)\big)=\cases{
\cF(U'\times V)&for $q=0$,\cr
              0&for $q\ge 1$.\cr}\eqno{\square}$$
\endproof

\begstat{(5.34) Properties} \smallskip
\item{\rm a)} If $0\to E_1\to E_2\to E_3\to 0$ is an exact sequence of
nuclear $A$-modules and if $E_2$, $E_3$ are transverse to $F$, then $E_1$
is transverse to $F$.
\smallskip
\item{\rm b)} Let $A\to A_1\to A_2$ be homomorphisms of nuclear algebras and 
let $E$ be a nuclear $A$-module. if $A_1$ and $A_2$ are transverse
to $E$ over $A$, then $A_2$ is tranverse to $A_1\hotimes_A E$ over $A_1$.
\smallskip
\item{\rm c)} Let $E^\bu$ be a complex of nuclear $A$-modules, bounded on
the right side, and let $M$ be a nuclear $A$-module which is transverse to
all $E^n$. If $E^\bu$ is acyclic in degrees $\ge k$, then $M\hotimes_A E^\bu$
is also acyclic in degrees $\ge k$.
\smallskip
\item{\rm d)} Let $E^\bu$, $F^\bu$ be complexes of nuclear $A$-modules, 
bounded on the right side. Let $f^\bu:E^\bu\to F^\bu$ be a $A$-linear
morphism and let $M$ be a nuclear $A$-module which is transverse to
all $E^q$ and $F^q$. If $f^\bu$ induces an isomorphism
$H^q(f^\bu):H^q(E^\bu)\to H^q(F^\bu)$ in degrees $q\ge k$ and an
epimorphism in degree $q=k-1$, then 
$$\Id_M\hotimes_A f^\bu:M\hotimes_A E^\bu\to M\hotimes_A F^\bu$$
has the same property.\smallskip
\endstat

\begproof{} a) is an immediate consequence of the $\toor$ exact sequence.
\medskip
\noindent{}To prove b), we need only check that if $A_1$ is transverse to
$E$ over $A$, then
$$\toor_q^{A_1}(A_2,A_1\hotimes_A E)=\toor_q^A(A_2,E),~~~~\forall n\ge 0.$$
Indeed, if $L_\bu=A\hotimes V_\bu$ is a nuclearly free resolution of $E$
over $A$, then $A_1\hotimes_A L_\bu=A_1\hotimes V_\bu$ is a nuclearly free
resolution of $A_1\hotimes_A E$ over $A_1$, since $H_q(A_1\hotimes_A L_\bu)=
\toor_q^A(A_1,E)=0$ for $q\ge 1$. Hence
$$\eqalign{
\toor_q^{A_1}(A_2,A_1\hotimes_A{}\!E)&=H_q\big(A_2\hotimes_{A_1}{}\!(A_1
\hotimes_A L_\bu)\big)=H_q\big(A_2\hotimes_{A_1}{}\!(A_1\hotimes V_\bu)\big)\cr
&=H_q(A_2\hotimes V_\bu)=H_q(A_2\hotimes_A L_\bu)=\toor_q^A(A_2,E).\cr}$$
c) The short exact sequences $0\to Z^q(E^\bu)\lhra E^q\buildo d^q\over\lra 
Z^{q+1}(E^\bu)\to 0$ show by backward induction on $q$ that
$M$ is transverse to $Z^q(E^\bu)$ for \hbox{$q\ge k-1$}. Hence for
$q\ge k-1$ we obtain an exact sequence
$$0\lra M\hotimes_A Z^q(E^\bu)\lhra M\hotimes_A E^q\buildo d^q\over\lra 
M\hotimes_A Z^{q+1}(E^\bu)\lra 0,$$
which gives in particular $Z^q(M\hotimes_A E^\bu)=B^q(M\hotimes_A E^\bu)=
M\hotimes_A Z^q(E^\bu)$ for $q\ge k$, as desired.
\medskip
\noindent{d)} is obtained by applying c) to the {\it mapping cylinder}
$C(f^\bu)$, as defined in the following lemma (the proof is 
straightforward and left to the reader).\qed
\endproof

