% Complex Analytic and Differential Geometry, Chapter II % J.-P. Demailly, Universit\'e de Grenoble I, Saint Martin d'H\`eres, France \input analgeom.mac \def\ga{{\frak a}} \def\gm{{\frak m}} \def\Spm{\mathop{\rm Spm}\nolimits} \def\Proj{\mathop{\rm Proj}\nolimits} \def\Cycl{\mathop{\rm Cycl}\nolimits} \def\nn{{\rm n\hbox{\sevenrm-}n}} \def\norm{{\rm norm}} \def\alg{{\rm alg}} \def\red{{\rm red}} \def\stred{{\rm st\hbox{\sevenrm-}red}} \def\cHom{{\cal H}{\it om}} \def\GL{{\rm GL}} \def\Coker{\mathop{\rm Coker}\nolimits} \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} \titlea{Chapter II.}{\newline Coherent Sheaves and Analytic Spaces} \begpet The chapter starts with rather general and abstract concepts concerning sheaves and ringed spaces. Introduced in the decade 1950-1960 by Leray, Cartan, Serre and Grothendieck, sheaves and ringed spaces have since been recognized as the adequate tools to handle algebraic varieties and analytic spaces in a unified framework. We then concentrate ourselves on the theory of complex analytic functions. The second section is devoted to a proof of the Weierstrass preparation theorem, which is nothing but a division algorithm for holomorphic functions. It is used to derive algebraic properties of the ring $\cO_n$ of germs of holomorphic functions in~$\bbbc^n$. Coherent analytic sheaves are then introduced and the fundamental coherence theorem of Oka is proved. Basic properties of analytic sets are investigated in detail: local parametrization theorem, Hilbert's Nullstellensatz, coherence of the ideal sheaf of an analytic set, analyticity of the singular set. The formalism of complex spaces is then developed and gives a natural setting for the proof of more global properties (decomposition into global irreducible components, maximum principle). After a few definitions concerning cycles, divisors and meromorphic functions, we investigate the important notion of normal space and establish the Oka normalization theorem. Next, the Remmert-Stein extension theorem and the Remmert proper mapping theorem on images of analytic sets are proved by means of semi-continuity results on the rank of morphisms. As an application, we give a proof of Chow's theorem asserting that every analytic subset of $\bbbp^n$ is algebraic. Finally, the concept of analytic scheme with nilpotent elements is introduced as a generalization of complex spaces, and we discuss the concepts of bimeromorphic maps, modifications and blowing-up. \endpet \titleb{\S 1.}{Presheaves and Sheaves} \titlec{\S 1.A.}{Main Definitions} Sheaves have become a very important tool in analytic or algebraic geometry as well as in algebraic topology. They are especially useful when one wants to relate global properties of an object to its local properties (the latter being usually easier to establish). We first introduce the axioms of presheaves and sheaves in full generality and give some basic examples. \begstat{(1.1) Definition} Let $X$ be a topological space. A presheaf $\cA$ on $X$ consists of the following data: \medskip \item{\rm a)} a collection of non empty sets $\cA(U)$ associated with every open set $U\subset X$, \smallskip \item{\rm b)} a collection of maps $\rho_{U,V}:\cA(V)\lra\cA(U)$ defined whenever $U\subset V$ and satisfying the transitivity property \smallskip \item{\rm c)} $\rho_{U,V}\circ\rho_{V,W}=\rho_{U,W}$~~~for~~$U\subset V\subset W,\qquad\rho_{U,U}=\Id_U$~~~for every~$U$. \medskip\noindent The set $\cA(U)$ is called the set of sections of the presheaf $\cA$ over~$U$. \endstat Most often, the presheaf $\cA$ is supposed to carry an additional algebraic structure. For instance: \begstat{(1.2) Definition} A presheaf $\cA$ is said to be a presheaf of abelian groups $($resp.\ rings, $R$-modules, algebras$)$ if all sets $\cA(U)$ are abelian groups $($resp.\ rings, $R$-modules, algebras$)$ and if the maps $\rho_{U,V}$ are morphisms of these algebraic structures. In this case, we always assume that $\cA(\emptyset)=\{0\}$. \endstat \begstat{(1.3) Example} \rm If we assign to each open set $U\subset X$ the set $\cC(U)$ of all real valued continuous functions on~$U$ and let $\rho_{U,V}$ be the obvious restriction morphism $\cC(V)\to\cC(U)$, then $\cC$ is a presheaf of rings on~$X$. Similarly if $X$ is a differentiable (resp.\ complex analytic) manifold, there are well defined presheaves of rings $\cC^k$ of functions of class $C^k$ (resp.\ $\cO$) of holomorphic functions) on~$X$. Because of these examples, the maps $\rho_{U,V}$ in Def.~1.1 are often viewed intuitively as ``restriction homomorphisms'', although the sets $\cA(U)$ are not necessarily sets of functions defined over~$U$. For the simplicity of notation we often just write $\rho_{U,V}(f)= f_{\restriction U}$ whenever $f\in\cA(V)$, $V\supset U$.\qed \endstat For the above presheaves $\cC$, $\cC^k$, $\cO$, the properties of functions under consideration are purely local. As a consequence, these presheaves satisfy the following additional {\it gluing axioms}, where $(U_\alpha)$ and $U=\bigcup U_\alpha$ are arbitrary open subsets of $X\,$: $$\leqalignno{ &\hbox{\rm If $F_\alpha\in\cA(U_\alpha)$ are such that $\rho_{U_\alpha\cap U_\beta,U_\alpha}(F_\alpha)= \rho_{U_\alpha\cap U_\beta,U_\beta}(F_\beta)$}&(1.4')\cr &\hbox{\rm for all $\alpha,\beta$, there exists $F\in\cA(U)$ such that $\rho_{U_\alpha,U}(F)=F_\alpha\,$;}\cr}$$ $$\hbox{\rm If $F,G\in\cA(U)$ and $\rho_{U_\alpha,U}(F)= \rho_{U_\alpha,U}(G)$ for all $\alpha$, then $F=G\,$;}\leqno(1.4'')$$ in other words, local sections over the sets $U_\alpha$ can be glued together if they coincide in the intersections and the resulting section on $U$ is uniquely defined. Not all presheaves satisfy $(1.4')$ and $(1.4'')$: \begstat{(1.5) Example} \rm Let $E$ be an arbitrary set with a distinguished element~$0$ (e.g.\ an abelian group, a $R$-module,~$\ldots$). The {\it constant presheaf} $E_X$ on $X$ is defined to be $E_X(U)=E$ for all~$\emptyset\ne U\subset X$ and $E_X(\emptyset)=\{0\}$, with restriction maps $\rho_{U,V}=\Id_E$ if $\emptyset\ne U\subset V$ and $\rho_{U,V}=0$ if~$U=\emptyset$. Then axiom $(1.4')$ is not satisfied if $U$ is the union of two disjoint open sets $U_1$, $U_2$ and $E$ contains a non zero element. \endstat \begstat{(1.6) Definition} A presheaf $\cA$ is said to be a sheaf if it satisfies the gluing axioms $(1.4')$ and $(1.4'')$. \endstat If $\cA$, $\cB$ are presheaves of abelian groups (or of some other algebraic structure) on the same space~$X$, a presheaf morphism $\varphi:\cA\to\cB$ is a collection of morphisms $\varphi_U:\cA(U)\to\cB(U)$ commuting with the restriction morphisms, i.e.\ such that for each pair $U\subset V$ there is a commutative diagram $$\eqalign{ &~~\cA(V)\buildo{\displaystyle\varphi_V}\over\lra\cB(V)\cr &\rho^\cB_{U,V}\big\downarrow~~\phantom{\lra}~~\big\downarrow\rho^\cA_{U,V}\cr &~~\cA(U)\buildo{\displaystyle\varphi_U}\over\lra\cB(U).\cr}$$ We say that $\cA$ is a subpresheaf of $\cB$ in the case where $\varphi_U:\cA(U)\subset\cB(U)$ is the inclusion morphism; the commutation property then means that $\rho^\cB_{U,V}(\cA(V))\subset\cA(U)$ for all $U$, $V$, and that $\rho^\cA_{U,V}$ coincides with $\rho^\cB_{U,V}$ on~$\cA(V)$. If $\cA$ is a subpresheaf of a presheaf $\cB$ of abelian groups, there is a presheaf quotient $\cC=\cB/\cA$ defined by $\cC(U)=\cB(U)/\cA(U)$. In a similar way, one defines the presheaf kernel (resp.\ presheaf image, presheaf cokernel) of a presheaf morphism $\varphi:\cA\to\cB$ to be the presheaves $$U\mapsto\Ker\varphi_U,\qquad U\mapsto\Im\varphi_U,\qquad U\mapsto\Coker\varphi_U.$$ The direct sum $\cA\oplus\cB$ of presheaves of abelian groups $\cA$, $\cB$ is the presheaf $U\mapsto\cA(U)\oplus\cB(U)$, the tensor product $\cA\otimes\cB$ of presheaves of $R$-modules is $U\mapsto\cA(U)\otimes_R\cB(U)$, etc $\ldots$ \begstat{(1.7) Remark} \rm The reader should take care of the fact that the presheaf quotient of a sheaf by a subsheaf is not necessarily a sheaf. To give a specific example, let $X=S^1$ be the unit circle in~$\bbbr^2$, let $\cC$ be the sheaf of continuous complex valued functions and $\cZ$ the subsheaf of integral valued continuous functions (i.e.\ locally constant functions to $\bbbz$). The exponential map $$\varphi=\exp(2\pi\ii\bu):\cC\lra\cC^\star$$ is a morphism from $\cC$ to the sheaf $\cC^\star$ of invertible continuous functions, and the kernel of $\varphi$ is precisely~$\cZ$. However $\varphi_U$ is surjective for all $U\ne X$ but maps $\cC(X)$ onto the multiplicative subgroup of continuous functions of $\cC^\star(X)$ of degree~$0$. Therefore the quotient presheaf $\cC/\cZ$ is not isomorphic with $\cC^\star$, although their groups of sections are the same for all $U\ne X$. Since $\cC^\star$ is a sheaf, we see that $\cC/\cZ$ does not satisfy property~$(1.4')$.\qed \endstat In order to overcome the difficulty appearing in Example~1.7, it is necessary to introduce a suitable process by which we can produce a sheaf from a presheaf. For this, it is convenient to introduce a slightly modified viewpoint for sheaves. \begstat{(1.8) Definition} If $\cA$ is a presheaf, we define the set $\wt\cA_x$ of germs of $\cA$ at a point $x\in X$ to be the abstract inductive limit $$\wt\cA_x=\lim_{\displaystyle\lra\atop\scriptstyle U\ni x} \big(\cA(U),\rho_{U,V}\big).$$ More explicitely, $\wt\cA_x$ is the set of equivalence classes of elements in the disjoint union $\coprod_{U\ni x}\cA(U)$ taken over all open neighborhoods $U$ of~$x$, with two elements $F_1\in\cA(U_1)$, $F_2\in\cA(U_2)$ being equivalent, $F_1\sim F_2$, if and only if there is a neighborhood $V\subset U_1,\,U_2$ such that $F_{1\restriction V}= F_{2\restriction V}$, i.e., $\rho_{VU_1}(F_1)=\rho_{VU_2}(F_2)$. The germ of an element $F\in\cA(U)$ at a point $x\in U$ will be denoted by~$F_x$. \endstat Let $\cA$ be an arbitrary presheaf. The disjoint union $\wt\cA=\coprod_{x\in X}\wt\cA_x$ can be equipped with a natural topology as follows: for every $F\in\cA(U)$, we set $$\Omega_{F,U}=\big\{F_x~;~x\in U\big\}$$ and choose the $\Omega_{F,U}$ to be a basis of the topology of~$\wt\cA\,$; note that this family is stable by intersection: $\Omega_{F,U}\cap\Omega_{G,V}=\Omega_{H,W}$ where $W$ is the (open) set of points $x\in U\cap V$ at which $F_x=G_x$ and $H=\rho_{W,U}(F)$. The obvious projection map $\pi:\wt\cA\to X$ which sends $\wt\cA_x$ to $\{x\}$ is then a local homeomorphism (it is actually a homeomorphism from $\Omega_{F,U}$ onto~$U$). This leads in a natural way to the following definition: \begstat{(1.9) Definition} Let $X$ and $\cS$ be topological spaces (not necessarily Hausdorff), and let $\pi:\cS\lra X$ be a mapping such that \smallskip \item{\rm a)} $\pi$ maps $\cS$ onto $X~;$ \smallskip \item{\rm b)} $\pi$ is a local homeomorphism, that is, every point in $\cS$ has an open neigh\-borhood which is mapped homeomorphically by $\pi$ onto an open subset of~$X$. \smallskip \noindent Then $\cS$ is called a sheaf-space on $X$ and $\pi$ is called the projection of $\cS$ on~$X$. If $x\in X$, then $\cS_x=\pi^{-1}(x)$ is called the stalk of $\cS$ at $x$. \endstat If $Y$ is a subset of $X$, we denote by $\Gamma(Y,\cS)$ the set of sections of $\cS$ on $Y$, i.e.\ the set of continuous functions $F:Y\to\cS$ such that $\pi\circ F=\Id_Y$. It is clear that the presheaf defined by the collection of sets $\cS'(U):=\Gamma(U,\cS)$ for all open sets $U\subset X$ together with the restriction maps $\rho_{U,V}$ satisfies axioms $(1.4')$ and $(1.4'')$, hence $\cS'$ is a sheaf. The set of germs of $\cS'$ at $x$ is in one-to-one correspondence with the stalk $\cS_x=\pi^{-1}(x)$, thanks to the local homeomorphism assumption 1.9~b). This shows that one can associate in a natural way a sheaf $\cS'$ to every sheaf-space~$\cS$, and that the sheaf-space $(\cS')^\sim$ can be considered to be identical to the original sheaf-space $\cS$. Since the assignment $\cS\mapsto\cS'$ from sheaf-spaces to sheaves is an equivalence of categories, we will usually omit the prime sign in the notation of $\cS'$ and thus use the same symbols for a sheaf-space and its associated sheaf of sections; in a corresponding way, we write $\Gamma(U,\cS)=\cS(U)$ when $U$ is an open set. Conversely, given a presheaf $\cA$ on $X$, we have an associated sheaf-space $\wt\cA$ and an obvious presheaf morphism $$\cA(U)\lra\wt\cA'(U)=\Gamma(U,\wt\cA),~~~~ F\longmapsto \wt F=(U\ni x\mapsto F_x).\leqno(1.10)$$ This morphism is clearly injective if and only if $\cA$ satisfies axiom $(1.4'')$, and it is not difficult to see that $(1.4')$ and $(1.4'')$ together imply surjectivity. Therefore $\cA\to\wt\cA'$ is an isomorphism if and only if $\cA$ is a sheaf. According to the equivalence of categories between sheaves and sheaf-spaces mentioned above, we will use from now on the same symbol $\wt\cA$ for the sheaf-space and its associated sheaf $\wt\cA'$; one says that $\wt\cA$ is the {\it sheaf associated with the presheaf}~$\cA$. If $\cA$ itself is a sheaf, we will again identify $\wt\cA$ and $\cA$, but we will of course keep the notational difference for a presheaf $\cA$ which is not a sheaf. \begstat{(1.11) Example} \rm The sheaf associated to the constant presheaf of stalk $E$ over $X$ is the sheaf of locally constant functions~$X\to E$. This sheaf will be denoted merely by $E_X$ or $E$ if there is no risk of confusion with the corresponding presheaf. In Example~1.7, we have $\cZ=\bbbz_X$ and the sheaf $(\cC/\bbbz_X)^\sim$ associated with the quotient presheaf $\cC/\bbbz_X$ is isomorphic to $\cC^\star$ via the exponential map.