\begstat{(5.35) Lemma} If $f^\bu:E^\bu\to F^\bu$ is a morphism of complexes,
the mapping cylinder $C^\bu=C(f^\bu)$ is the complex defined by
$C^q=E^q\oplus F^{q-1}$ with differential 
$$\pmatrix{d^q_E&0\cr -f^q&d^{q-1}_F\cr}:E^q\oplus F^{q-1}\lra
E^{q+1}\oplus F^q.$$
Then there is a short exact sequence $0\to F^{\bu-1}\to C^\bu\to E^\bu\to 0$
and the associated connecting homomorphism $\partial^q:H^q(E^\bu)\to
H^q(F^\bu)$ is equal to $H^q(f^\bu)\,$; in particular,
$C^\bu$ is acyclic in degree $q$ if and only if $H^q(f^\bu)$ is
injective and $H^{q-1}(f^\bu)$ is surjective.\qed
\endstat

\titlec{5.D.}{$A$-Subnuclear Morphisms and Perturbations}
We now introduce a notion of nuclearity relatively to an algebra $A$. This
notion is needed for example to describe the properties of the $\cO(S)$-linear
restriction map $\cO(S\times U)\to\cO(S\times U')$ when $U'\compact U$.

\begstat{(5.36) Definition} Let $E$ and $F$ be Fr\'echet $A$-modules over a
Fr\'echet algebra $A$ and let $f:E\to F$ be a $A$-linear map. We say that
\smallskip
\item{\rm a)} $f$ is $A$-nuclear if there exist a scalar sequence
$(\lambda_j)$ with $\sum|\lambda_j|<+\infty$, an equicontinuous family
of $A$-linears maps $\xi_j:E\to A$ and a bounded sequence $y_j$ in $F$
such that for all $x\in E$
$$f(x)=\sum\lambda_j\,\xi_j(x)y_j.$$
\item{\rm b)} $f$ is $A$-subnuclear if there exists a Fr\'echet $A$-module
$M$ and an epimorphism $p:M\to E$ such that $f\circ p$ is $A$-nuclear; if
$E$ is nuclear, we also require $M$ to be nuclear.
\smallskip
\endstat

If $f:E\to F$ is $A$-nuclear and if $u:S\to E$ and $v:F\to T$ are continuous
$A$-linear maps then $v\circ f\circ u$ is $A$-nuclear; the same is true for
$A$-subnuclear maps. If $V$ and $W$ are nuclear spaces and if 
$u:V\to W$ is $\bbbc$-nuclear, then $\Id_A\hotimes u:A\hotimes V\to 
A\hotimes W$ is $A$-nuclear. From this we infer:

\begstat{(5.37) Proposition} Let $S$, $Z$ be Stein spaces and let
$U'\compact U\compact Z$ be Stein open subsets. Then the
restriction $\rho:\cO(S\times U)\to\cO(S\times U')$ is
\hbox{$\cO(S)$-nuclear}. If $\cF$ is a coherent sheaf over $Y\times Z$ with
$Y$ Stein and $S\compact Y$, then the restriction map
$\rho:\cF(S\times U)\to\cF(S\times U')$ is $\cO(S)$-subnuclear.
\endstat

\begproof{} As $\cO(S\times U)=\cO(S)\hotimes\cO(U)$ and $\cO(U)\to\cO(U')$
is $\bbbc$-nuclear, only the second statement needs a proof. By Cartan's
theorem A, there exists a free resolution $\cL_\bu\to \cF$ over $S\times U$.
Then there is a commutative diagram
$$\cmalign{
\hfill\cL_0(S&\times U)&\lra\cF(S&\times U)\cr
\hfill\rho&\big\downarrow&&\big\downarrow\rho\cr
\hfill\cL_0(S&\times U')&\lra\cF(S&\times U')\cr}$$
in which the top horizontal arrow is an $\cO(S)$-epimorphism and the left
vertical arrow is an $\cO(S)$-nuclear map; its composition with the
bottom horizontal arrow is thus also $\cO(S)$-nuclear.\qed
\endproof

Let $f:E\to F$ be a $A$-linear morphism of Fr\'echet $A$-modules. Suppose
that $f(E)\subset F_1$ where $F_1$ is a {\it closed} $A$-submodule of $F$
and let $f_1:E\to F_1$ be the map induced by $f$. If $f$ is $A$-nuclear,
it is not true in general that $f_1$ is $A$-nuclear or $A$-subnuclear,
even if $A$, $E$, $F$ are nuclear. However:

\begstat{(5.38) Proposition} With the above notations, suppose
$A$, $E$, $F$ nuclear. Let $B$ be a nuclear Fr\'echet algebra and let
$\rho:A\to B$ be a $\bbbc$-nuclear \hbox{homomorphism}. Suppose that $B$ is
transverse to $E$, $F$ and $F/F_1$ over $A$. If~$f:E\to F$ is
$A$-subnuclear, then $\Id_B\hotimes_A f_1:B\hotimes_A E\to B
\hotimes_A F_1$ is $B$-subnuclear.
\endstat

\begproof{} We first show that $\rho\hotimes_A f_1:E=A\hotimes_A E\to
B\hotimes_A F_1$ is $\bbbc$-nuclear. Since a quotient of a $\bbbc$-nuclear
map is $\bbbc$-nuclear by Cor.~5.16~b), we may suppose for this that
$f$ is $A$-nuclear. Write
$$\eqalign{
f(x)&=\sum\lambda_j\,\xi_j(x)y_j,~~~~\xi_j:E\to A,~~\sum|\lambda_j|<+\infty,
~~y_j\in F,\cr
\rho(t)&=\sum\mu_k\,\eta_k(t)b_k,~~~~\eta_k:A\to\bbbc,~~\sum|\mu_k|<+\infty,
~~b_k\in B\cr}$$
as in the definition of ($A$-)nuclearity. Then $\rho\hotimes_A f:E\lra
B\hotimes_A F$ is \hbox{$\bbbc$-nuclear}: for any $x\in E$, we have
$\rho(\xi_j(x))=\xi_j(x)\rho(1)$ in the $A$-module structure of $B$,
hence
$$\eqalign{
\rho\hotimes_A f(x)=\rho\otimes f(1\otimes x)
&=\sum\lambda_j\,\rho(\xi_j(x))\hotimes_A y_j\cr
&=\sum\lambda_j\mu_k\,(\eta_k\circ\xi_j)(x)\,b_k\hotimes_A y_j.\cr}$$
By our transversality assumptions, $B\hotimes_A F_1$ is a closed subspace
of $B\hotimes_A F$. As $\Im(\rho\hotimes_A f)\subset B\hotimes_A F_1$,
the induced map $\rho\hotimes_A f_1:E\to B\hotimes_A F_1$ is $\bbbc$-nuclear
by Cor.~5.16~a). Finally, there is a commutative diagram
$$\cmalign{
B&\hotimes E&\buildo\Id_B\hotimes(\rho\hotimes_A f_1)\over{\larex 60 }
B\hotimes(&B\hotimes_A F_1)\cr
&~\big\downarrow&&~\big\downarrow\cr
B&\hotimes_A E&\buildo\Id_B\hotimes_A f_1\over{\larex 60 }
\hfill B&\hotimes_A F_1\cr}$$
in which the vertical arrows are $B$-linear epimorphisms. The top horizontal
arrow is $B$-nuclear by the $\bbbc$-nuclearity of $\rho\hotimes_A f_1$,
hence $\Id_B\hotimes_A f_1$ is \hbox{$B$-subnuclear}.\qed
\endproof

Example~5.12  suggests the following definition (which is somewhat less
general than some other in current use, but sufficient for our purposes).

\begstat{(5.39) Definition} We say that a Fr\'echet algebra $A$ is fully nuclear
if the topology of $A$ is defined by an increasing family
$(p_t)_{t\in{}]0,1[}$ of multiplicative semi-norms $\big($that is,
$p_t(xy)\le p_t(x)\,p_t(y)\,\big)$, such that the Banach algebra homomorphism 
$\wh A_{p_{t'}}\to\wh A_{p_t}$ is nuclear for all $t<t'<1$.
\endstat