\qed \endstat In the sequel, we usually work in the category of sheaves rather than in the category of presheaves themselves. For instance, the quotient $\cB/\cA$ of a sheaf $\cB$ by a subsheaf $\cA$ generally refers to the sheaf associated with the quotient presheaf: its stalks are equal to $\cB_x/\cA_x$, but a section $G$ of $\cB/\cA$ over an open set $U$ need not necessarily come from a global section of $\cB(U)\,$; what can be only said is that there is a covering $(U_\alpha)$ of $U$ and local sections $F_\alpha\in\cB(U_\alpha)$ representing $G_{\restriction U_\alpha}$ such that $(F_\beta-F_\alpha)_{\restriction U_\alpha\cap U_\beta}$ belongs to $\cA(U_\alpha\cap U_\beta)$. A sheaf morphism $\varphi:\cA\to\cB$ is said to be injective (resp.\ surjective) if the germ morphism $\varphi_x:\cA_x\to\cB_x$ is injective (resp.\ surjective) for every $x\in X$. Let us note again that a surjective sheaf morphism $\varphi$ does not necessarily give rise to surjective morphisms $\varphi_U:\cA(U)\to\cB(U)$. \titlec{\S 1.B.}{Direct and Inverse Images of Sheaves} Let $X$, $Y$ be topological spaces and let $f:X\to Y$ be a continuous map. If $\cA$ is a presheaf on $X$, the {\it direct image} $f_\star\cA$ is the presheaf on $Y$ defined by $$f_\star\cA(U)=\cA\big(f^{-1}(U)\big)\leqno(1.12)$$ for all open sets $U\subset Y$. When $\cA$ is a sheaf, it is clear that $f_\star\cA$ also satisfies axioms $(1.4')$ and $(1.4'')$, thus $f_\star\cA$ is a sheaf. Its stalks are given by $$(f_\star\cA)_y=\lim_{\displaystyle\lra\atop\scriptstyle V\ni y} \cA\big(f^{-1}(V)\big)\leqno(1.13)$$ where $V$ runs over all open neighborhoods of $y\in Y$. Now, let $\cB$ be a sheaf on~$Y$, viewed as a sheaf-space with projection map $\pi:\cB\to Y$. We define the {\it inverse image} $f^{-1}\cB$ by $$f^{-1}\cB=\cB\times_Y X=\big\{(s,x)\in\cB\times X\,;~\pi(s)=f(x)\big\} \leqno(1.14)$$ with the topology induced by the product topology on $\cB\times X$. It is then easy to see that the projection $\pi'={\rm pr}_2:f^{-1}\cB\to X$ is a local homeomorphism, therefore $f^{-1}\cB$ is a sheaf on~$X$. By construction, the stalks of $f^{-1}\cB$ are $$(f^{-1}\cB)_x=\cB_{f(x)},\leqno(1.15)$$ and the sections $\sigma\in f^{-1}\cB(U)$ can be considered as continuous mappings \hbox{$s:U\to\cB$} such that $\pi\circ\sigma=f$. In particular, any section $s\in\cB(V)$ on an open set $V\subset Y$ has a {\it pull-back} $$f^\star s=s\circ f~~\in~~f^{-1}\cB\big(f^{-1}(V)\big).\leqno(1.16)$$ There are always natural sheaf morphisms $$f^{-1}f_\star\cA\lra\cA,~~~~\cB\lra f_\star f^{-1}\cB\leqno(1.17)$$ defined as follows. A germ in $(f^{-1}f_\star\cA)_x=(f_\star\cA)_{f(x)}$ is defined by a local section $s\in (f_\star\cA)(V)=\cA(f^{-1}(V))$ for some neighborhood $V$ of $f(x)\,$; this section can be mapped to the germ $s_x\in\cA_x$. In the opposite direction, the pull-back $f^\star s$ of a section $s\in\cB(V)$ can be seen by (1.16) as a section of $f_\star f^{-1}\cB(V)$. It is not difficult to see that these natural morphisms are not isomorphisms in general. For instance, if $f$ is a finite covering map with $q$ sheets and if we take $\cA=E_X$, $\cB=E_Y$ to be constant sheaves, then $f_\star E_X\simeq E^q_Y$ and $f^{-1}E_Y=E_X$, thus $f^{-1}f_\star E_X\simeq E^q_X$ and $f_\star f^{-1}E_Y\simeq E_Y^q$. \titlec{\S 1.C.}{Ringed Spaces} Many natural geometric structures considered in analytic or algebraic geometry can be described in a convenient way as topological spaces equipped with a suitable ``structure sheaf'' which, most often, is a sheaf of commutative rings. For instance, a lot of properties of $C^k$ differentiable (resp.\ real analytic, complex analytic) manifolds can be described in terms of their sheaf of rings $\cC^k_X$ of differentiable functions (resp.\ $\cC^\omega_X$ of real analytic functions, $\cO_X$ of holomorphic functions). We first recall a few standard definitions concerning rings, referring to textbooks on algebra for more details (see e.g.\ Lang 1965). \begstat{(1.18) Some definitions and conventions about rings} All our rings $R$ are supposed implicitly to have a unit element $1_R$ $($if $R=\{0\}$, we agree that \hbox{$1_R=0_R\,)$}, and a ring morphism $R\to R'$ is supposed to map $1_R$ to $1_{R'}$. In the subsequent definitions, we assume that all rings under consideration are commutative. \smallskip \item{\rm a)} An ideal $I\subset R$ is said to be prime if $xy\in I$ implies $x\in I$ or $y\in I$, i.e., if the quotient ring $R/I$ is entire. \smallskip \item{\rm b)} An ideal $I\subset R$ is said to be maximal if $I\ne R$ and there are no ideals $J$ such that $I\subsetneq J\subsetneq R$ $($equivalently, if the quotient ring $R/I$ is a field$)$. \smallskip \item{\rm c)} The ring $R$ is said to be a local ring if $R$ has a unique maximal ideal $\gm$ $($equivalently, if $R$ has an ideal $\gm$ such that all elements of $R\ssm\gm$ are invertible$)$. Its residual field is defined to be the quotient field~$R/\gm$. \smallskip \item{\rm d)} The ring $R$ is said to be Noetherian if every ideal $I\subset R$ is finitely gene\-rated $($equivalently, if every increasing sequence of ideals $I_1\subset I_2\subset\ldots$ is~stationary$)$. \smallskip \item{\rm e)} The radical $\sqrt{I}$ of an ideal $I$ is the set of all elements $x\in R$ such that some power $x^m$, $m\in\bbbn^\star$, lies in in~$I$. Then $\sqrt{I}$ is again an ideal of~$R$. \smallskip \item{\rm f)} The nilradical $N(R)=\sqrt{\{0\}}$ is the ideal of nilpotent elements of~$R$. The ring $R$ is said to be reduced if $N(R)=\{0\}$. Otherwise, its reduction is defined to be the reduced ring~$R/N(R)$. \vskip0pt \endstat We now introduce the general notion of a ringed space. \begstat{(1.19) Definition} A ringed space is a pair $(X,\cR_X)$ consisting of a topolo\-gical space $X$ and of a sheaf of rings $\cR_X$ on $X$, called the structure sheaf. A~morphism $$F:(X,\cR_X)\to(Y,\cR_Y)$$ of ringed spaces is a pair $(f,F^\star)$ where $f:X\to Y$ is a continuous map and $$F^\star~:~~f^{-1}\cR_Y\to\cR_X,~~~~F^\star_x~:~~\cR_{Y,f(x)} \to\cR_{X,x}$$ a homomorphism of sheaves of rings on $X$, called the comorphism of $F$. \endstat If $F:(X,\cR_X)\to(Y,\cR_Y)$ and $G:(Y,\cR_Y)\to(Z,\cR_Z)$ are morphisms of ringed spaces, the composite $G\circ F$ is the pair consisting of the map $g\circ f:X\to Z$ and of the comorphism $(G\circ F)^\star=F^\star\circ f^{-1}G^\star\,$: $$\cmalign{ \hfill&F^\star\circ f^{-1}G^\star~:~~&f^{-1}g^{-1}\cR_Z~\,&\kern-1.9mm \buildo f^{-1}G^\star\over{\relbar\mkern-4mu\relbar\mkern-4mu\lra}f^{-1}\cR_Y &\kern-1.9mm\buildo F^\star\over{\relbar\mkern-4mu\lra}\cR_X,\cr &F^\star_x\circ G^\star_{f(x)}~:~~~&\cR_{Z,g\circ f(x)}~\,&{\relbar\mkern-4mu \relbar\mkern-4mu\lra}~\cR_{Y,f(x)}~&{\relbar\mkern-4mu\lra}~\cR_{X,x}.\cr} \leqno(1.20)$$ We say of course that $F$ is an isomorphism of ringed spaces if there exists $G$ such that $G\circ F=\Id_X$ and $F\circ G=\Id_Y$. If $(X,\cR_X)$ is a ringed space, the nilradical of $\cR_X$ defines an ideal subsheaf $\cN_X$ of $\cR_X$, and the identity map $\Id_X:X\to X$ together with the ring homomorphism $\cR_X\to\cR_X/\cN_X$ defines a ringed space morphism $$(X,\cR_X/\cN_X)\to (X,\cR_X)\leqno(1.21)$$ called the {\it reduction morphism}. Quite often, the letter $X$ by itself is used to denote the ringed space $(X,\cR_X)\,$; we then denote by $X_\red=(X,\cR_X/\cN_X)$ its reduction. The ringed space $X$ is said to be {\it reduced} if $\cN_X=0$, in which case the reduction morphism $X_\red\to X$ is an isomorphism. In all examples considered later on in this book, the structure sheaf $\cR_X$ will be a sheaf of {\it local rings} over some field~$k$. The relevant definition is as follows. \begstat{(1.22) Definition} \smallskip \item{\rm a)} A local ringed space is a ringed space $(X,\cR_X)$ such that all stalks $\cR_{X,x}$ are local rings. The maximal ideal of~$\cR_{X,x}$ will be denoted by $\gm_{X,x}$. A morphism $F=(f,F^\star):(X,\cR_X)\to(Y,\cR_Y)$ of local ringed spaces is a morphism of ringed spaces such that $F^\star_x(\gm_{Y,f(x)}) \subset\gm_{X,x}$ at any point $x\in X$ $($i.e., $F^\star_x$ is a ``local'' homomorphism of rings$)$. \smallskip \item{\rm b)} A local ringed space over a field $k$ is a local ringed space $(X,\cR_X)$ such that all rings $\cR_{X,x}$ are local $k$-algebras with residual field $\cR_{X,x}/\gm_{X,x}\simeq k$. A morphism $F$ between such spaces is supposed to have its comorphism defined by local $k$-homomorphisms $F^\star_x:\cR_{Y,f(x)}\to\cR_{X,x}$. \vskip0pt \endstat If $(X,\cR_X)$ is a local ringed space over~$k$, we can associate to each section $s\in\cR_X(U)$ a function $$\ovl s:U\to k,\qquad x\mapsto\ovl s(x)\in k=\cR_{X,x}/\gm_{X,x},$$ and we get a sheaf morphism $\cR_X\to\ovl\cR_X$ onto a subsheaf of rings $\ovl\cR_X$ of the sheaf of functions from $X$ to $k$. We clearly have a factorization $$\cR_X\to\cR_X/\cN_X\to\ovl\cR_X,$$ and thus a corresponding factorization of ringed space morphisms (with $\Id_X$ as the underlying set theoretic map) $$X_\stred\to X_\red\to X$$ where $X_\stred=(X,\ovl R_X)$ is called the {\it strong reduction} of $(X,\cR_X)$. It is easy to see that $X_\stred$ is actually a reduced local ringed space over~$k$. We say that $X$ is strongly reduced if $\cR_X\to\ovl\cR_X$ is an isomorphism, that is, if $\cR_X$ can be identified with a subsheaf of the sheaf of functions $X\to k$ (in our applications to the theory of algebraic or analytic schemes, the concepts of reduction and strong reduction will actually be the same$\,$; in general, these notions differ, see Exercise ??.??). It is important to observe that reduction (resp.\ strong reduction) is a fonctorial process:\newline if $F=(f,F^\star):(X,\cR_X)\to(Y,\cR_Y)$ is a morphism of ringed spaces (resp.\ of local ringed spaces over~$k$), there are natural reductions $$\eqalign{ F_\red&=(f,F^\star_\red):X_\red\to Y_\red,\quad F^\star_\red:\cR_{Y,f(x)}/\cN_{Y,f(x)}\to\cR_{X,x}/\cN_{X,x},\cr F_\stred&=(f,f^\star):X_\stred\to Y_\stred,\quad f^\star:\ovl\cR_{Y,f(x)}\to\ovl\cR_{X,x},\quad\ovl s\mapsto \ovl s\circ f\cr}$$ where $f^\star$ is the usual pull-back comorphism associated with~$f$. Therefore, if $(X,\cR_X)$ and $(Y,\cR_Y)$ are strongly reduced, the morphism $F$ is completely determined by the underlying set-theoretic map~$f$. Our first basic examples of (strongly reduced) ringed spaces are the various types of mani\-folds already defined in Chapter~I. The language of ringed spaces provides an equivalent but more elegant and more intrinsic definition. \begstat{(1.23) Definition} Let $X$ be a Hausdorff separable topological space. One can define the category of $C^k$, $k\in\bbbn\cup\{\infty,\omega\}$, differentiable manifolds $($resp.\ complex analytic manifolds$)$ to be the category of reduced local ringed spaces $(X,\cR_X)$ over~$\bbbr$ $($resp.\ over~$\bbbc)$, such that every point $x\in X$ has a neighborhood $U$ on which the restriction $(U,\cR_{X\restriction U})$ is isomorphic to a ringed space $(\Omega,\cC^k_\Omega)$ where $\Omega\subset\bbbr^n$ is an open set and $\cC^k_\Omega$ is the sheaf of $C^k$ differentiable functions $($resp.\ $(\Omega,\cO_\Omega)$, where $\Omega\subset\bbbc^n$ is an open subset, and $\cO_\Omega$ is the sheaf of holomorphic functions on~$\Omega)$. \endstat We say that the ringed spaces $(\Omega,\cC^k_\Omega)$ and $(\Omega,\cO_\Omega)$ are the {\it models} of the category of differentiable (resp.\ complex analytic) manifolds, and that a general object $(X,\cR_X)$ in the category is {\it locally isomorphic} to one of the given model spaces. It is easy to see that the corresponding ringed spaces morphisms are nothing but the usual concepts of differentiable and holomorphic maps. \titlec{\S 1.D.}{Algebraic Varieties over a Field} As a second illustration of the notion of ringed space, we present here a brief introduction to the formalism of algebraic varieties, referring to (Hartshorne 1977) or (EGA 1967) for a much more detailed exposition. Our hope is that the reader who already has some background of analytic or algebraic geometry will find some hints of the strong interconnections between both theories. Beginners are invited to skip this section and proceed directly to the theory of complex analytic sheaves in~\S$,$2. All rings or algebras occurring in this section are supposed to be commutative rings with unit. \titled{\S 1.D.1. Affine Algebraic Sets.} Let $k$ be an algebraically closed field of any characteristic. An {\it affine algebraic set} is a subset $X\subset k^N$ of the affine space $k^N$ defined by an arbitrary collection $S\subset k[T_1\ld T_N]$ of polynomials, that is, $$X=V(S)=\big\{(z_1\ld z_N)\in k^N\,;\,P(z_1\ld z_N)=0,\, \forall P\in S\big\}.$$ Of course, if $J\subset k[T_1\ld T_N]$ is the ideal generated by $S$, then $V(S)=V(J)$. As $k[T_1\ld T_N]$ is Noetherian, $J$ is generated by finitely many elements $(P_1\ld P_m)$, thus $X=V(\{P_1\ld P_m\})$ is always defined by finitely many equations. Conversely, for any subset $Y\subset k^N$, we consider the ideal $I(Y)$ of $k[T_1\ld T_N]$, defined by $$I(Y)=\big\{P\in k[T_1\ld T_N]\,;\,P(z)=0,~\forall z\in Y\big\}.$$ Of course, if $Y\subset k^N$ is an algebraic set, we have $V(I(Y))=Y$. In the opposite direction, we have the following fundamental result. \begstat{(1.24) Hilbert's Nullstellensatz {\rm (see Lang 1965)}} If $J\subset k[T_1\ld T_N]$ is an ideal, then $I(V(J))=\sqrt{J}$. \endstat If $X=V(J)\subset k^N$ is an affine algebraic set, we define the (reduced) ring $\cO(X)$ of algebraic functions on $X$ to be the set of all functions $X\to k$ which are restrictions of polynomials, i.e., $$\cO(X)=k[T_1\ld T_N]/I(X)=k[T_1\ld T_N]/\sqrt{J}.\leqno(1.25)$$ This is clearly a reduced $k$-algebra. An (algebraic) morphism of affine algebraic sets $X=V(J)\subset k^N$, $Y=V(J')\subset k^{N'}$ is a map $f:Y\to X$ which is the restriction of a polynomial map $k^{N'}to k^N$. We then get a $k$-algebra homomomorphism $$f^\star:\cO(X)\to\cO(Y),\qquad s\mapsto s\circ f,$$ called the {\it comorphism} of~$f$. In this way, we have defined a contravariant fonctor $$X\mapsto\cO(X),\qquad f\mapsto f^\star\leqno(1.26)$$ from the category of affine algebraic sets to the category of finitely generated reduced $k$-algebras. We are going to show the existence of a natural fonctor going in the opposite direction. In fact, let us start with an arbitrary finitely generated algebra~$A$ (not necessarily reduced at this moment). For any choice of gene\-rators $(g_1\ld g_N)$ of $A$ we get a surjective morphism of the polynomial ring $k[T_1\ld T_N]$ onto~$A$, $$k[T_1\ld T_N]\to A,\qquad T_j\mapsto g_j,$$ and thus $A\simeq k[T_1\ld T_N]/J$ with the ideal $J$ being the kernel of this morphism. It is well-known that every maximal ideal $\gm$ of $A$ has codimension~$1$ in $A$ (see Lang 1965), so that $\gm$ gives rise to a $k$-algebra homomorphism $A\to A/\gm=k$. We thus get a bijection $$\Hom_\alg(A,k)\to\Spm(A),\qquad u\mapsto\Ker u$$ between the set of $k$-algebra homomorphisms and the set $\Spm(A)$ of maximal ideals of~$A$. In fact, if $A=k[T_1\ld T_N]/J$, an element $\varphi\in\Hom_\alg(A,k)$ is completely determined by the values $z_j=\varphi(T_j\mod J)$, and the corresponding algebra homomorphism $k[T_1\ld T_N]\to k$, $P\mapsto P(z_1\ld z_N)$ can be factorized mod $J$ if and only if $z=(z_1\ld z_N)\in k^N$ satisfies the equations $$P(z_1\ld z_N)=0,\qquad\forall P\in J.$$ We infer from this that $$\Spm(A)\simeq V(J)=\big\{(z_1\ld z_N)\in k^N\,;\,P(z_1\ld z_N)=0,\, \forall P\in J\big\}$$ can be identified with the {\it affine algebraic set} $V(J)\subset k^N$. If we are given an algebra homomorphism $\Phi:A\to B$ of finitely generated $k$-algebras we get a corresponding map $\Spm(\Phi):\Spm(B)\to\Spm(A)$ described either as $$\eqalign{ &\Spm(B)\to\Spm(A),\quad\gm\mapsto\Phi^{-1}(\gm)\quad\hbox{or}\cr &\Hom_\alg(B,k)\to\Hom_\alg(A,k),\quad v\mapsto v\circ\Phi.\cr}$$ If $B=k[T'_1\ld T'_{N'}]/J'$ and $\Spm(B)=V(J')\subset k^{N'}$, it is easy to see that $\Spm(\Phi):\Spm(B)\to\Spm(A)$ is the restriction of the polynomial map $$f:k^{N'}\to k^N,\qquad w\mapsto f(w)=(P_1(w)\ld P_N(w)),$$ where $P_j\in k[T'_1\ld T'_{N'}]$ are polynomials such that $P_j=\Phi(T_j)\mod J'$ in~$B$. We have in this way defined a contravariant fonctor $$A\mapsto\Spm(A),\qquad \Phi\mapsto\Spm(\Phi)\leqno(1.27)$$ from the category of finitely generated $k$-algebras to the category of affine algebraic sets. Since $A=k[T_1\ld T_N]/J$ and its reduction $A/N(A)=k[T_1\ld T_N]/\sqrt{J}$ give rise to the same algebraic set $$V(J)=\Spm(A)=\Spm(A/N(A))=V(\sqrt{J}),$$ we see that the category of affine algebraic sets is actually equivalent to the subcategory of {\it reduced} finitely generated $k$-algebras. \begstat{(1.28) Example} \rm The simplest example of an affine algebraic set is the affine space $$k^N=\Spm(k[T_1\ld T_N]),$$ in particular $\Spm(k)=k^0$ is just one point. We agree that $\Spm(\{0\})=\emptyset$ (observe that $V(J)=\emptyset$ when $J$ is the unit ideal in $k[T_1\ld T_N])$. \endstat \titled{\S 1.D.2. Zariski Topology and Affine Algebraic Schemes.} Let $A$ be a finitely generated algebra and $X=\Spm(A)$. To each ideal $\ga\subset A$ we associate the zero variety $V(\ga)\subset X$ which consists of all elements $\gm\in X=\Spm(A)$ such that $\gm\supset \ga\,$; if $$A\simeq k[T_1\ld T_N]/J\quad\hbox{and}\quad X\simeq V(J)\subset k^N,$$ then $V(\ga)$ can be identified with the zero variety $V(J_\ga)\subset X$ of the inverse image $J_\ga$ of $\ga$ in $k[T_1\ld T_N]$. For any family $(\ga_\alpha)$ of ideals in $A$ we have $$V(\sum\ga_\alpha)=\bigcap V(\ga_\alpha),\qquad V(\ga_1)\cup V(\ga_2)=V(\ga_1\ga_2),$$ hence there exists a unique topology on $X$ such that the closed sets consist precisely of all algebraic subsets $(V(\ga))_{\ga\subset A}$ of~$X$. This topology is called the Zariski topology. The Zariski topology is almost never Hausdorff (for example, if $X=k$ is the affine line, the open sets are $\emptyset$ and complements of finite sets, thus any two nonempty open sets have nonempty intersection). However, $X$ is a {\it Noetherian space}, that is, a topological space in which every decreasing sequence of closed sets is stationary; an equivalent definition is to require that every open set is quasi-compact (from any open covering of an open set, one can extract a finite covering). We now come to the concept of affine open subsets. For $s\in A$, the open set $D(s)=X\ssm V(s)$ can be given the structure of an affine algebraic variety. In fact, if $A=k[T_1\ld T_N]/J$ and $s$ is represented by a polynomial in $k[T_1\ld T_N]$, the localized ring $A[1/s]$ can be written as $A[1/s]=k[T_1\ld T_N,T_{N+1}]/J_s$ where $J_s=J[T_{N+1}]+(sT_{N+1}-1)$, thus $$V(J_s)=\{(z,w)\in V(J)\times k\,;\, s(z)\,w=1\}\simeq V(I)\ssm s^{-1}(0)$$ and $D(s)$ can be identified with $\Spm(A[1/s])$. We have $D(s_1)\cap D(s_2)=D(s_1s_2)$, and the sets $(D(s))_{s\in A}$ are easily seen to be a basis of the Zariski topology on~$X$. The open sets $D(s)$ are called {\it affine open sets}. Since the open sets $D(s)$ containing a given point $x\in X$ form a basis of neighborhoods, one can define a sheaf space $\cO_X$ such that the ring of germs $\cO_{X,x}$ is the inductive limit $$\cO_{X,x}=\lim_{\displaystyle\lra\atop\scriptstyle D(s)\ni x}A[1/s]= \{\hbox{fractions}~p/q\,;\,p,q\in A,\,q(x)\ne 0\}.$$ This is a local ring with maximal ideal $$\gm_{X,x}=\{p/q\,;\,p,q\in A,\,p(x)=0,\,q(x)\ne 0\},$$ and residual field $\cO_{X,x}/\gm_{X,x}=k$. In this way, we get a ringed space $(X,\cO_X)$ over~$k$. It is easy to see that $\Gamma(X,\cO_X)$ coincides with the finitely generated $k$-algebra~$A$. In fact, from the definition of~$\cO_X$, a global section is obtained by gluing together local sections $p_j/s_j$ on affine open sets $D(s_j)$ with \hbox{$\bigcup D(s_j)=X$}, $1\le j\le m$. This means that the ideal $\ga=(s_1\ld s_m)\subset A$ has an empty zero variety $V(\ga)$, thus $\ga=A$ and there are elements $u_j\in A$ with $\sum u_js_j=1$. The compatibility condition $p_j/s_j=p_k/s_k$ implies that these elements are induced by $$\sum u_jp_j/\sum u_js_j=\sum u_jp_j\in A,$$ as desired. More generally, since the open sets $D(s)$ are affine, we get $$\Gamma(D(s),\cO_X)=A[1/s].$$ It is easy to see that the ringed space $(X,\cO_X)$ is reduced if and only if $A$ itself is reduced; in this case, $X$ is even strongly reduced as Hilbert's Nullstellensatz shows. Otherwise, the reduction $X_\red$ can obtained from the reduced algebra $A_\red=A/N(A)$. Ringed spaces $(X,\cO_X)$ as above are called {\it affine algebraic schemes} over~$k$ (although substantially different from the usual definition, our definition can be shown to be equivalent in this special situation; compare with (Hartshorne 1977); see also Exercise ??.??). The category of affine algebraic schemes is equivalent to the category of finitely generated $k$-algebras (with the arrows reversed). \titled{1.D.3. Algebraic Schemes.} Algebraic schemes over $k$ are defined to be ringed spaces over $k$ which are locally isomorphic to affine algebraic schemes, modulo an ad hoc separation condition. \begstat{(1.29) Definition} An algebraic scheme over $k$ is a local ringed space $(X,\cO_X)$ over $k$ such that \smallskip \item{\rm a)} $X$ has a finite covering by open sets $U_\alpha$ such that $(U_\alpha,\cO_{X\restriction U_\alpha})$ is isomorphic as a ringed space to an affine algebraic scheme $(\Spm(A_\alpha),\cO_{\Spm(A_\alpha)})$. \smallskip \item{\rm b)} $X$ satisfies the algebraic separation axiom, namely the diagonal $\Delta_X$ of $X\times X$ is closed for the Zariski topology. \medskip\noindent A morphism of algebraic schemes is just a morphism of the underlying local ringed spaces. An $($abstract$)$ algebraic variety is the same as a reduced algebraic scheme. \vskip0pt \endstat In the above definition, some words of explanation are needed for b), since the product $X\times Y$ of algebraic schemes over $k$ {\it is not} the ringed space theoretic product, i.e., the product topological space equipped with the structure sheaf $\pr_1^\star\cO_X\otimes_k\pr_2^\star\cO_Y$. Instead, we define the product of two affine algebraic schemes $X=\Spm(A)$ and $Y=\Spm(B)$ to be $X\times Y=\Spm(A\otimes_k B)$, equipped with the Zariski topology and the structural sheaf associated with~$A\otimes_k B$. Notice that the Zariski topology on $X\times Y$ {\it is not the product topology} of the Zariski topologies on $X$,~$Y$, as the example $k^2=k\times k$ shows; also, the rational function $1/(1-z_1-z_2)\in \cO_{k^2,(0,0)}$ is not in $\cO_{k,0}\otimes_k\cO_{k,0}$. In general, if $X$, $Y$ are written as $X=\bigcup U_\alpha$ and $Y=\bigcup V_\beta$ with affine open sets $U_\alpha$, $V_\beta$, we define $X\times Y$ to be the union of all open affine charts $U_\alpha\times V_\beta$ with their associated structure sheaves of affine algebraic varieties, the open sets of $X\times Y$ being all unions of open sets in the various charts $U_\alpha\times V_\beta$. The separation axiom b) is introduced for the sake of excluding pathological examples such as an affine line $k\amalg\{0'\}$ with the origin changed into a double point. \titled{1.D.4. Subschemes.} If $(X,\cO_X)$ is an affine algebraic scheme and $A=\Gamma(X,\cO_X)$ is the associated algebra, we say that $(Y,\cO_Y)$ is a {\it subscheme} of $(X,\cO_X)$ if there is an ideal $\ga$ of $A$ such that $Y\hookrightarrow X$ is the morphism defined by the algebra morphism $A\to A/\ga$ as its comorphism. As $\Spm(A/\ga)\to\Spm(A)$ has for image the set $V(\ga)$ of maximal ideals $\gm$ of $A$ containing $\ga$, we see that $Y=V(\ga)$ as a set; let us introduce the ideal subsheaf $\cJ=\ga\cO_X\subset\cO_X$. Since the structural sheaf $\cO_Y$ is obtained by taken localizations $A/\ga[1/s]$, it is easy to see that $\cO_Y$ coincides with the quotient sheaf $\cO_X/\cJ$ restricted to~$Y$. Since $\ga$ has finitely many generators, the ideal sheaf $\cJ$ is locally finitely generated (see \S~2 below). This leads to the following definition. \begstat{(1.30) Definition} If $(X,\cO_X)$ is an algebraic scheme, a $($closed$)$ subscheme is an algebraic scheme $(Y,\cO_Y)$ such that $Y$ is a Zariski closed subset of~$X$, and there is a locally finitely generated ideal subsheaf $\cJ\subset\cO_X$ such that $Y=V(\cJ)$ and $\cO_Y=(\cO_X/\cJ)_{\restriction Y}$. \endstat If $(Y,\cO_Y)$, $(Z,\cO_Z)$ are subschemes of $(X,\cO_X)$ defined by ideal subsheaves $\cJ,\,\cJ'\subset\cO_X$, there are corresponding subschemes $Y\cap Z$ and $Y\cup Z$ defined as ringed spaces $$(Y\cap Z,\cO_X/(\cJ+\cJ')),\qquad(Y\cup Z,\cO_X/\cJ\cJ').$$ \titled{\S 1.D.5. Projective Algebraic Varieties.} A very important subcategory of the category of algebraic varieties is provided by {\it projective algebraic varieties}. Let $\bbbp^N_k$ be the projective $N$-space, that is, the set $k^{N+1}\ssm\{0\}/k^\star$ of equivalence classes of $(N+1)$-tuples $(z_0\ld z_N)\in k^{N+1}\ssm\{0\}$ under the equivalence relation given by $(z_0\ld z_N)\sim\lambda (z_0\ld z_N)$, $\lambda\in k^\star$. The corresponding element of $\bbbp^N_k$ will be denoted $[z_0:z_1:\ldots:z_N]$. It is clear that $\bbbp_N^k$ can be covered by the $(N+1)$ affine charts $U_\alpha$, $0\le\alpha\le N$, such that $$U_\alpha=\big\{[z_0:z_1:\ldots:z_N]\in\bbbp^N_k\,\;\,z_\alpha\ne 0\big\}.$$ The set $U_\alpha$ can be identified with the affine $N$-space $k^N$ by the map $$U_\alpha\to k^N,\qquad[z_0:z_1:\ldots:z_N]\mapsto \Big({z_0\over z_\alpha},\,{z_1\over z_\alpha}\ld {z_{\alpha-1}\over z_\alpha},\,{z_{\alpha+1}\over z_\alpha}\ld {z_N\over z_\alpha}\Big).$$ With this identification, $\cO(U_\alpha)$ is the algebra of homogeneous rational functions of degree $0$ in $z_0\ld z_N$ which have just a power of $z_\alpha$ in their denominator. It is easy to see that the structure sheaves $\cO_{U_\alpha}$ and $\cO_{U_\beta}$ coincide in the intersections $U_\alpha\cap U_\beta\,$; they can be glued together to define an algebraic variety structure $(\bbbp^N_k,\cO_{\bbbp^N})$, such that $\cO_{\bbbp^N,[z]}$ consists of all homogeneous rational functions $p/q$ of degree $0$ (i.e., $\deg p=\deg q$), such that $q(z)\ne 0$. \begstat{(1.30) Definition} An algebraic scheme or variety $(X,\cO_X)$ is said to be projective if it is isomorphic to a closed subscheme of some projective space $(\bbbp^N_k,\cO_{\bbbp^N})$. \endstat We now indicate a standard way of constructing projective schemes. Let $S$ be a collection of homogeneous polynomials $P_j\in k[z_0\ld z_N]$, of degree $d_j\in\bbbn$. We define an associated {\it projective algebraic set} $$\wt V(S)=\big\{[z_0:\ldots:z_N]\in\bbbp^N_k\,;\,P(z)=0,~\forall P\in S\big\}.$$ Let $J$ be the {\it homogeneous ideal} of $k[z_0\ld z_N]$ generated by $S$ (recall that an ideal $J$ is said to be homogeneous if $J=\bigoplus J_m$ is the direct sum of its homogeneous components, or equivalently, if $J$ is generated by homogeneous elements). We have an associated graded algebra $$B=k[z_0\ld z_N]/J=\bigoplus B_m,\qquad B_m=k[z_0\ld z_N]_m/J_m$$ such that $B$ is generated by $B_1$ and $B_m$ is a finite dimensional vector space over $k$ for each~$k$. This is enough to construct the desired scheme structure on $\wt V(J):=\bigcap\wt V(J_m)$, as we see in the next subsection. \titled{1.D.6. Projective Scheme Associated with a Graded Algebra.} Let us start with a reduced graded $k$-algebra $$B=\bigoplus_{m\in\bbbn}B_m$$ such that $B$ is generated by $B_0$ and $B_1$ as an algebra, and $B_0$, $B_1$ are finite dimensional vector spaces over $k$ (it then follows that $B$ is finitely generated and that all $B_m$ are finite dimensional vector spaces). Given $s\in B_m$, $m>0$, we define a $k$-algebra $A_s$ to be the ring of all fractions of homogeneous degree~$0$ with a power of $s$ as their denominator, i.e., $$A_s=\big\{p/s^d\,;\,p\in B_{dm},\,d\in\bbbn\big\}.