If $A$ is fully nuclear and $t\in{}]0,1]$, we define $A_t$ to be the
completion of $A$ equipped with the family of semi-norms $p_{\lambda t}$,
$\lambda\in{}]0,1[$. Then $A_t$ is again a fully nuclear algebra, and for
all $t<t'<1$ the canonical map $A_{t'}\to A_t$ is nuclear: indeed, for
$t\le u<u'<t'$, there is a factorization
$$A_{t'}\lra\wh A_{p_{u'}}\lra\wh A_{p_u}\lra A_t.$$
If $E$ is a nuclear $A$-module, we say that $E$ is fully $A$-transverse
if $E$ is transverse to all $A_t$ over $A$. Then by 5.34~b), each
nuclear space
$$E_t=A_t\hotimes_A E\leqno(5.40)$$
is a fully $A_t$-transverse $A_t$-module. If $f:E\to F$ is a morphism
of fully $A$-transverse nuclear modules, there is an induced map
$$f_t=\Id_{A_t}\hotimes_A f:E_t\lra F_t,~~~~\forall t\in{}]0,1].
\leqno(5.40')$$

\begstat{(5.41) Example} \rm Let $X$ be a closed analytic subscheme of an open set
$\Omega\subset\bbbc^N$, $D=D(a,R)\compact\Omega$ a polydisk and
$U=D\cap X$. We have an epimorphism $\cO(D)\to\cO(U)$. Denote by
$\wt p_t$ the quotient semi-norm of $p_t(f)=\sup_{D(a,tR)}|f|$ on $\cO(U)$.
Then $\cO(U)$ equipped with $(\wt p_t)_{t\in{}]0,1[}$ is a fully nuclear
algebra, and $\cO(U)_t=\cO\big(D(a,tR)\cap X\big)$.

Now, let $Y$ be a Stein space, $V\compact Y$ a Stein open subset and $\cF$ a
coherent sheaf over $X\times Y$. Then Prop.~5.33 shows that
$\cF(U\times V)$ is a fully transverse nuclear $\cO(U)$-module.
\endstat

\begstat{(5.42) Subnuclear perturbation theorem} Let $A$ be a fully
nuclear algebra,
let $E$ and $F$ be two fully $A$-transverse nuclear $A$-modules and let
$f,u:E\to F$ be $A$-linear maps. Suppose that $u$ is $A$-subnuclear and
that $f$ is an epimorphism. Then for every $t<1$, the cokernel of
$$f_t-u_t:E_t\lra F_t$$
is a finitely generated $A_t$-module $($as an algebraic module; we do not
assert that the cokernel is Hausdorff$)$.
\endstat

\begproof{} We argue in several steps. The first step is the following special
case.
\endproof

\begstat{(5.43) Lemma} Let $B$ be a Banach algebra, $S$ a Fr\'echet $B$-module and
$v:S\to S$ a $B$-nuclear morphism. Then $\Coker(\Id_S-v)$ is a finitely
generated $B$-module.
\endstat

\begproof{} Let $v(x)=\sum\lambda_j\,\xi_j(x)y_j$ be a $B$-nuclear
decomposition of~$v$. We have a factorization
$$v=\beta\circ\alpha:S\buildo\alpha\over\lra\ell^1(B)\buildo\beta\over\lra S$$
where $\alpha(x)=\big(\lambda_j\xi_j(x)\big)$ and $\beta(t_j)=\sum t_jy_j$.
Set $w=\alpha\circ\beta:\ell^1(B)\to\ell^1(B)$. As $\alpha$ is $B$-nuclear,
so is $w$, and $\alpha$, $\beta$ induce isomorphisms
$$\Coker(\Id_S-v)~{\raise-4pt\hbox{
$\scriptstyle\wt\alpha\atop\displaystyle\relbar\joinrel\lra$}
\atop\raise4pt\hbox{
$\displaystyle\longleftarrow\joinrel\relbar\atop\scriptstyle\wt\beta$}}
~\Coker\big(\Id_{\ell^1(B)}-w\big).$$
We are thus reduced to the case when $S$ is a Banach module. Then we write
$v=v'+v''$ with
$$v'(x)=\sum_{1\le j\le N}\lambda_j\,\xi_j(x)y_j,~~~~
v''(x)=\sum_{j>N}\lambda_j\,\xi_j(x)y_j.$$
For $N$ large enough, we have $||v''||<1$, hence $\Id_S-v''$ is an
automorphism and $\Coker(\Id_S-v'-v'')$ is generated by the classes
of $y_1\ld y_N$.\qed
\endproof