\leqno(1.31)$$ Since $A_s$ is generated by ${1\over s}B_1^m$ over $B_0$, $A_s$ is a finitely generated algebra. We define $U_s=\Spm(A_s)$ to be the associated affine algebraic variety. For $s\in B_m$ and $s'\in B_{m'}$, we clearly have algebra homomorphisms $$A_s\to A_{ss'},\qquad A_{s'}\to A_{ss'},$$ since $A_{ss'}$ is the algebra of all $0$-homogeneous fractions with powers of $s$ and $s'$ in the denominator. As $A_{ss'}$ is the same as the localized ring $A_s[s^{m'}/s^{\prime m}]$, we see that $U_{ss'}$ can be identified with an affine open set in $U_s$, and we thus get canonical injections $$U_{ss'}\hookrightarrow U_s,\qquad U_{ss'}\hookrightarrow U_{s'}.$$ \begstat{(1.32) Definition} If $B=\bigoplus_{m\in\bbbn}B_m$ is a reduced graded algebra generated by its finite dimensional vector subspaces $B_0$ and $B_1$, we associate an algebraic scheme $(X,\cO_X)=\Proj(B)$ as follows. To each finitely generated algebra $A_s=\big\{p/s^d\,;\,p\in B_{dm},\,d\in\bbbn\big\}$ we associate an affine algebraic variety $U_s=\Spm(A_s)$. We let $X$ be the union of all open charts $U_s$ with the identifications $U_s\cap U_{s'}=U_{ss'}\,$; then the collection $(U_s)$ is a basis of the topology of $X$, and $\cO_X$ is the unique sheaf of local $k$-algebras such that $\Gamma(U_s,\cO_X)=A_s$ for each~$U_s$. \endstat The following proposition shows that only finitely many open charts are actually needed to describe~$X$ (as required in Def.~1.29~a)). \begstat{(1.33) Lemma} If $s_0\ld s_N$ is a basis of $B_1$, then $\Proj(B)=\bigcup\limits_{0\le j\le N}U_{s_j}$. \endstat \begproof{} In fact, if $x\in X$ is contained in a chart $U_s$ for some $s\in B_m$, then $U_s=\Spm(A_s)\ne\emptyset$, and therefore $A_s\ne\{0\}$. As $A_s$ is generated by ${1\over s}B_1^m$, we can find a fraction $f=s_{j_1}\ldots s_{j_m}/s$ representing an element $f\in\cO(U_s)$ such that $f(x)\ne 0$. Then $x\in U_s\ssm f^{-1}(0)$, and $U_s\ssm f^{-1}(0)=\Spm(A_s[1/f])=U_s\cap U_{s_{j_1}}\cap\ldots\cap U_{s_{j_m}}$. In particular $x\in U_{s_{j_1}}$.\qed \endproof \begstat{(1.34) Example} \rm One can consider the {\it projective space} $\bbbp^N_k$ to be the algebraic scheme $$\bbbp^N_k=\Proj(k[T_0\ld T_N]).$$ \endstat The Proj construction is fonctorial in the following sense: if we have a graded homomorphism $\Phi:B\to B'$ (i.e.\ an algebra homomorphism such that $\Phi(B_m)\subset B'_m$, then there are corresponding morphisms $A_s\to A'_{\Phi(s)}$, $U'_{\Phi(s)}\to U_s$, and we thus find a scheme morphism $$F:\Proj(B')\to\Proj(B).$$ Also, since $p/s^d=ps^l/s^{d+l}$, the algebras $A_s$ depend only on components $B_m$ of large degree, and we have $A_s=A_{s^l}$. It follows easily that there is a canonical isomorphism $$\Proj(B)\simeq\Proj\Big(\bigoplus_m B_{lm}\Big).$$ Similarly, we may if we wish change a finite number of components $B_m$ without affecting $\Proj(B)$. In particular, we may alway assume that $B_0=k\,1_B$. By selecting finitely many generators $g_0\ld g_N$ in $B_1$, we then find a surjective graded homomorphism $k[T_0\ld T_N]\to B$, thus $B\simeq k[T_0\ld T_N]/J$ for some graded ideal $J\subset B$. The algebra homomorphism $k[T_0\ld T_N]\to B$ therefore yields a scheme embedding $\Proj(B)\to\bbbp^N$ onto $V(J)$. We will not pursue further the study of algebraic varieties from this point of view$\,$; in fact we are mostly interested in the case $k=\bbbc$, and algebraic varieties over $\bbbc$ are a special case of the more general concept of complex analytic space. \titleb{\S 2.}{The Local Ring of Germs of Analytic Functions} \titlec{\S 2.A.}{The Weierstrass Preparation Theorem} Our first goal is to establish a basic factorization and division theorem for analytic functions of several variables, which is essentially due to Weierstrass. We follow here a simple proof given by C.L. Siegel, based on a clever use of the Cauchy formula. Let $g$ be a holomorphic function defined on a neighborhood of $0$ in $\bbbc^n$, $g\not\equiv 0$. There exists a dense set of vectors $v\in\bbbc^n\ssm\{0\}$ such that the function $\bbbc\ni t\longmapsto g(tv)$ is not identically zero. In fact the Taylor series of $g$ at the origin can be written $$g(tv)=\sum_{k=0}^{+\infty}{1\over k!}\,t^k\,g^{(k)}(v)$$ where $g^{(k)}$ is a homogeneous polynomial of degree $k$ on $\bbbc^n$ and $g^{(k_0)}\not\equiv 0$ for some index $k_0$. Thus it suffices to select $v$ such that $g^{(k_0)}(v)\ne 0$. After a change of coordinates, we may assume that $v=(0\ld 0,1)$. Let $s$ be the vanishing order of $z_n\longmapsto g(0\ld 0,z_n)$ at $z_n=0$. There exists $r_n>0$ such that $g(0\ld 0,z_n)\ne 0$ when $0<|z_n|\le r_n$. By continuity of $g$ and compactness of the circle $|z_n|=r_n$, there exists $r'>0$ and $\varepsilon>0$ such that $$g(z',z_n)\ne 0~~~~\hbox{\rm for}~z'\in\bbbc^{n-1}, ~~~|z'|\le r',~~~r_n-\varepsilon\le|z_n|\le r_n+\varepsilon.$$ For every integer $k\in\bbbn$, let us consider the integral $$S_k(z')={1\over 2\pi\ii}\int_{|z_n|=r_n}{1\over g(z',z_n)}{\partial g\over \partial z_n}(z',z_n)\,z_n^k\,dz_n.$$ Then $S_k$ is holomorphic in a neighborhood of $|z'|\le r'$. Rouch\'e's theorem shows that $S_0(z')$ is the number of roots $z_n$ of $g(z',z_n)=0$ in the disk $|z_n|0$ for all~$j$. Then $$P(0,z_n)=z_n^s=P_1(0,z_n)\ldots P_N(0,z_n),$$ hence $P_j(0,z_n)=z_n^{s_j}$ and therefore $P_j$ is a Weierstrass polynomial. \smallskip \noindent b) Set $s=\deg\,P$ and $P(0,z_n)=z_n^s$. Assume that $P$ is reducible in $\cO_n$, with $P(z',z_n)=g_1(z)g_2(z)$ for non invertible elements $g_1,g_2\in\cO_n$. Then $g_1(0,z_n)$ and $g_2(0,z_n)$ have vanishing orders $s_1,s_2>0$ with $s_1+s_2=s$, and $$g_j=u_jP_j,~~~~\deg\,P_j=s_j,~~~j=1,2,$$ where $P_j$ is a Weierstrass polynomial and $u_j\in\cO_n$ is invertible. Therefore $P_1P_2=uP$ for an invertible germ $u\in\cO_n$. Lemma~2.8 shows that $P$ divides $P_1P_2$ in $\cO_{n-1}[z_n]\,$; since $P_1$, $P_2$ are unitary and $s=s_1+s_2$, we get $P=P_1P_2$, hence $P$ is reducible in $\cO_{n-1}[z_n]$. The converse implication is obvious from~a). \qed \endproof \begstat{(2.10) Theorem} $\cO_n$ is a factorial ring, i.e.\ $\cO_n$ is entire and: \smallskip \item{\rm a)} every non zero germ $f\in\cO_n$ admits a factorization $f=f_1\ldots f_N$ in irreducible elements$\,;$ \smallskip \item{\rm b)} the factorization is unique up to invertible elements. \endstat \begproof{} The existence part a) follows from Lemma~2.9 if we take $f$ to be a Weierstrass polynomial and $f=f_1\ldots f_N$ be a decomposition of maximal length $N$ into polynomials of positive degree. In order to prove the uniqueness, it is sufficient to verify the following statement: \smallskip \item{\rlap{\hbox{${\rm b}')$}}} {\it If $g$ is an irreducible element that divides a product $f_1f_2$, then $g$ divides either $f_1$ or $f_2$.} \smallskip \noindent By Th.~2.1, we may assume that $f_1$, $f_2$, $g$ are Weierstrass polynomials in $z_n$. Then $g$ is irreducible and divides $f_1f_2$ in $\cO_{n-1}[z_n]$ thanks to Lemmas 2.8 and 2.9~b). By induction on $n$, we may assume that $\cO_{n-1}$ is factorial. The standard Gauss lemma (Lang 1965, Chapter V) says that the polynomial ring $A[T]$ is factorial if the ring $A$ is factorial. Hence $\cO_{n-1}[z_n]$ is factorial by induction and thus $g$ must divide $f_1$ or $f_2$ in $\cO_{n-1}[z_n]$.\qed \endproof \begstat{(2.11) Theorem} If $f,g\in\cO_n$ are relatively prime, then the germs $f_z,~g_z$ at every point $z\in\bbbc^n$ near $0$ are again relatively prime. \endstat \begproof{} One may assume that $f=P,~g=Q$ are Weierstrass polynomials. Let us recall that unitary polynomials $P,Q\in {\cal A}[X]$ (${\cal A}={\rm a}$ factorial ring) are relatively prime if and only if their resultant $R\in{\cal A}$ is non zero. Then the resultant $R(z')\in\cO_{n-1}$ of $P(z',z_n)$ and $Q(z',z_n)$ has a non zero germ at $0$. Therefore the germ $R_{z'}$ at points $z'\in\bbbc^{n-1}$ near $0$ is also non zero.\qed \endproof \titleb{\S 3.}{Coherent Sheaves} \titlec{\S 3.1.}{Locally Free Sheaves and Vector Bundles} Section~9 will greatly develope this philosophy. Before introducing the more general notion of a coherent sheaf, we discuss the notion of locally free sheaves over a sheaf a ring. All rings occurring in the sequel are supposed to be commutative with unit (the non commutative case is also of considerable interest, e.g.\ in view of the theory of $\cD$-modules, but this subject is beyond the scope of the present book). \begstat{(3.1) Definition} Let $\cA$ be a sheaf of rings on a topological space $X$ and let $\cS$ a sheaf of modules over $\cA$ (or briefly a $\cA$-module). Then $\cS$ is said to be locally free of rank $r$ over $\cA$, if $\cS$ is locally isomorphic to $\cA^{\oplus r}$ on a neighborhood of every point, i.e.\ for every $x_0\in X$ one can find a neighborhood $\Omega$ and sections $F_1\ld F_r\in\cS(\Omega)$ such that the sheaf homomorphism $$F:\cA_{\restriction\Omega}^{\oplus r}\lra\cS_{\restriction\Omega},~~~~ \cA_x^{\oplus r}\ni(w_1\ld w_r)\longmapsto \sum_{1\le j\le r}w_jF_{j,x}\in\cS_x$$ is an isomorphism. \endstat By definition, if $\cS$ is locally free, there is a covering $(U_\alpha)_{\alpha\in I}$ by open sets on which $\cS$ admits free generators $F_\alpha^1\ld F_\alpha^r\in\cS(U_\alpha)$. Because the generators can be uniquely expressed in terms of any other system of independent generators, there is for each pair $(\alpha,\beta)$ a $r\times r$ matrix $$G_{\alpha\beta}=(G_{\alpha\beta}^{jk})_{1\le j,k\le r},\qquad G_{\alpha\beta}^{jk}\in\cA(U_\alpha\cap U_\beta),$$ such that $$F_\beta^k=\sum_{1\le j\le r}F_\alpha^jG^{jk}_{\alpha\beta}\qquad \hbox{\rm on}\quad U_\alpha\cap U_\beta.$$ In other words, we have a commutative diagram $$\eqalign{ &\cA^{\oplus r}_{\restriction U_\alpha\cap U_\beta} \buildo{\displaystyle F_\alpha}\over\lra \cS_{\restriction U_\alpha\cap U_\beta}\cr &G_{\alpha\beta}\Big\uparrow\quad\phantom{\lra}\qquad \Big|~\Big|\cr &\cA^{\oplus r}_{\restriction U_\alpha\cap U_\beta} \buildu{\displaystyle F_\beta}\under\lra \cS_{\restriction U_\alpha\cap U_\beta}\cr}$$ It follows easily from the equality $G_{\alpha\beta}=F_\alpha^{-1}\circ F_\beta$ that the {\it transition matrices} $G_{\alpha\beta}$ are invertible matrices satisfying the transition relation $$G_{\alpha\gamma}=G_{\alpha\beta}G_{\beta\gamma}\qquad\hbox{\rm on}\quad U_\alpha\cap U_\beta\cap U_\gamma\leqno(3.2)$$ for all indices $\alpha,\beta,\gamma\in I$. In particular $G_{\alpha\alpha}= \Id$ on $U_\alpha$ and $G_{\alpha\beta}^{-1}=G_{\beta\alpha}$ on $U_\alpha\cap U_\beta$. Conversely, if we are given a system of invertible $r\times r$ matrices $G_{\alpha\beta}$ with coefficients in $\cA(U_\alpha\cap U_\beta)$ satisfying the transition relation~(3.2), we can define a locally free sheaf $\cS$ of rank $r$ over $\cA$ by taking $\cS\simeq\cA^{\oplus r}$ over each $U_\alpha$, the identification over $U_\alpha\cap U_\beta$ being given by the isomorphism $G_{\alpha\beta}$. A~section $H$ of $\cS$ over an open set $\Omega\subset X$ can just be seen as a collection of sections $H_\alpha=(H_\alpha^1\ld H_\alpha^r)$ of $\cA^{\oplus r}(\Omega\cap U_\alpha)$ satisfying the transition relations $H_\alpha=G_{\alpha\beta} H_\beta$ over $\Omega\cap U_\alpha\cap U_\beta$. The notion of locally free sheaf is closely related to another essential notion of differential geometry, namely the notion of vector bundle (resp.\ topological, differentiable, holomorphic $\ldots$, vector bundle). To describe the relation between these notions, we assume that the sheaf of rings $\cA$ is a subsheaf of the sheaf $\cC_\bbbk$ of continous functions on $X$ with values in the field $\bbbk=\bbbr$ or $\bbbk=\bbbc$, containing the sheaf of locally constant functions $X\to\bbbk$. Then, for each $x\in X$, there is an evaluation map $$\cA_x\to\bbbk,\qquad w\mapsto w(x)$$ whose kernel is a maximal ideal $\gm_x$ of $\cA_x$, and $\cA_x/\gm_x=\bbbk$. Let $\cS$ be a locally free sheaf of rank $r$ over~$\cA$. To each $x\in X$, we can associate a $\bbbk$-vector space $E_x=\cS_x/\gm_x\cS_x$: since $\cS_x\simeq\cA_x^{\oplus r}$, we have $E_x\simeq(\cA_x/\gm_x)^{\oplus r}=\bbbk^r$. The set $E=\coprod_{x\in X} E_x$ is equipped with a natural projection $$\pi:E\to X,\qquad \xi\in E_x\mapsto \pi(\xi):=x,$$ and the fibers $E_x=\pi^{-1}(x)$ have a structure of $r$-dimensional $\bbbk$-vector space: such a structure $E$ is called a {\it $\bbbk$-vector bundle of rank $r$} over~$X$. Every section $s\in \cS(U)$ gives rise to a {\it section} of $E$ over $U$ by setting $s(x)=s_x$~mod~$\gm_x$. We obtain a function (still denoted by the same symbol) $s:U\to E$ such that $s(x)\in E_x$ for every $x\in U$, i.e.\ $\pi\circ s=\Id_U$. It is clear that $\cS(U)$ can be considered as a $\cA(U)$-submodule of the $\bbbk$-vector space of functions $U\to E$ mapping a point $x\in U$ to an element in the fiber~$E_x$. Thus we get a subsheaf of the sheaf of $E$-valued sections, which is in a natural way a $\cA$-module isomorphic to~$\cS$. This subsheaf will be denoted by $\cA(E)$ and will be called the {\it sheaf of $\cA$-sections} of~$E$. If we are given a $\bbbk$-vector bundle $E$ over $X$ and a subsheaf $\cS=\cA(E)$ of the sheaf of all sections of $E$ which is in a natural way a locally free $\cA$-module of rank~$r$, we say that $E$ (or more precisely the pair $(E,\cA(E))$) is a $\cA$-vector bundle of rank~$r$ over~$X$. \begstat{(3.3) Example} \rm In case $\cA=\cC_{X,\bbbk}$ is the sheaf of all $\bbbk$-valued continuous functions on~$X$, we say that $E$ is a {\it topological} vector bundle over~$X$. When $X$ is a manifold and $\cA=\cC^p_{X,\bbbk}$, we say that $E$ is a {\it $C^p$-differentiable} vector bundle; finally, when $X$ is complex analytic and $\cA=\cO_X$, we say that $E$ is a {\it holomorphic} vector bundle. \endstat Let us introduce still a little more notation. Since $\cA(E)$ is a locally free sheaf of rank $r$ over any open set $U_\alpha$ in a suitable covering of $X$, a choice of generators $(F_\alpha^1\ld F_\alpha^r)$ for $\cA(E)_{\restriction U_\alpha}$ yields corresponding generators $(e_\alpha^1(x)\ld e_\alpha^r(x))$ of the fibers~$E_x$ over~$\bbbk$. Such a system of gene\-rators is called a {\it $\cA$-admissible frame} of $E$ over~$U_\alpha$. There is a corresponding isomorphism $$\theta_\alpha:E_{\restriction U_\alpha}:=\pi^{-1}(U_\alpha)\lra U_\alpha\times\bbbk^r\leqno(3.4)$$ which to each $\xi\in E_x$ associates the pair $(x,(\xi_\alpha^1\ld\xi_\alpha^r))\in U_\alpha\times\bbbk^r$ composed of $x$ and of the components $(\xi_\alpha^j)_{1\le j\le r}$ of $\xi$ in the basis $(e_\alpha^1(x)\ld e_\alpha^r(x))$ of~$E_x$. The bundle $E$ is said to be {\it trivial} if it is of the form $X\times\bbbk^r$, which is the same as saying that $\cA(E)=\cA^{\oplus r}$. For this reason, the isomorphisms $\theta_\alpha$ are called {\it trivializations} of $E$ over~$U_\alpha$. The corresponding {\it transition automorphisms} are $$\eqalign{ &\theta_{\alpha\beta}:=\theta_\alpha\circ\theta_\beta^{-1}: (U_\alpha\cap U_\beta)\times \bbbk^r\lra (U_\alpha\cap U_\beta)\times \bbbk^r,\cr &\theta_{\alpha\beta}(x,\xi)=(x,g_{\alpha\beta}(x)\cdot \xi), \qquad (x,\xi)\in (U_\alpha\cap U_\beta)\times \bbbk^r,\cr}$$ where $(g_{\alpha\beta})\in\GL_r(\cA)(U_\alpha\cap U_\beta)$ are the transition matrices already described (except that they are just seen as matrices with coefficients in $\bbbk$ rather than with coefficients in a sheaf). Conversely, if we are given a collection of matrices $g_{\alpha\beta}=(g_{\alpha\beta}^{jk})\in\GL_r(\cA)(U_\alpha\cap U_\beta)$ satisfying the transition relation $$g_{\alpha\gamma}=g_{\alpha\beta}g_{\beta\gamma}\qquad\hbox{\rm on}\quad U_\alpha\cap U_\beta\cap U_\gamma,$$ we can define a $\cA$-vector bundle $$E=\big(\coprod_{\alpha\in I}U_\alpha\times\bbbk^r\big)/\sim$$ by gluing the charts $U_\alpha\times\bbbk^r$ via the identification $(x_\alpha,\xi_\alpha)\sim(x_\beta,\xi_\beta)$ if and only if $x_\alpha=x_\beta=x\in U_\alpha\cap U_\beta$ and $\xi_\alpha=g_{\alpha\beta}(x)\cdot\xi_\beta$. \begstat{(3.5) Example} \rm When $X$ is a real differentiable manifold, an interesting example of real vector bundle is the {\it tangent bundle} $T_X$~; if $\tau_\alpha:U_\alpha\to\bbbr^n$ is a collection of coordinate charts on $X$, then \hbox{$\theta_\alpha=\pi\times d\tau_\alpha:T_{X\restriction U_\alpha}\to U_\alpha\times\bbbr^m$} define trivializations of $T_X$ and the transition matrices are given by $g_{\alpha\beta}(x)= d\tau_{\alpha\beta}(x^\beta)$ where $\tau_{\alpha\beta}= \tau_\alpha\circ\smash{\tau_\beta^{-1}}$ and $x^\beta=\tau_\beta(x)$. The dual $T^\star_X$ of $T_X$ is called the {\it cotangent bundle} of~$X$. If $X$ is complex analytic, then $T_X$ has the structure of a holomorphic vector bundle. \endstat We now briefly discuss the concept of sheaf and bundle morphisms. If $\cS$ and $\cS'$ are sheaves of $\cA$-modules over a topological space~$X$, then by a morphism $\varphi:\cS\to\cS'$ we just mean a $\cA$-linear sheaf morphism. If $\cS=\cA(E)$ and $\cS'=\cA(E')$ are locally free sheaves, this is the same as a $\cA$-linear bundle morphism, that is, a fiber preserving $\bbbk$-linear morphism $\varphi(x):E_x\to E'_x$ such that the matrix representing $\varphi$ in any local $\cA$-admissible frames of $E$ and $E'$ has coefficients in~$\cA$. \begstat{(3.6) Proposition} Suppose that $\cA$ is a sheaf of local rings, i.e.\ that a section of $\cA$ is invertible in $\cA$ if and only if it never takes the zero value in~$\bbbk$. Let $\varphi:\cS\to \cS'$ be a $\cA$-morphism of locally free $\cA$-modules of rank $r$,~$r'$. If the rank of the $r'\times r$ matrix $\varphi(x)\in M_{r'r}(\bbbk)$ is constant for all $x\in X$, then $\Ker\varphi$ and $\Im\varphi$ are locally free subsheaves of $\cS$, $\cS'$ respectively, and $\Coker\varphi=\cS'/\Im\varphi$ is locally free. \endstat \begproof{} This is just a consequence of elementary linear algebra, once we know that non zero determinants with coefficients in $\cA$ can be inverted. \qed \endproof Note that all three sheaves $\cC_{X,\bbbk}$, $\cC^p_{X,\bbbk}$, $\cO_X$ are sheaves of local rings, so Prop.~3.6 applies to these cases. However, even if we work in the holomorphic category ($\cA=\cO_X$), a difficulty immediately appears that the kernel or cokernel of an arbitrary morphism of locally free sheaves is in general not locally free. \begstat{(3.7) Examples} {\rm\smallskip \item{a)} Take $X=\bbbc$, let $\cS=\cS'=\cO$ be the trivial sheaf, and let $\varphi:\cO\to\cO$ be the morphism $u(z)\mapsto z\,u(z)$. It is immediately seen that $\varphi$ is injective as a sheaf morphism ($\cO$ being an entire ring), and that $\Coker\varphi$ is the {\it skyscraper sheaf} $\bbbc_0$ of stalk $\bbbc$ at $z=0$, having zero stalks at all other points $z\ne 0$. Thus $\Coker\varphi$ is not a locally free sheaf, although $\varphi$ is everywhere injective (note however that the corresponding morphism \hbox{ $\varphi:E\to E'$}, $(z,\xi)\mapsto (z,z\xi)$ of trivial rank~$1$ vector bundles $E=E'=\bbbc\times\bbbc$ is {\it not injective} on the zero fiber $E_0$). \smallskip \item{b)} Take $X=\bbbc^3$, $\cS=\cO^{\oplus 3}$, $\cS'=\cO$ and $$\varphi:\cO^{\oplus 3}\to\cO,\qquad (u_1,u_2,u_3)\mapsto \sum_{1\le j\le 3}z_ju_j(z_1,z_2,z_3).$$ Since $\varphi$ yields a surjective bundle morphism on $\bbbc^3\ssm\{0\}$, one easily sees that $\Ker\varphi$ is locally free of rank~$2$ over $\bbbc^3\ssm\{0\}$. However, by looking at the Taylor expansion of the $u_j$'s at $0$, it is not difficult to check that $\Ker\varphi$ is the $\cO$-submodule of $\cO^{\oplus 3}$ generated by the three sections $(-z_2,z_1,0)$, $(-z_3,0,z_1)$ and $(0,z_3,-z_2)$, and that any two of these three sections cannot generate the $0$-stalk $(\Ker\varphi)_0$. Hence $\Ker\varphi$ is not locally free.\smallskip} \endstat Since the category of locally free $\cO$-modules is not stable by taking kernels or cokernels, one is led to introduce a more general category which will be stable under these operations. This leads to the notion of {\it coherent sheaves}. \titlec{\S 3.2.}{Notion of Coherence} The notion of coherence again deals with sheaves of modules over a sheaf of rings. It is a semi-local property which says roughly that the sheaf of modules locally has a finite presentation in terms of generators and relations. We describe here some general properties of this notion, before concentrating ourselves on the case of coherent $\cO_X$-modules. \begstat{(3.8) Definition} Let $\cA$ be a sheaf of rings on a topological space $X$ and $\cS$ a sheaf of modules over $\cA$ (or briefly a $\cA$-module). Then $\cS$ is said to be locally finitely generated if for every point $x_0\in X$ one can find a neighborhood $\Omega$ and sections $F_1\ld F_q\in\cS(\Omega)$ such that for every $x\in\Omega$ the stalk $\cS_x$ is generated by the germs $F_{1,x}\ld F_{q,x}$ as an $\cA_x$-module. \endstat \begstat{(3.9) Lemma} Let $\cS$ be a locally finitely generated sheaf of $\cA$-modules on $X$ and $G_1\ld G_N$ sections in $\cS(U)$ such that $G_{1,x_0}\ld G_{N,x_0}$ generate $\cS_{x_0}$ at $x_0\in U$. Then $G_{1,x}\ld G_{N,x}$ generate $\cS_x$ for $x$ near $x_0$. \endstat \begproof{} Take $F_1\ld F_q$ as in Def.~3.8. As $G_1\ld G_N$ generate $\cS_{x_0}$, one can find a neighborhood $\Omega'\subset\Omega$ of $x_0$ and $H_{jk}\in\cA(\Omega')$ such that $F_j=\sum H_{jk}G_k$ on $\Omega'$. Thus $G_{1,x}\ld G_{N,x}$ generate $\cS_x$ for all $x\in\Omega'$.\qed \endproof \titled{\S 3.2.1.}{Definition of Coherent Sheaves.} If $U$ is an open subset of $X$, we denote by $\cS_{\restriction U}$ the restriction of $\cS$ to $U$, i.e.\ the union of all stalks $\cS_x$ for $x\in U$. If $F_1\ld F_q\in\cS(U)$, the kernel of the sheaf homomorphism $F:\cA_{\restriction U}^{\oplus q}\lra\cS_{\restriction U}$ defined by $$\cA_x^{\oplus q}\ni(g^1\ld g^q)\longmapsto\sum_{1\le j\le q}g^jF_{j,x} \in\cS_x,~~~~x\in U\leqno(3.10)$$ is a subsheaf $\cR(F_1\ld F_q)$ of $\cA_{\restriction U}^{\oplus q}$, called the {\it sheaf of relations} between $F_1\ld F_q$. \begstat{(3.11) Definition} A sheaf $\cS$ of $\cA$-modules on $X$ is said to be coherent if: \smallskip \item{\rm a)} $\cS$ is locally finitely generated~$;$ \smallskip \item{\rm b)} for any open subset $U$ of $X$ and any $F_1\ld F_q\in\cS(U)$, the sheaf of relations $\cR(F_1\ld F_q)$ is locally finitely generated.\smallskip \endstat Assumption a) means that every point $x\in X$ has a neighborhood $\Omega$ such that there is a surjective sheaf morphism $F:\cA_{\restriction\Omega}^{\oplus q}\lra \cS_{\restriction \Omega}$, and assumption b) implies that the kernel of $F$ is locally finitely generated. Thus, after shrinking $\Omega$, we see that $\cS$ admits over $\Omega$ a finite presentation under the form of an exact sequence $$\cA_{\restriction\Omega}^{\oplus p}\buildo G\over\lra\cA_{\restriction\Omega}^{\oplus q} \buildo F\over\lra\cS_{\restriction\Omega}\lra 0,\leqno(3.12)$$ where $G$ is given by a $q\times p$ matrix $(G_{jk})$ of sections of $\cA(\Omega)$ whose columns $(G_{j1})\ld (G_{jp})$ are generators of $\cR(F_1\ld F_q)$. It is clear that every locally finitely generated subsheaf of a coherent sheaf is coherent. From this we easily infer: \begstat{(3.13) Theorem} Let $\varphi:\cF\lra\cG$ be a $\cA$-morphism of coherent sheaves. Then $\Im\varphi$ and $\ker\varphi$ are coherent. \endstat \begproof{} Clearly $\Im\varphi$ is a locally finitely generated subsheaf of $\cG$, so it is coherent. Let $x_0\in X$, let $F_1\ld F_q\in\cF(\Omega)$ be generators of $\cF$ on a neighborhood $\Omega$ of $x_0$, and $G_1\ld G_r\in \cA(\Omega')^{\oplus q}$ be generators of $\cR\big(\varphi(F_1)\ld\varphi(F_q)\big)$ on a neighborhood $\Omega'\subset\Omega$ of $x_0$. Then $\ker\varphi$ is generated over $\Omega'$ by the sections $$H_j=\sum_{k=1}^q G_j^kF_k\in\cF(\Omega'),~~~~1\le j\le r.\eqno\square$$ \endproof \begstat{(3.14) Theorem} Let $0\lra\cF\lra\cS\lra\cG\lra 0$ be an exact sequence of $\cA$-modules. If two of the sheaves $\cF,\cS,\cG$ are coherent, then all three are coherent. \endstat \begproof{} If $\cS$ and $\cG$ are coherent, then $\cF=\ker(\cS\to\cG)$ is coherent by Th.~3.13. If $\cS$ and $\cF$ are coherent, then $\cG$ is locally finitely generated; to prove the coherence, let $G_1\ld G_q\in\cG(U)$ and $x_0\in U$. Then there is a neighborhood $\Omega$ of $x_0$ and sections $\tilde G_1\ld\tilde G_q\in\cS(\Omega)$ which are mapped to $G_1\ld G_q$ on $\Omega$. After shrinking $\Omega$, we may assume also that $\cF_{\restriction\Omega}$ is generated by sections $F_1\ld F_p\in\cF(\Omega)$. Then $\cR(G_1\ld G_q)$ is the projection on the last $q$-components of $\cR(F_1\ld F_p,\tilde G_1\ld\tilde G_q)\subset\cA^{p+q}$, which is finitely generated near $x_0$ by the coherence of $\cS$. Hence $\cR(G_1\ld G_q)$ is finitely generated near $x_0$ and $\cG$ is coherent. Finally, assume that $\cF$ and $\cG$ are coherent. Let $x_0\in X$ be any point, let $F_1\ld F_p\in\cF(\Omega)$ and $G_1\ld G_q\in\cG(\Omega)$ be generators of $\cF$, $\cG$ on a neighborhood $\Omega$ of $x_0$. There is a neighborhood $\Omega'$ of $x_0$ such that $G_1\ld G_q$ admit liftings $\tilde G_1\ld\tilde G_q\in\cS(\Omega')$. Then $(F_1\ld F_q,\tilde G_1\ld\tilde G_q)$ generate $\cS_{\restriction\Omega'}$, so $\cS$ is locally finitely generated. Now, let $S_1\ld S_q$ be arbitrary sections in $\cS(U)$ and $\ovl S_1\ld\ovl S_q$ their images in $\cG(U)$. For any $x_0\in U$, the sheaf of relations $\cR(\ovl S_1\ld\ovl S_q)$ is generated by sections $P_1\ld P_s\in\cA(\Omega)^{\oplus q}$ on a small neighborhood $\Omega$ of $x_0$. Set $P_j=(P_j^k)_{1\le k\le q}$. Then $H_j=P_j^1S_1+\ldots+P_j^qS_q$, $1\le j\le s$, are mapped to $0$ in $\cG$ so they can be seen as sections of $\cF$. The coherence of $\cF$ shows that $\cR(H_1\ld H_s)$ has generators $Q_1\ld Q_t\in\cA(\Omega')^s$ on a small neighborhood $\Omega'\subset\Omega$ of $x_0$. Then $\cR(S_1\ld S_q)$ is generated over $\Omega'$ by $R_j=\sum Q_j^kP_k\in\cA(\Omega')$, $1\le j\le t$, and $\cS$ is coherent.\qed \endproof \begstat{(3.15) Corollary} If $\cF$ and $\cG$ are coherent subsheaves of a coherent analytic sheaf $\cS$, the intersection $\cF\cap\cG$ is a coherent sheaf. \endstat \begproof{} Indeed, the intersection sheaf $\cF\cap\cG$ is the kernel of the composite morphism $\cF\lhra\cS\lra\cS/\cG,$ and $\cS/\cG$ is coherent.\qed \endproof \titled{\S 3.2.2. Coherent Sheaf of Rings.} A sheaf of rings $\cA$ is said to be coherent if it is coherent as a module over itself. By Def.~3.11, this means that for any open set $U\subset X$ and any sections $F_j\in\cA(U)$, the sheaf of relations $\cR(F_1\ld F_q)$ is finitely generated. The above results then imply that all free modules $\cA^{\oplus p}$ are coherent. As a consequence: \begstat{(3.16) Theorem} If $\cA$ is a coherent sheaf of rings, any locally finitely gene\-rated subsheaf of $\cA^{\oplus p}$ is coherent. In particular, if $\cS$ is a coherent \hbox{\it$\cA$-module} and $F_1\ld F_q\in\cS(U)$, the sheaf of relations $\cR(F_1\ld F_q)\subset\cA^{\oplus q}$ is also coherent. \endstat Let $\cS$ be a coherent sheaf of modules over a coherent sheaf of ring $\cA$. By an iteration of construction (3.12), we see that for every integer $m\ge 0$ and every point $x\in X$ there is a neighborhood $\Omega$ of $x$ on which there is an exact sequence of sheaves $$\cA_{\restriction\Omega}^{\oplus p_m}\buildo F_m\over\lra \cA_{\restriction\Omega}^{\oplus p_{m-1}}\lra\cdots\lra \cA_{\restriction\Omega}^{\oplus p_1}\buildo F_1\over\lra \cA_{\restriction\Omega}^{\oplus p_0}\buildo F_0\over\lra \cS_{\restriction\Omega}\lra 0,\leqno(3.17)$$ where $F_j$ is given by a $p_{j-1}\times p_j$ matrix of sections in $\cA(\Omega)$. \titlec{\S 3.3.