\begproof{of Theorem 5.42.} a) We may suppose that $E$ is
nuclearly free and that $u$ is $A$-nuclear, otherwise we replace
$f$, $u$ by their composition with
\hbox{$A\hotimes M\lra M\buildo p\over\lra E$,}
where $M$ is nuclear and $p:M\to E$ is an epimorphism such that
$u\circ p$ is $A$-nuclear.
\medskip
\noindent{b)} As in (5.9), there is a $A$-nuclear decomposition
$u(x)=\sum\lambda_j\,\xi_j(x)y_j$ where $(y_j)$ converges to $0$ in~$F$.
Since $f$ is an epimorphism, we can find a sequence $(x_j)$ converging to
$0$ in $E$ such that $f(x_j)=y_j$. Hence we have $u=f\circ v$ where
$v(x)=\sum\lambda_j\,\xi_j(x)x_j$ is a $A$-nuclear endomorphism of $E$,
and the cokernel of $f-u$ is the image by $f$ of the cokernel of
$\Id_E-v$.
\medskip
\noindent{c)} By a), b) we may suppose that $F=E=A\hotimes M$, $f=\Id_E$
and that $u$ is $A$-nuclear. Let $B$ be the Banach algebra
$B=\wh A_{p_t}$. Then \hbox{$B\hotimes_A E=B\hotimes M$} is a Fr\'echet
$B$-module and $\Id_B\hotimes_A u$ is $B$-nuclear. By Lemma~5.42,
\hbox{$\Id_B\hotimes_A\Id_E-\Id_B\hotimes_A u$} has a finitely generated
cokernel over $B$. Now, there is an obvious morphism $B\to A_t$,
hence by taking the tensor product with $A_t\hotimes_B\bu$
we get
$$A_t\hotimes_B(B\hotimes_A E)=A_t\hotimes_B(B\hotimes M)=
A_t\hotimes M=A_t\hotimes_A E=E_t$$
and we see that
$$\Id_{E_t}-u_t=\Id_{A_t}\hotimes_A\Id_E-\Id_{A_t}\hotimes_A u$$
has a finitely generated cokernel over $A_t$.\qed
\endproof

\titlec{5.E.}{Proof of the Direct Image Theorem}
We first prove a functional analytic version of the result, which appears
as a relative version of Schwartz' theorem~1.9.

\begstat{(5.44) Theorem} Let $A$ be a fully nuclear algebra, $E^\bu$ and $F^\bu$
complexes of fully $A$-transverse nuclear $A$-modules. Let
$f^\bu:E^\bu\to F^\bu$ be a morphism of complexes such that each $f^q$
is $A$-subnuclear. Suppose that $E^\bu$ and $F^\bu$ are bounded on the
right and that $H^q(f^\bu)$ is an isomorphism for each $q$. Then for
every $t<1$, there is a complex $L^\bu$ of finitely generated free
$A_t$-modules and a complex morphism $h^\bu:L^\bu\to E_t^\bu$ which induces
an isomorphism on cohomology.
\endstat