}{Analytic Sheaves and the Oka Theorem} Many properties of holomorphic functions which will be considered in this book can be expressed in terms of sheaves. Among them, analytic sheaves play a central role. The Oka theorem (Oka 1950) asserting the coherence of the sheaf of holomorphic functions can be seen as a far-reaching deepening of the noetherian property seen in Sect.~1. The theory of analytic sheaves could not be presented without it. \begstat{(3.18) Definition} Let $M$ be a $n$-dimensional complex analytic manifold and let $\cO_M$ be the sheaf of germs of analytic functions on~$M$. An analytic sheaf over $M$ is by definition a sheaf $\cS$ of modules over~$\cO_M$. \endstat \begstat{(3.19) Coherence theorem of Oka} The sheaf of rings $\cO_M$ is coherent for any complex manifold~$M$. \endstat Let $F_1\ld F_q\in\cO(U)$. Since $\cO_{M,x}$ is Noetherian, we already know that every stalk $\cR(F_1\ld F_q)_x\subset\cO_{M,x}^{\oplus q}$ is finitely generated, but the important new fact expressed by the theorem is that the sheaf of relations is locally finitely generated, namely that the ``same'' generators can be chosen to generate each stalk in a neighborhood of a given point. \begproof{} By induction on $n=\dim_\bbbc M$. For $n=0$, the stalks $\cO_{M,x}$ are equal to $\bbbc$ and the result is trivial. Assume now that $n\ge 1$ and that the result has already been proved in dimension $n-1$. Let $U$ be an open set of $M$ and $F_1\ld F_q\in\cO_M(U)$. To show that $\cR(F_1\ld F_q)$ is locally finitely generated, we may assume that $U=\Delta=\Delta'\times\Delta_n$ is a polydisk in $\bbbc^n$ centered at $x_0=0$~; after a change of coordinates and multiplication of $F_1\ld F_q$ by invertible functions, we may also suppose that $F_1\ld F_q$ are Weierstrass polynomials in $z_n$ with coefficients in $\cO(\Delta')$. We need a lemma. \endproof \begstat{(3.20) Lemma} If $x=(x',x_n)\in\Delta$, the $\cO_{\Delta,x}$-module $\cR(F_1\ld F_q)_x$ is generated by those of its elements whose components are germs of analytic polynomials in $\cO_{\Delta',x'}[z_n]$ with a degree in $z_n$ at most equal to $\mu$, the maximum of the degrees of $F_1\ld F_q$. \endstat \begproof{} Assume for example that $F_q$ is of the maximum degree $\mu$. By the Weierstrass preparation Th.~1.1 and Lemma~1.9 applied at $x$, we can write $F_{q,x}=f'f''$ where $f',f''\in\cO_{\Delta',x'} [z_n]$, $f'$ is a Weierstrass polynomial in $z_n-x_n$ and $f''(x)\ne 0$. Let $\mu'$ and $\mu''$ denote the degrees of $f'$ and $f''$ with respect to $z_n$, so $\mu'+\mu''=\mu$. Now, take $(g^1\ld g^q)\in \cR(F_1\ld F_q)_x$. The Weierstrass division theorem gives $$g^j=F_{q,x}t^j+r^j,~~~~j=1\ld q-1,$$ where $t^j\in\cO_{\Delta,x}$ and $r^j\in\cO_{\Delta',x'}[z_n]$ is a polynomial of degree $<\mu'$. For $j=q$, define $r^q=g^q+\sum_{1\le j\le q-1} t^jF_{j,x}$. We can write $$(g^1\ld g^q)=\sum_{1\le j\le q}t^j(0\ld F_q\ld 0,-F_j)_x+(r^1\ld r^q) \leqno(3.21)$$ where $F_q$ is in the $j$-th position in the $q$-tuples of the summation. Since these\break $q$-tuples are in $\cR(F_1\ld F_q)_x$, we have $(r^1\ld r^q)\in \cR(F_1\ld F_q)_x$, thus $$\sum_{1\le j\le q-1}F_{j,x}r^j+f'f''r^q=0.$$ As the sum is a polynomial in $z_n$ of degree $<\mu+\mu'$, it follows from Lemma~1.9 that $f''r^q$ is a polynomial in $z_n$ of degree $<\mu$. Now we have $$(r^1\ld r^q)=1/f''(f''r^1\ld f''r^q)$$ where $f''r^j$ is of degree $<\mu'+\mu''=\mu$. In combination with (3.21) this proves the lemma.\qed \endproof \begproof{of Theorem 3.19 (end)} If $g=(g^1\ld g^q)$ is one of the polynomials of $\cR(F_1\ld F_q)_x$ described in Lemma~3.20, we can write $$g^j=\sum_{0\le k\le\mu}u^{jk}z_n^k,~~~~u^{jk}\in\cO_{\Delta',x'}.$$ The condition for $(g^1\ld g^q)$ to belong to $\cR(F_1\ld F_q)_x$ therefore consists of $2\mu+1$ linear conditions for the germ $u=(u^{jk})$ with coefficients in $\cO(\Delta')$. By the induction hypothesis, $\cO_{\Delta'}$ is coherent and Th.~3.16 shows that the corresponding modules of relations are generated over $\cO_{\Delta',x'}$, for $x'$ in a neighborhood $\Omega'$ of $0$, by finitely many $(q\times\mu)$-tuples $U_1\ld U_N\in\cO(\Omega')^{q\mu}$. By Lemma~3.20, $\cR(F_1\ld F_q)_x$ is generated at every point $x\in\Omega=\Omega'\times\Delta_n$ by the germs of the corresponding polynomials $$G_l(z)=\Big(\sum_{0\le k\le\mu}U^{jk}_l(z')z_n^k\Big)_{1\le j\le q}, ~~~~z\in\Omega,~~~1\le l\le N.\eqno\square$$ \endproof \begstat{(3.22) Strong Noetherian property} Let $\cF$ be a coherent analytic sheaf on a complex manifold $M$ and let $\cF_1\subset\cF_2\subset \ldots$ be an increasing sequence of coherent subsheaves of~$\cF$. Then the sequence $(\cF_k)$ is stationary on every compact subset of~$M$. \endstat \begproof{} Since $\cF$ is locally a quotient of a free module $\cO_M^{\oplus q}$, we can pull back the sequence to $\cO_M^{\oplus q}$ and thus suppose $\cF=\cO_M$ (by easy reductions similar to those in the proof of Th.~3.14). Suppose $M$ connected and $\cF_{k_0}\ne\{0\}$ for some index $k_0$ (otherwise, there is nothing to prove). By the analytic continuation theorem, we easily see that $\cF_{k_0,x}\ne\{0\}$ for every $x\in M$. We can thus find a non zero Weierstrass polynomial $P\in\cF_{k_0}(V)$, ${\rm deg}_{z_n}P(z',z_n)=\mu$, in a coordinate neighborhood $V=\Delta'\times\Delta_n$ of any point $x\in M$. A division by $P$ shows that for $k\ge k_0$ and $x\in V$, all stalks $\cF_{k,x}$ are generated by $P_x$ and by polynomials of degree $<\mu$ in $z_n$ with coefficients in $\cO_{\Delta',x'}$. Therefore, we can apply induction on $n$ to the coherent $\cO_{\Delta'}$-module $$\cF'=\cF\cap\big\{Q\in\cO_{\Delta'}[z_n]\,;\,{\rm deg}\,Q \le\mu\big\}$$ and its increasing sequence of coherent subsheaves $\cF'_k= \cF_k\cap\cF'$.\qed \endproof \titleb{\S 4.}{Complex Analytic Sets. Local Properties} \titlec{\S 4.1.}{Definition. Irreducible Components} A complex analytic set is a set which can be defined locally by finitely many holomorphic equations; such a set has in general singular points, because no assumption is made on the differentials of the equations. We are interested both in the description of the singularities and in the study of algebraic properties of holomorphic functions on analytic sets. For a more detailed study than ours, we refer to H.~Cartan's seminar (Cartan 1950), to the books of (Gunning-Rossi 1965), (Narasimhan 1966) or the recent book by (Grauert-Remmert 1984). \begstat{(4.1) Definition} Let $M$ be a complex analytic manifold. A subset $A\subset M$ is said to be an analytic subset of $M$ if $A$ is closed and if for every point $x_0\in A$ there exist a neighborhood $U$ of $x_0$ and holomorphic functions $g_1\ld g_n$ in $\cO(U)$ such that $$A\cap U=\{z\in U~;~g_1(z)=\ldots=g_N(z)=0\}.$$ Then $g_1\ld g_N$ are said to be $($local$)$ equations of $A$ in $U$. \endstat It is easy to see that a finite union or intersection of analytic sets is analytic: if $(g'_j)$, $(g''_k)$ are equations of $A'$, $A''$ in the open set $U$, then the family of all products $(g'_jg''_k)$ and the family $(g'_j)\cup(g''_k)$ define equations of $A'\cup A''$ and $A'\cap A''$ respectively. \begstat{(4.2) Remark} \rm Assume that $M$ is connected. The analytic continuation theorem shows that either $A=M$ or $A$ has no interior point. In the latter case, each piece $A\cap U=g^{-1}(0)$ is the set of points where the function $\log|g|^2=\log(|g_1|^2+\cdots+|g_N|^2)\in\Psh(U)$ takes the value $-\infty$, hence $A$ is pluripolar. In particular $M\ssm A$ is connected and every function $f\in\cO(M\ssm A)$ that is locally bounded near $A$ can be extended to a function $\tilde f\in\cO(M)$.\qed \endstat We focus now our attention on local properties of analytic sets. By definition, a germ of set at a point $x\in M$ is an equivalence class of elements in the power set ${\cal P}(M)$, with $A\sim B$ if there is an open neighborhood $V$ of $x$ such that $A\cap V=B\cap V$. The germ of a subset $A\subset M$ at $x$ will be denoted by $(A,x)$. We most often consider the case when $A\subset M$ is a analytic set in a neighborhood $U$ of~$x$, and in this case we denote by $\cI_{A,x}$ the ideal of germs $f\in\cO_{M,x}$ which vanish on $(A,x)$. Conversely, if $\cJ=(g_1\ld g_N)$ is an ideal of $\cO_{M,x}$, we denote by $\big(V(\cJ),x\big)$ the germ at $x$ of the zero variety $V(\cJ)=\{z\in U~;~g_1(z)=\ldots=g_N(z)=0\}$, where $U$ is a neighborhood of $x$ such that $g_j\in\cO(U)$. It is easy to check that the germ $(V(cJ),x)$ does not depend on the choice of generators. Moreover, it is clear that $$\leqalignno{ &\hbox{\rm for every ideal $\cJ$ in the ring $\cO_{M,x}$,}\qquad~~\, \cI_{V(\cJ),x}\supset\cJ,&(4.3')\cr &\hbox{\rm for every germ of analytic set $(A,x)$,}\qquad \big(V(\cI_{A,x}),x\big)=(A,x).&(4.3'')\cr}$$ \begstat{(4.4) Definition} A germ $(A,x)$ is said to be irreducible if it has no decom\-po\-sition $(A,x)=(A_1,x)\cup(A_2,x)$ with analytic sets \hbox{$(A_j,x)\ne(A,x)$, $j=1,2$}. \endstat \begstat{(4.5) Proposition} A germ $(A,x)$ is irreducible if and only if $\cI_{A,x}$ is a prime ideal of the ring $\cO_{M,x}$. \endstat \begproof{} Let us recall that an ideal $\cJ$ is said to be {\it prime} if $fg\in\cJ$ implies $f\in\cJ$ or $g\in\cJ$. Assume that $(A,x)$ is irreducible and that $fg\in\cI_{A,x}$. As we can write $(A,x)=(A_1,x)\cup(A_2,x)$ with $A_1=A\cap f^{-1}(0)$ and $A_2=A\cap g^{-1}(0)$, we must have for example $(A_1,x)=(A,x)$~; thus $f\in\cI_{A,x}$ and $\cI_{A,x}$ is prime. Conversely, if $(A,x)=(A_1,x)\cup(A_2,x)$ with $(A_j,x)\ne(A,x)$, there exist $f\in\cI_{A_1,x}$, $g\in\cI_{A_2,x}$ such that $f,g\notin \cI_{A,x}$. However $fg\in\cI_{A,x}$, thus $\cI_{A,x}$ is not prime.\qed \endproof \begstat{(4.6) Theorem} Every decreasing sequence of germs of analytic sets $(A_k,x)$ is stationary. \endstat \begproof{} In fact, the corresponding sequence of ideals $\cJ_k=\cI_{A_k,x}$ is increa\-sing, thus $\cJ_k=\cJ_{k_0}$ for $k\ge k_0$ large enough by the Noetherian property of~$\cO_{M,x}$. Hence $(A_k,x)=\big(V(\cJ_k),x\big)$ is constant for $k\ge k_0$. This result has the following straightforward consequence:\qed \endproof \begstat{(4.7) Theorem} Every analytic germ $(A,x)$ has a finite decomposition $$(A,x)=\bigcup_{1\le k\le N}(A_k,x)$$ where the germs $(A_j,x)$ are irreducible and $(A_j,x)\not\subset(A_k,x)$ for $j\ne k$. The decomposition is unique apart from the ordering. \endstat \begproof{} If $(A,x)$ can be split in several components, we split repeatedly each component as long as one of them is reducible. The process must stop by Th.~4.6, whence the existence. For the uniqueness, assume that\break $(A,x)=\bigcup(A'_l,x)$, $1\le l\le N'$, is another decomposition. Since $(A_k,x)=\bigcup_l(A_k\cap A'_l,x)$, we must have $(A_k,x)=(A_k\cap A'_l,x)$ for some $l=l(k)$, i.e.\ $(A_k,x)\subset (A'_{l(k)},x)$, and likewise $(A'_{l(k)},x)\subset(A_j,x)$ for some~$j$. Hence $j=k$ and $(A'_{l(k)},x)=(A_k,x)$.\qed \endproof \titlec{\S 4.2.}{Local Structure of a Germ of Analytic Set} We are going to describe the local structure of a germ, both from the holomorphic and topological points of view. By the above decomposition theo\-rem, we may restrict ourselves to the case of irreducible germs Let~$\cJ$ be a prime ideal of $\cO_n=\cO_{\bbbc^n,0}$ and let $A=V(\cJ)$ be its zero variety. We set $\cJ_k=\cJ\cap\bbbc\{z_1\ld z_k\}$ for each $k=0,1\ld n$. \begstat{(4.8) Proposition} There exist an integer $d$, a basis $(e_1\ld e_n)$ of $\bbbc^n$ and associated coordinates $(z_1\ld z_n)$ with the following properties: $\cJ_d=\{0\}$ and for every integer $k=d+1\ld n$ there is a Weierstrass polynomial $P_k\in\cJ_k$ of the form $$P_k(z',z_k)=z_k^{s_k}+\sum_{1\le j\le s_k}a_{j,k}(z')\,z_k^{s_k-j},~~~~ a_{j,k}(z')\in\cO_{k-1},\leqno(4.9)$$ where $a_{j,k}(z')=O(|z'|^j)$. Moreover, the basis $(e_1\ld e_n)$ can be chosen arbitrarily close to any preassigned basis $(e^0_1\ld e^0_n)$. \endstat \begproof{} By induction on $n$. If $\cJ=\cJ_n=\{0\}$, then $d=n$ and there is nothing to prove. Otherwise, select a non zero element $g_n\in\cJ$ and a vector $e_n$ such that $\bbbc\ni w\longmapsto g_n(we_n)$ has minimum vanishing order $s_n$. This choice excludes at most the algebraic set $g_n^{(s_n)}(v)=0$, so $e_n$ can be taken arbitrarily close to $e^0_n$. Let $(\tilde z_1\ld\tilde z_{n-1},z_n)$ be the coordinates associated to the basis $(e^0_1\ld e^0_{n-1},e_n)$. After multiplication by an invertible element, we may assume that $g_n$ is a Weierstrass polynomial $$P_n(\tilde z,z_n)=z_n^{s_n}+\sum_{1\le j\le s_n}a_{j,n}(\tilde z)\,z_n^{s_n-j}, ~~~~a_{j,n}\in\cO_{n-1},$$ and $a_{j,n}(\tilde z)=O(|\tilde z|^j)$ by Remark~2.2. If $\cJ_{n-1}=\cJ\cap\bbbc\{\tilde z\}=\{0\}$ then $d=n-1$ and the construction is finished. Otherwise we apply the induction hypothesis to the ideal $\cJ_{n-1}\subset\cO_{n-1}$ in order to find a new basis $(e_1\ld e_{n-1})$ of $\Vect(e^0_1\ld e^0_{n-1})$, associated coordinates $(z_1\ld z_{n-1})$ and Weierstrass polynomials $P_k\in\cJ_k$, $d+1\le k\le n-1$, as stated in the lemma.\qed \endproof \begstat{(4.10) Lemma} If $w\in\bbbc$ is a root of $w^d+a_1w^{d-1}+\cdots+a_d=0$, $a_j\in\bbbc$, then $|w|\le 2\max|a_j|^{1/j}$. \endstat \begproof{} Otherwise $|w|>2|a_j|^{1/j}$ for all $j=1\ld d$ and the given equation $-1=a_1/w+\cdots+a_d/w^d$ implies $1\le 2^{-1}+\cdots+2^{-d}$, a contradiction.\qed \endproof \begstat{(4.11) Corollary} Set $z'=(z_1\ld z_d)$, $z''=(z_{d+1}\ld z_n)$, and let $\Delta'$ in $\bbbc^d$, $\Delta''$ in $\bbbc^{n-d}$ be polydisks of center $0$ and radii $r',r''>0$. Then the germ $(A,0)$ is contained in a cone $|z''|\le C|z'|$, $C=\hbox{\it constant}$, and the restriction of the projection map $\bbbc^n\lra\bbbc^d$, $(z',z'')\longmapsto z'~:$ $$\pi:A\cap(\Delta'\times\Delta'')\lra\Delta'$$ is proper if $r''$ is small enough and $r'\le r''/C$. \endstat \begproof{} The polynomials $P_k(z_1\ld z_{k-1}\,;\,z_k)$ vanish on $(A,0)$. By Lemma~4.10 and (4.9), every point $z\in A$ sufficiently close to $0$ satisfies $$|z_k|\le C_k(|z_1|+\cdots+|z_{k-1}|),~~~~d+1\le k\le n,$$ thus $|z''|\le C|z'|$ and the Corollary follows.