\begproof{} a) We first show the following statement:
\smallskip
\item{}{\it Suppose that $E^\bu_t$ and $F^\bu_t$ are acyclic in degrees $>q$.
Then for every $t'<t$, the cohomology space $H^q(E^\bu_{t'})\simeq
H^q(F^\bu_{t'})$ is a finitely generated \hbox{$A_{t'}$-module}.}
\smallskip
Indeed, the exact sequences $0\to Z^k(E^\bu_t)\to E^k_t\to Z^{k+1}(E^\bu_t)
\to 0$ show by backward induction on $k$ that $Z^k(E^\bu_t)$ is fully
$A_t$-transverse for $k\ge q$. The same is true for $Z^k(F^\bu_t)$.
Then $f^q_t$ is a $A_t$-subnuclear map from $Z^q(E^\bu_t)$ into
$F^q_t$, and its image is contained in the closed subspace $Z^q(F^\bu_t)$.
By Prop.~5.38, for all $t''<t$, the map $f^q_{t''}=\Id_{A_{t''}}
\hotimes_{A_t}f^q_t$ is a $A_{t''}$-subnuclear map $Z^q(E^\bu_{t''})\to
Z^q(F^\bu_{t''})$. By Prop.~5.34~d), $H^\bu(f^\bu_{t''})$ is an
isomorphism in all degrees, hence
$$d^q_{t''}\oplus f^q_{t''}:F^{q-1}_{t''}\oplus Z^q(E^\bu_{t''})\lra
Z^q(F^\bu_{t''})$$
is surjective. By the subnuclear perturbation theorem, the map
$$d^q_{t'}\oplus 0=\Id_{A_{t'}}\hotimes_{A_{t''}}\big(
(d^q_{t''}\oplus f^q_{t''})-(0\oplus f^q_{t''})\big)$$
has a finitely generated $A_{t'}$-cokernel for $t'<t''<t$, as desired.
\medskip
\noindent{b)} Let $N$ be an index such that $E^k=F^k=0$ for $k>N$. Fix a
sequence $t<\ldots<t_q<t_{q+1}<\ldots<t_N<1$. To prove the theorem, we
construct by backward induction on $q$ a finitely generated free module
$L^q$ over $A_{t_q}$ and morphisms $d^q:L^q\to L^{q+1}_{t_q}$,
$h^q:L^q\to E^q_{t_q}$ such that
\smallskip
\itemitem{\llap{\hbox{i)}}}{\it $L_{\gge q,\,t_q}^\bu:0\to L^q\to
L^{q+1}_{t_q}\to\cdots\to L^N_{t_q}\to 0$ is a complex and\hfill\break
$h^\bu_{\gge q,\,t_q}:L_{\gge q,\,t_q}^\bu\to E^\bu_{t_q}$
is a complex morphism.}
\smallskip
\itemitem{\llap{\hbox{ii)}}}{\it The mapping cylinder $M^\bu_q=
C(h^\bu_{\gge q,\,t_q})$ defined by\hfill\break
$M^k_q=\bigoplus_{k\in\bbbz}\big(L^k_{\gge q,\,t_q}\oplus E^{k-1}_{t_q}\big)$
is acyclic in degrees $k>q$.}
\smallskip
Suppose that $L^k$ has been constructed for $k\ge q$. Consider the
mapping cylinder $N^\bu_q=C(f^\bu_{t_q}\circ h^\bu_{\gge q,\,t_q})$
and the complex morphism
$$M^\bu_q\lra N^\bu_q,~~~~
L^k_{\gge q,\,t_q}\oplus E^{k-1}_{t_q}\lra
L^k_{\gge q,\,t_q}\oplus F^{k-1}_{t_q}$$
given by $\Id\oplus f^{k-1}_{t_q}$. This morphism is
$A_{t_q}$-subnuclear in each degree and induces an isomorphism
in cohomology (compare the cohomology of the short exact sequences
associated to each mapping cylinder, with the obvious morphism
between them). Moreover, $M^\bu_q$ and $N^\bu_q$ are acyclic in degrees
\hbox{$k>q$}. By step a), the cohomology space $H^q(M^\bu_{q,\,t_{q-1}})$
is a finitely generated \hbox{$A_{t_{q-1}}$-module}.
Therefore, we can find a finitely generated free $A_{t_{q-1}}$-module
$L^{q-1}$ and a morphism
$$d^{q-1}\oplus h^{q-1}:L^{q-1}\to M^q_{q,\,t_{q-1}}=L^q_{t_{q-1}}\oplus
E^{q-1}_{t_{q-1}}$$
such that the image is contained in $Z^q(M^\bu_{q,\,t_{q-1}})$ and
generates the cohomology space $H^q(M^\bu_{q,\,t_{q-1}})$. As
$M^{q-1}_{q,\,t_{q-1}}=E^{q-2}_{t_{q-1}}$, this means that
$M^\bu_{q-1}$ is also acyclic in degree $q$. Thus $L^{q-1}$, together
with the maps $(d^{q-1},h^{q-1})$ satisfies the induction hypotheses
for $q-1$, and $L^\bu_t$ together with the induced map
$h^\bu_t:L^\bu_t\to E^\bu_t$ is the required morphism of complexes.\qed
\endproof