\qed \endproof Since $\cJ_d=\{0\}$, we have an injective ring morphism $$\cO_d=\bbbc\{z_1\ld z_d\}\lhra\cO_n/\cJ.\leqno(4.12)$$ \begstat{(4.13) Proposition} $\cO_n/\cJ$ is a finite integral extension of $\cO_d$. \endstat \begproof{} Let $f\in\cO_n$. A division by $P_n$ yields $f=P_nq_n+R_n$ with a remainder $R_n\in\cO_{n-1}[z_n]$, $\deg_{z_n}R_nk$. Hence $$f=R_d+\sum_{d+1\le k\le n}P_kq_k=R_d~~~\hbox{\rm mod}~~(P_{d+1}\ld P_n) \subset\cJ\leqno(4.14)$$ and $\cO_n/\cJ$ is finitely generated as an $\cO_d$-module by the family of monomials $z_{d+1}^{\alpha_{d+1}}\ldots z_n^{\alpha_n}$ with $\alpha_jd}c_kz_k$. If $\Delta',\Delta''$ are polydisks of sufficiently small radii $r',r''$ and if $r'\le r''/C$ with $C$ large, the projection map $\pi:A\cap(\Delta'\times\Delta'')\lra\Delta'$ is a ramified covering with $q$ sheets, whose ramification locus is contained in $S=\{z'\in\Delta';\delta(z')=0\}$. This means that: \smallskip \item{\rm a)} the open subset $A_S=A\cap\big((\Delta'\ssm S) \times\Delta''\big)$ is a smooth $d$-dimensional manifold, dense in $A\cap(\Delta'\times\Delta'')~;$ \smallskip \item{\rm b)} $\pi:A_S\lra\Delta'\ssm S$ is a covering~$;$ \smallskip \item{\rm c)} $\,$the fibers $\pi^{-1}(z')$ have exactly $q$ elements if $z'\notin S$ and at most $q$ if $z'\in S.$ \smallskip \noindent Moreover, $A_S$ is a connected covering of $\Delta'\ssm S$, and $A\cap(\Delta'\times\Delta'')$ is contained in a cone $|z''|\le C|z'|$~ $($see Fig. 1$)$. \endstat \Input epsfiles/fig_2_1.tex \vskip0mm \centerline{{\bf Fig.~II-1} Ramified covering from $A$ to $\Delta'\subset\bbbc^p$.} \vskip6mm \begproof{} After a linear change in the coordinates $z_{d+1}\ld z_n$, we may assume $u(z'')=z_{d+1}$, so $W_u=W_{d+1}$ and $B_{d+1}(z'\,;T)=\delta(z')T$. By Lemma~4.18, we have $$\cG=\big(W_{d+1}(z',z_{d+1})\,,\,\delta(z')z_k-B_k(z',z_{d+1})\big)_{k\ge d+2} \subset\cJ,~~~~\delta^m\cJ\subset\cG.$$ We can thus find a polydisk $\Delta=\Delta'\times\Delta''$ of sufficiently small radii $r',r''$ such that $V(\cJ)\subset V(\cG)\subset V(\delta^m\cJ)$ in~$\Delta$. As $V(\cJ)=A$ and $V(\delta)\cap\Delta=S\times \Delta''$, this implies $$A\cap\Delta\subset V(\cG)\cap\Delta\subset(A\cap\Delta)\cup (S\times\Delta'').$$ In particular, the set $A_S=A\cap\big((\Delta'\ssm S)\times\Delta''\big)$ lying above $\Delta'\ssm S$ coincides with $V(\cG)\cap\big((\Delta'\ssm S)\times\Delta''\big)$, which is the set of points $z\in\Delta$ parametrized by the equations $$\cases{ \delta(z')\ne 0,~~~W_{d+1}(z',z_{d+1})=0,\cr z_k=B_k(z',z_{d+1})/\delta(z'),~~d+2\le k\le n.\cr}\leqno(4.20)$$ As $\delta(z')$ is the resultant of $W_{d+1}$ and $\partial W_{d+1}/\partial T$, we have $$\partial W_{d+1}/\partial T(z',z_{d+1})\ne 0~~~~\hbox{\rm on}~~A_S.$$ The implicit function theorem shows that $z_{d+1}$ is locally a holomorphic function of $z'$ on $A_S$, and the same is true for $z_k=B_k(z',z_{d+1})/\delta(z')$, $k\ge d+2$. Hence $A_S$ is a smooth manifold, and for $r'\le r''/C$ small, the projection map $\pi:A_S\lra\Delta'\ssm S$ is a proper local diffeomorphism; by (4.20) the fibers $\pi^{-1}(z')$ have at most $q$ points corresponding to some of the $q$ roots $w$ of $W_{d+1}(z'\,;w)=0$. Since $\Delta'\ssm S$ is connected (Remark~4.2), either $A_S=\emptyset$ or the map $\pi$ is a covering of constant sheet number $q'\le q$. However, if $w$ is a root of $W_{d+1}(z',w)=0$ with $z'\in\Delta'\ssm S$ and if we set $z_{d+1}=w$, $z_k=B_k(z',w)/\delta(z')$, $k\ge d+2$, relation $(4.17)$ shows that $W_k(z',z_k)=0$, in particular $|z_k|=O(|z'|^{1/q})$ by Lemma~4.10. For $z'$ small enough, the $q$ points $z=(z',z'')$ defined in this way lie in $\Delta$, thus $q'=q$. Property b) and the first parts of a) and c) follow. Now, we need the following lemma. \endproof \begstat{(4.21) Lemma} If $\cJ\subset\cO_n$ is prime and $A=V(\cJ)$, then $\cI_{A,0}=\cJ$. \endstat \begproof It is obvious that $\cI_{A,0}\supset\cJ$. Now, for any $f\in\cI_{A,0}$, Prop.~4.13 implies that $\tilde f$ satisfies in $\cO_n/\cI$ an irreducible equation $$f^r+b_1(z')\,f^{r-1}+\cdots+b_r(z')=0~~~~\hbox{\rm (mod}~\cJ).$$ Then $b_r(z')$ vanishes on $(A,0)$ and the first part of c) gives $b_r=0$ on $\Delta'\ssm S$. Hence $\tilde b_r=0$ and the irreducibility of the equation of~$\tilde f$ implies $r=1$, so \hbox{$f\in\cJ$,} as desired.\qed \endproof \begproof{of Theorem 4.19 (end).} It only remains to prove that $A_S$ is connected and dense in $A\cap\Delta$ and that the fibers $\pi^{-1}(z')$, $z'\in S$, have at most $q$ elements. Let $A_{S,1}\ld A_{S,N}$ be the connected components of $A_S$. Then $\pi:A_{S,j}\lra\Delta'\ssm S$ is a covering with $q_j$ sheets, $q_1+\cdots+q_N=q$. For every point $\zeta'\in\Delta'\ssm S$, there exists a neighborhood $U$ of $\zeta'$ such that $A_{S,j}\cap\pi^{-1}(U)$ is a disjoint union of graphs $z''=g_{j,k}(z')$ of analytic functions $g_{j,k}\in \cO(U)$, $1\le k\le q_j$. If $\lambda(z'')$ is an arbitrary linear form in $z_{d+1}\ld z_n$ and $z'\in\Delta'\ssm S$, we set $$P_{\lambda,j}(z'\,;T)=\prod_{\{z''\,;\,(z',z'')\in A_{S,j}\}}\big(T-\lambda (z'')\big)=\prod_{1\le k\le k_j}\big(T-\lambda\circ g_{j,k}(z')\big).$$ This defines a polynomial in $T$ with bounded analytic coefficients on $\Delta'\ssm S$. These coefficients have analytic extensions to $\Delta'$ (Remark~4.2), thus $P_{\lambda,j}\in\cO(\Delta')[T]$. By construction, $P_{\lambda,j}\big(z'\,;\lambda(z'')\big)$ vanishes identically on $A_{S,j}$. Set $$P_\lambda=\prod_{1\le j\le N}P_{\lambda,j},~~~~ f(z)=\delta(z')\,P_\lambda\big(z'\,;\lambda(z'')\big)~;$$ $f$ vanishes on $A_{S,1}\cup\ldots\cup A_{S,N}\cup(S\times\Delta'')\supset A\cap\Delta$. Lemma~4.21 shows that $\cI_{A,0}$ is prime; as $\delta\notin\cI_{A,0}$, we get $P_{\lambda,j}\big(z'\,;\lambda(z'')\big)\in\cI_{A,0}$ for some~$j$. This is a contradiction if $N\ge 2$ and if $\lambda$ is chosen in such a way that $\lambda$ separates the $q$ points $z''_\nu$ in each fiber $\pi^{-1}(z'_\nu)$, for a sequence $z'_\nu\to 0$ in $\Delta'\ssm S$. Hence $N=1$, $A_S$ is connected, and for every $\lambda\in(\bbbc^{n-d})^\star$ we have $P_\lambda\big(z',\lambda(z'')\in\cI_{(A,0)}$. By construction $P_\lambda\big(z',\lambda(z'')\big)$ vanishes on $A_S$, so it vanishes on $\ovl A_S$~; hence, for every $z'\in S$, the fiber $\ovl A_S\cap\pi^{-1}(z')$ has at most $q$ elements, otherwise selecting $\lambda$ which separates $q+1$ of these points would yield $q+1$ roots $\lambda(z'')$ of $P_\lambda(z'\,;T)$, a contradiction. Assume now that $A_S$ is not dense in $A\cap\Delta$ for arbitrarily small polydisks $\Delta$. Then there exists a sequence $A\ni z_\nu=(z'_\nu,z''_\nu)\to 0$ such that $z'_\nu\in S$ and $z''_\nu$ is not in $F_\nu:=\hbox{\rm pr}''\big(\ovl A_S\cap \pi^{-1}(z'_\nu)\big)$. The continuity of the roots of the polynomial $P_\lambda(z'\,;T)$ as $\Delta'\ssm S\ni z'\to z'_\nu$ implies that the set of roots of $P_\lambda(z'_\nu\,;T)$ is $\lambda(F_\nu)$. Select $\lambda$ such that $\lambda(z''_\nu)\notin\lambda(F_\nu)$ for all $\nu$. Then $P_\lambda\big(z'_\nu\,;\lambda(z''_\nu)\big)\ne 0$ for every $\nu$ and $P_\lambda\big(z'\,;\lambda(z'')\big)\notin\cI_{A,0}$, a contradiction.\qed \endproof At this point, it should be observed that many of the above statements completely fail in the case of real analytic sets. In $\bbbr^2$, for example, the prime ideal $\cJ=(x^5+y^4)$ defines an irreducible germ of curve $(A,0)$ and there is an injective integral extension of rings $\bbbr\{x\}\lhra\bbbr\{x,y\}/\cJ$ of degree 4; however, the projection of $(A,0)$ on the first factor, $(x,y)\mapsto x$, has not a constant sheet number near $0$, and this number is not related to the degree of the extension. Also, the prime ideal $\cJ=(x^2+y^2)$ has an associated variety $V(\cJ)$ reduced to $\{0\}$, hence $\cI_{A,0}=(x,y)$ is strictly larger than $\cJ$, in contrast with Lemma~4.21. Let us return to the complex situation, which is much better behaved. The result obtained in Lemma~4.21 can then be extended to non prime ideals and we get the following important result: \begstat{(4.22) Hilbert's Nullstellensatz} For every ideal $\cJ\subset\cO_n$ $$\cI_{V(\cJ),0}=\sqrt{\cJ},$$ where $\sqrt{\cJ}$ is the radical of $\cJ$, i.e.\ the set of germs $f\in\cO_n$ such that some power $f^k$ lies in $\cJ$. \endstat \begproof{} Set $B=V(\cJ)$. If $f^k\in\cJ$, then $f^k$ vanishes on $(B,0)$ and $f\in\cI_{B,0}$. Thus $\sqrt{\cJ}\subset\cI_{B,0}$. Conversely, it is well known that $\sqrt{\cJ}$ is the intersection of all prime ideals $\cP\supset\cJ$ (Lang~1965, Chapter~VI). For such an ideal $(B,0)=\big(V(\cJ),0)\supset\big(V(\cP),0\big)$, thus $\cI_{B,0}\subset\cI_{V(\cP),0}=\cP$ in view of Lemma~4.21. Therefore $\cI_{B,0}\subset\bigcap_{\cP\supset\cJ}\cP=\sqrt{\cJ}$ and the Theorem is proved.\qed \endproof In other words, if a germ $(B,0)$ is defined by an arbitrary ideal $\cJ\subset\cO_n$ and if $f\in\cO_n$ vanishes on $(B,0)$, then some power $f^k$ lies in $\cJ$. \titlec{\S 4.3.}{Regular and Singular Points. Dimension} The above powerful results enable us to investigate the structure of singularities of an analytic set. We first give a few definitions. \begstat{(4.23) Definition} Let $A\subset M$ be an analytic set and $x\in A$. We say that $x\in A$ is a regular point of $A$ if $A\cap\Omega$ is a $\bbbc$-analytic submanifold of $\Omega$ for some neighborhood $\Omega$ of $x$. Otherwise $x$ is said to be singular. The corresponding subsets of $A$ will be denoted respectively $A_\reg$ and $A_\sing$. \endstat It is clear from the definition that $A_\reg$ is an open subset of $A$ (thus $A_\sing$ is closed), and that the connected components of $A_\reg$ are $\bbbc$-analytic submanifolds of $M$ (non necessarily closed). \begstat{(4.24) Proposition} If $(A,x)$ is irreducible, there exist arbitrarily small neighborhoods $\Omega$ of $x$ such that $A_\reg\cap\Omega$ is dense and connected in $A\cap\Omega$. \endstat \begproof{} Take $\Omega=\Delta$ as in Th.~4.19. Then $A_S\subset A_\reg\cap\Omega\subset A\cap\Omega$, where $A_S$ is connected and dense in $A\cap\Omega$~; hence $A_\reg\cap\Omega$ has the same properties.\qed \endproof \begstat{(4.25) Definition} The dimension of an irreducible germ of analytic set $(A,x)$ is defined by $\dim(A,x)=\dim(A_\reg,x)$. If $(A,x)$ has several irreducible components $(A_l,x)$, we set $$\dim (A,x)=\max\{\dim(A_l,x)\},~~~~\codim(A,x)=n-\dim(A,x).$$ \endstat \begstat{(4.26) Proposition} Let $(B,x)\subset(A,x)$ be germs of analytic sets. If $(A,x)$ is irreducible and $(B,x)\ne(A,x)$, then $\dim(B,x)<\dim(A,x)$ and $B\cap\Omega$ has empty interior in $A\cap\Omega$ for all sufficiently small neighborhoods $\Omega$ of $x$. \endstat \begproof{} We may assume $x=0$, $(A,0)\subset(\bbbc^n,0)$ and $(B,0)$ irreducible. Then $\cI_{A,0} \subset\cI_{B,0}$ are prime ideals. When we choose suitable coordinates for the ramified coverings, we may at each step select vectors $e_n,e_{n-1},\ldots$ that work simultaneously for $A$ and $B$. If $\dim B=\dim A$, the process stops for both at the same time, i.e.\ we get ramified coverings $$\pi:A\cap(\Delta'\times\Delta'')\lra\Delta',~~~~ \pi:B\cap(\Delta'\times\Delta'')\lra\Delta'$$ with ramification loci $S_A,S_B$. Then $B\cap\big((\Delta'\ssm (S_A\cup S_B))\times\Delta''\big)$ is an open subset of the manifold $A_S=A\cap\big((\Delta'\ssm S_A)\times\Delta''\big)$, therefore $B\cap A_S$ is an analytic subset of $A_S$ with non empty interior. The same conclusion would hold if $B\cap\Delta$ had non empty interior in $A\cap\Delta$. As $A_S$ is connected, we get $B\cap A_S=A_S$, and as $B\cap\Delta$ is closed in $\Delta$ we infer $B\cap\Delta\supset\ovl A_S=A\cap\Delta$, hence $(B,0)=(A,0)$, in contradiction with the hypothesis.\qed \endproof \begstat{(4.27) Example: parametrization of curves} {\rm Suppose that $(A,0)$ is an irreducible germ of curve ($\dim(A,0)=1$). If the disk $\Delta'\subset\bbbc$ is chosen so small that $S=\{0\}$, then $A_S$ is a connected covering of $\Delta'\ssm\{0\}$ with $q$ sheets. Hence, there exists a covering isomorphism between $\pi$ and the standard covering $$\bbbc\supset\Delta(r)\ssm\{0\}\lra \Delta(r^q)\ssm\{0\},~~~~ t\longmapsto t^q,~~~~r^q=\hbox{\rm radius of~~}\Delta',$$ i.e.\ a map $\gamma:\Delta(r)\ssm\{0\}\lra A_S$ such that $\pi\circ\gamma(t)=t^q$. This map extends into a bijective holomorphic map $\gamma:\Delta(r)\lra A\cap\Delta$ with $\gamma(0)=0$. This means that every irreducible germ of curve can be parametrized by a bijective holomorphic map defined on a disk in $\bbbc$ (see also Exercise~10.8).} \endstat \titlec{\S 4.4.}{Coherence of Ideal Sheaves} Let $A$ be an analytic set in a complex manifold $M$. The {\it sheaf of ideals} $\cI_A$ is the subsheaf of $\cO_M$ consisting of germs of holomorphic functions on $M$ which vanish on $A$. Its stalks are the ideals $\cI_{A,x}$ already considered; note that $\cI_{A,x}=\cO_{M,x}$ if $x\notin A$. If $x\in A$, we let $\cO_{A,x}$ be the ring of germs of functions on $(A,x)$ which can be extended as germs of holomorphic functions on $(M,x)$. By definition, there is a surjective morphism $\cO_{M,x}\lra\cO_{A,x}$ whose kernel is $\cI_{A,x}$, thus $$\cO_{A,x}=\cO_{M,x}/\cI_{A,x},~~~~\forall x\in A,\leqno(4.28)$$ i.e.\ $\cO_A=(\cO_M/\cI_A)_{\restriction A}$. Since $\cI_{A,x}=\cO_{M,x}$ for $x\notin A$, the quotient sheaf $\cO_M/\cI_A$ is zero on $M\ssm A$. \begstat{(4.29) Theorem {\rm (Cartan 1950)}} For any analytic set $A\subset M$, the sheaf of ideals $\cI_A$ is a coherent analytic sheaf. \endstat \begproof{} It is sufficient to prove the result when $A$ is an analytic subset in a neighborhood of $0$ in $\bbbc^n$. If $(A,0)$ is not irreducible, there exists a neighborhood $\Omega$ such that $A\cap\Omega=A_1\cup\ldots\cup A_N$ where $A_k$ are analytic sets in $\Omega$ and $(A_k,0)$ is irreducible. We have $\cI_{A\cap\Omega}=\bigcap\cI_{A_k}$, so by Cor.