\begproof{of theorem 5.1.} Let $X$, $Y$ be complex analytic
schemes, let \hbox{$F:X\to Y$} be a proper analytic morphism and let
$\cS$ be a coherent sheaf over $X$. Fix a point $y_0\in Y$, a neighborhood
of $y_0$ which is isomorphic to a closed analytic subscheme of a Stein
open set $W\subset\bbbc^n$ and a polydisk $D^0=D(y_0,R_0)\compact W$.
The compact set $K=F^{-1}(\ovl D^0\cap Y)$ can be covered
by finitely many open subsets $U_\alpha^0\compact X$ which possess
embeddings as closed analytic subschemes of Stein open sets
$\Omega_\alpha^0\subset\bbbc^{N_\alpha}$. Let
$\Omega'_\alpha\compact\Omega_\alpha\compact\Omega_\alpha^0$ be
Stein open subsets such that $U_\alpha=U^0_\alpha\cap\Omega_\alpha$
and $U'_\alpha=U^0_\alpha\cap\Omega'_\alpha$ still cover $K$.
Let $i_\alpha:U^0_\alpha\to\Omega^0_\alpha$ and $j:Y\cap D^0\to D^0$ be the
embeddings and $\cS_\alpha=\big(i_\alpha\times(j\circ F)\big)_\star\cS$ the
image sheaf of $\cS$ on $\Omega^0_\alpha\times D^0$. Let $D\compact D^0$
be a concentric polydisk. Then $\cS\big(U_\alpha\cap F^{-1}(D)\big)=
\cS_\alpha(\Omega_\alpha\times D)$ is a fully transverse $\cO(D)$-module
by Ex.~5.41, and so is $\cS\big(U'_\alpha\cap F^{-1}(D)\big)=
\cS_\alpha(\Omega'_\alpha\times D)$. Moreover, the restriction map
$$\cS\big(U_\alpha\cap F^{-1}(D)\big)\lra\cS\big(U'_\alpha\cap F^{-1}(D)\big)$$
is $\cO(D)$-subnuclear by Prop.~5.37 applied to $\cF=\cS_\alpha$. For
every Stein open set $V\subset D$, Prop.~5.33  shows that
$$\cO(V)\hotimes_{\cO(D)}\cS\big(U_\alpha\cap F^{-1}(D)\big)
=\cS\big(U_\alpha\cap F^{-1}(V)\big).$$

Denote by $\cU\cap F^{-1}(D)$ the collection $\big(U_\alpha\cap F^{-1}(D)
\big)$ and use a similar notation with $\cU'=(U'_\alpha)$.
As $\cU\cap F^{-1}(D)$, $\cU'\cap F^{-1}(D)$ are Stein coverings of
$F^{-1}(D)$, the Leray theorem applied to the alternate \v Cech complex of
$\cS$ over $\cU\cap F^{-1}(D)$ and $\cU'\cap F^{-1}(D)$ gives an isomorphism
$$H^\bu\big(AC^\bu(\cU\cap F^{-1}(D),\cS)\big)=
H^\bu\big(AC^\bu(\cU'\cap F^{-1}(D),\cS)\big)=H^\bu\big(F^{-1}(D),\cS\big).$$
By the above discussion, $AC^\bu(\cU\cap F^{-1}(D),\cS)$ and
$AC^\bu(\cU'\cap F^{-1}(D),\cS)$ are finite complexes of fully
transverse nuclear $\cO(D)$-modules, the restriction map
$$AC^\bu(\cU\cap F^{-1}(D),\cS)\lra AC^\bu(\cU'\cap F^{-1}(D),\cS)$$
is $\cO(D)$-subnuclear and induces an isomorphism on cohomology groups.
Set $D=D(y_0,R)$ and $D_t=D(y_0,tR)$. Theorem~5.44 shows that for every $t<1$
there is a complex of finitely generated free $\cO$-modules $\cL^\bu$ and a
\hbox{$\cO(D_t)$-linear} morphism of complexes
$$\cL^\bu(D_t)\to AC^\bu(\cU\cap F^{-1}(D_t),\cS)$$ which
induces an isomorphism on cohomology. Let $V\subset D_t$ be an arbitrary
Stein open set. By Prop.~5.34~d) applied with $M=\cO(V)$, we conclude
that $\cL^\bu(V)\to AC^\bu(\cU\cap F^{-1}(V),\cS)$ induces an
isomorphism on cohomology. If we take the direct limit as $V$ runs
over all Stein neighborhoods of a point $y\in Y\cap D_t$, we see that
$\cH^q(\cL^\bu)\simeq R^qF_\star\cS$ over $Y\cap D_t$, hence
$R^qF_\star\cS$ is $\cO_Y$-coherent near $y_0$.\qed
\endproof

\end