~3.15 we may assume that $(A,0)$ is irreducible. Then we can choose coordinates \hbox{$z'$, $z''$}, polydisks $\Delta',\Delta''$ and a primitive element $u(z'')=c_{d+1} z_{d+1}+\cdots+c_nz_n$ such that Th.~4.19 is valid. Since $\delta(z')= \prod_{jp$ for all $x\in Z$. Then the closure $\ovl Z$ of $Z$ in $X$ is an analytic subset. \endstat \begstat{(8.8) Remmert's proper mapping theorem} Let $F:X\lra Y$ be a proper holomorphic map. Then $F(X)$ is an analytic subset of $Y$. \endstat \begproof{} We let $(8.7_m)$ denote statement (8.7) for $\dim Z\le m$ and $(8.8_m)$ denote statement (8.8) for $\dim X\le m$. We proceed by induction on $m$ in two steps:\medskip \noindent{\it Step 1.} $(8.7_m)$ and $(8.8_{m-1})$ imply $(8.8_m)$.\newline \noindent{\it Step 2.} $(8.8_{m-1})$ implies $(8.7_m)$. \medskip \noindent As $(8.8_m)$ is obvious for $m=0$, our statements will then be valid for all $m$, i.e.\ for all complex spaces of bounded dimension. However, Th.~8.7 is local on $X$ and Th.~8.8 is local on $Y$, so the general case is immediately reduced to the finite dimensional case. \medskip \noindent{\it Proof of step 1.} The analyticity of $F(X)$ is a local question in $Y$. Since $F:F^{-1}(U)\lra U$ is proper for any open set $U\subset Y$ and $F^{-1}(U)\compact X$ if $U\compact Y$, we may suppose that $Y$ is embedded in an open set $\Omega\subset\bbbc^n$ and that $X$ only has finitely many irreducible components $X_\alpha$. Then we have $F(X)=\bigcup F(X_\alpha)$ and we are reduced to the case when $X$ is irreducible, $\dim X=m$ and $Y=\Omega$. First assume that $X$ is a manifold and that the rank of $dF$ is constant. The constant rank theorem implies that every point in $X$ has a neighborhood $V$ such that $F(V)$ is a closed submanifold in a neighborhood $W$ of $F(x)$ in $Y$. For any point $y\in Y$, the fiber $F^{-1}(y)$ can be covered by finitely many neighborhoods $V_j$ of points $x_j\in F^{-1}(y)$ such that $F(V_j)$ is a closed submanifold in a neighborhood $W_j$ of $y$. Then there is a neighborhood of $y$ $W\subset\bigcap W_j$ such that $F^{-1}(W)\subset\bigcup V_j$, so $F(X)\cap W=\bigcup F(V_j)\cap W$ is a finite union of closed submanifolds in $W$ and $F(X)$ is analytic in $Y$. Now suppose that $X$ is a manifold, set $r=\ovl\rho(F)$ and let $Z\subset X$ be the analytic subset of points $x$ where the rank of $dF_x$ is $s}\lambda_jz_j$ be a linear form on $\bbbc^n$ in the coordinates of index $j>s$. For $z'\in B'\ssm(A'\cup S'_1\cup S'_2)$, we let $\sigma_j(z')$ be the elementary symmetric functions in the $q$ complex numbers $\lambda(z)$ corresponding to $z\in\pi^{-1}(z')$. Then these functions can be extended as bounded holomorphic functions on $B'$ and we get a polynomial $P_\lambda(z'\,;\,T)$ such that $P_\lambda\big(z'\,;\,\lambda(z'')\big)$ vanishes identically on $Z\ssm\pi^{-1}(A'\cup S'_1\cup S'_2)$. Since $\pi$ is finite, $Z\cap\pi^{-1}(A'\cup S'_1\cup S'_2)$ is a union of three (non necessarily closed) analytic subsets of dimension $\le s-1$, thus has empty interior in $Z$. It follows that the closure $\ovl Z\cap(B'\times B'')$ is contained in the analytic set $W\subset B'\times B''$ equal to the common zero set of all functions $P_\lambda\big(z'\,;\,\lambda(z'')\big)$. Moreover, by construction, $$Z\ssm\pi^{-1}(A'\cup S'_1\cup S'_2)= W\ssm\pi^{-1}(A'\cup S'_1\cup S'_2).$$ As in the proof of Cor.~5.4, we easily conclude that $\ovl Z\cap(B'\times B'')$ is equal to the union of all irreducible components of $W$ that are not contained in $\pi^{-1}(A'\cup S'_1\cup S'_2)$. Hence $\ovl Z$ is analytic.\qed\medskip Finally, we give two interesting applications of the Remmert-Stein theorem. We assume here that the reader knows what is the complex projective space $\bbbp^n$. For more details, see Sect. 5.15. \begstat{(8.10) Chow's theorem {\rm(Chow 1949)}} Let $A$ be an analytic subset of the complex projective space $\bbbp^n$. Then $A$ is algebraic, i.e.\ $A$ is the common zero set of finitely many homogeneous polynomials $P_j(z_0\ld z_n)$, $1\le j\le N$. \endstat \begproof{} Let $\pi:\bbbc^{n+1}\ssm\{0\}\lra\bbbp^n$ be the natural projection and $Z=\pi^{-1}(A)$. Then $Z$ is an analytic subset of $\bbbc^{n+1}\ssm\{0\}$ which is invariant by homotheties and $\dim Z=\dim A+1\ge 1$. The Remmert-Stein theorem implies that $\ovl Z=Z\cup\{0\}$ is an analytic subset of $\bbbc^{n+1}$. Let $f_1\ld f_N$ be holomorphic functions on a small polydisk $\Delta\subset\bbbc^{n+1}$ of center $0$ such that $\ovl Z\cap\Delta=\bigcap f_j^{-1}(0)$. The Taylor series at $0$ gives an expansion $f_j=\sum_{k=0}^{+\infty}P_{j,k}$ where $P_{j,k}$ is a homogeneous polynomial of degree $k$. We claim that $\ovl Z$ coincides with the common zero $W$ set of the polynomials $P_{j,k}$. In fact, we clearly have $W\cap\Delta \subset\bigcap f_j^{-1}(0)=\ovl Z\cap\Delta$. Conversely, for $z\in\ovl Z\cap\Delta$, the invariance of $Z$ by homotheties shows that $f_j(tz)=\sum P_{j,k}(z)t^k$ vanishes for every complex number $t$ of modulus $<1$, so all coefficients $P_{j,k}(z)$ vanish and $z\in W\cap\Delta$. By homogeneity $\ovl Z=W$~; since $\bbbc[z_0\ld z_n]$ is Noetherian, $W$ can be defined by finitely many polynomial equations.\qed \endproof \begstat{(8.11) E.E.\ Levi's continuation theorem} Let $X$ be a normal complex space and $A$ an analytic subset such that $\dim(A,x)\le\dim(X,x)-2$ for all $x\in A$. Then every meromorphic function on $X\ssm A$ has a meromorphic extension to $X$. \endstat \begproof{} We may suppose $X$ irreducible, $\dim X=n$. Let $f$ be a meromorphic function on $X\ssm A$. By Th.~6.13, the pole set $P_f$ has pure dimension $(n-1)$, so the Remmert-Stein theorem implies that $\ovl P_f$ is analytic in $X$. Fix a point $x\in A$. There is a connected neighborhood $V$ of $x$ and a non zero holomorphic function $h\in\cO_X(V)$ such that $\ovl P_f\cap V$ has finitely many irreducible components $\ovl P_{f,j}$ and $\ovl P_f\cap V\subset h^{-1}(0)$. Select a point $x_j$ in $\ovl P_{f,j}\ssm(X_\sing\cup(\ovl P_f)_\sing\cup A)$. As $x_j$ is a regular point on $X$ and on $\ovl P_f$, there is a local coordinate $z_{1,j}$ at $x_j$ defining an equation of $\ovl P_{f,j}$, such that $z_{1,j}^{m_j}f\in\cO_{X,x_j}$ for some integer $m_j$. Since $h$ vanishes along $P_f$, we have $h^{m_j}f\in\cO_{X,x}$. Thus, for $m=\max\{m_j\}$, the pole set $P_g$ of $g=h^m f$ in $V\ssm A$ does not contain $x_j$. As $P_g$ is $(n-1)$-dimensional and contained in $P_f\cap V$, it is a union of irreducible components $\ovl P_{f,j}\ssm A$. Hence $P_g$ must be empty and $g$ is holomorphic on $V\ssm A$. By Cor.~7.8, $g$ has an extension to a holomorphic function $\tilde g$ on $V$. Then $\tilde g/h^m$ is the required meromorphic extension of $f$ on~$V$.\qed \endproof \titleb{\S 9.}{Complex Analytic Schemes} Our goal is to introduce a generalization of the notion of complex space given in Def.~5.2. A complex space is a space locally isomorphic to an analytic set $A$ in an open subset $\Omega\subset\bbbc^n$, together with the sheaf of rings $\cO_A=(\cO_\Omega/\cI_A)_{\restriction A}$. Our desire is to enrich the structure sheaf $\cO_A$ by replacing $\cI_A$ with a possibly smaller ideal $\cJ$ defining the same zero variety $V(\cJ)=A$. In this way holomorphic functions are described not merely by their values on $A$, but also possibly by some ``transversal derivatives'' along $A$. \titlec{\S 9.1.}{Ringed Spaces} We start by an abstract notion of ringed space on an arbitrary topological space. \begstat{(9.1) Definition} A ringed space is a pair $(X,\cR_X)$ consisting of a topolo\-gical space $X$ and of a sheaf of rings $\cR_X$ on $X$, called the structure sheaf. A~morphism $$F:(X,\cR_X)\lra(Y,\cR_Y)$$ of ringed spaces is a pair $(f,F^\star)$ where $f:X\lra Y$ is a continuous map and $$F^\star~:~~f^{-1}\cR_Y\lra\cR_X,~~~~F^\star_x~:~~(\cR_Y)_{f(x)} \lra(\cR_X)_x$$ a homomorphism of sheaves of rings on $X$, called the comorphism of $F$. \endstat If $F:(X,\cR_X)\lra(Y,\cR_Y)$ and $G:(Y,\cR_Y)\lra(Z,\cR_Z)$ are morphisms of ringed spaces, the composite $G\circ F$ is the pair consisting of the map $g\circ f:X\lra Z$ and of the comorphism $(G\circ F)^\star=F^\star\circ f^{-1}G^\star\,$: $$\cmalign{ \hfill&F^\star\circ f^{-1}G^\star~:~~&f^{-1}g^{-1}\cR_Z~\,&\kern-1.9mm \buildo f^{-1}G^\star\over{\relbar\mkern-4mu\relbar\mkern-4mu\lra}f^{-1}\cR_Y &\kern-1.9mm\buildo F^\star\over{\relbar\mkern-4mu\lra}\cR_X,\cr &F^\star_x\circ G^\star_{f(x)}~:~~~&(\cR_Z)_{g\circ f(x)}~\,&{\relbar\mkern-4mu \relbar\mkern-4mu\lra}~(\cR_Y)_{f(x)}~&{\relbar\mkern-4mu\lra}~(\cR_X)_x.\cr} \leqno(9.2)$$ \titlec{\S 9.2.}{Definition of Complex Analytic Schemes} We begin by a description of what will be the local model of an analytic scheme. Let $\Omega\subset\bbbc^n$ be an open subset, $\cJ\subset\cO_\Omega$ a coherent sheaf of ideals and $A=V(\cJ)$ the analytic set in $\Omega$ defined locally as the zero set of a system of generators of $\cJ$. By Hilbert's Nullstellensatz~4.22 we have $\cI_A=\sqrt\cJ$, but $\cI_A$ differs in general from $\cJ$. The sheaf of rings $\cO_\Omega/\cJ$ is supported on $A$, i.e.\ $(\cO_\Omega/\cJ)_x=0$ if $x\notin A$. Ringed spaces of the type $(A,\cO_\Omega/\cJ)$ will be used as the local models of analytic schemes. \begstat{(9.3) Definition} A morphism $$F=(f,F^\star):(A,\cO_\Omega/\cJ_{\restriction A})\lra(A',\cO_{\Omega'}/\cJ' _{\restriction A'})$$ is said to be analytic if for every point $x\in A$ there exists a neighborhood $W_x$ of $x$ in $\Omega$ and a holomorphic function $\Phi:W_x\lra\Omega'$ such that $f_{\restriction A\cap W_x}= \Phi_{\restriction A\cap W_x}$ and such that the comorphism $$F^\star_x:(\cO_{\Omega'}/\cJ')_{f(x)}\lra(\cO_\Omega/\cJ)_x$$ is induced by $\Phi^\star:\cO_{\Omega',f(x)}\ni u\longmapsto u\circ\Phi \in\cO_{\Omega,x}$ with $\Phi^\star\cJ'\subset\cJ$. \endstat \begstat{(9.4) Example} \rm Take $\Omega=\bbbc^n$ and $\cJ=(z_n^2)$. Then $A$ is the hyperplane $\bbbc^{n-1}\times\{0\}$, and the sheaf $\cO_{\bbbc^n}/\cJ$ can be identified with the sheaf of rings of functions $u+z_nu'$, $u,u'\in\cO_{\bbbc^{n-1}}$, with the relation $z_n^2=0$. In particular, $z_n$ is a nilpotent element of $\cO_{\bbbc^n}/\cJ$. A morphism $F$ of $(A,\cO_{\bbbc^n}/\cJ)$ into itself is induced (at least locally) by a holomorphic map $\Phi=(\wt\Phi,\Phi_n)$ defined on a neighborhood of $A$ in $\bbbc^n$ with values in $\bbbc^n$, such that $\Phi(A)\subset A$, i.e.\ $\Phi_{n\restriction A}=0$. We see that $F$ is completely determined by the data $$\cmalign{ \hfill f(z_1\ld z_{n-1})&=\wt\Phi(z_1\ld z_{n-1},0),~~~~\hfill f~&:~~\bbbc^{n-1}\lra\bbbc^{n-1},\cr f'(z_1\ld z_{n-1})&=\displaystyle{\partial\Phi\over\partial z_n} (z_1\ld z_{n-1},0),~~~~f'~&:~~\bbbc^{n-1}\lra\bbbc^n,\cr}$$ which can be chosen arbitrarily. \endstat \begstat{(9.5) Definition} A complex analytic scheme is a ringed space $(X,\cO_X)$ over a separable Hausdorff topological space $X$, satisfying the following property: there exist an open covering $(U_\lambda)$ of $X$ and isomorphisms of ringed spaces $$G_\lambda:(U_\lambda,\cO_{X\restriction U_\lambda}) \lra(A_\lambda,\cO_{\Omega_\lambda}/\cJ_{\lambda\,\restriction A_\lambda})$$ where $A_\lambda$ is the zero set of a coherent sheaf of ideals $\cJ_\lambda$ on an open subset $\Omega_\lambda\subset\bbbc^{N_\lambda}$, such that every transition morphism $G_\lambda\circ G_\mu^{-1}$ is a holomorphic isomorphism from $g_\mu(U_\lambda\cap U_\mu)\subset A_\mu$ onto $g_\lambda(U_\lambda\cap U_\mu)\subset A_\lambda$, equipped with the respective structure sheaves $\cO_{\Omega_\mu}/\cJ_{\mu\,\restriction A_\mu}$, $\cO_{\Omega_\lambda}/\cJ_{\lambda\,\restriction A_\lambda}$. \endstat We shall often consider the maps $G_\lambda$ as identifications and write simply $U_\lambda=A_\lambda$. A morphism $F:(X,\cO_X)\lra(Y,\cO_Y)$ of analytic schemes obtained by gluing patches $(A_\lambda,\cO_{\Omega_\lambda}/\cJ_{\lambda\, \restriction A_\lambda})$ and $(A'_\mu,\cO_{\Omega'_\mu}/\cJ'_{\mu\,A'_\mu})$, respectively, is a morphism $F$ of ringed spaces such that for each pair $(\lambda,\mu)$, the restriction of $F$ from $A_\lambda\cap f^{-1}(A'_\mu)\subset X$ to $A'_\mu\subset Y$ is holomorphic in the sense of Def.~9.3. \titlec{\S 9.3.}{Nilpotent Elements and Reduced Schemes} Let $(X,\cO_X)$ be an analytic scheme. The set of {\it nilpotent elements} is the sheaf of ideals of $\cO_X$ defined by $$\cN_X=\{u\in\cO_X\,;\,u^k=0~~\hbox{\rm for~some}~\,k\in\bbbn\}.\leqno(9.6)$$ Locally, we have $\cO_{X\restriction A_\lambda}= (\cO_{\Omega_\lambda}/\cJ_\lambda)_{\restriction A_\lambda}$, thus $$\leqalignno{ &\cN_{X\restriction A_\lambda}=(\sqrt{\cJ_\lambda}/\cJ_\lambda )_{\restriction A_\lambda},&(9.7)\cr &(\cO_X/\cN_X)_{\restriction A_\lambda}\simeq(\cO_{\Omega_\lambda}/ \sqrt{\cJ_\lambda})_{\restriction A_\lambda}= (\cO_{\Omega_\lambda}/\cI_{A_\lambda})_{\restriction A_\lambda}= \cO_{A_\lambda}.&(9.8)\cr}$$ The scheme $(X,\cO_X)$ is said to be {\it reduced} if $\cN_X=0$. The associated ringed space $(X,\cO_X/\cN_X)$ is reduced by construction; it is called the {\it reduced scheme} of $(X,\cO_X)$. We shall often denote the original scheme by the letter $X$ merely, the associated reduced scheme by $X_\red$, and let $\cO_{X,\red}=\cO_X/\cN_X$. There is a canonical morphism $X_\red\to X$ whose comorphism is the reduction morphism $$\cO_X(U)\lra\cO_{X,\red}(U)=(\cO_X/\cN_X)(U),\qquad \hbox{$\forall U$ open set in $X$}.\leqno(9.9)$$ By~(9.8), the notion of reduced scheme is equivalent to the notion of complex space introduced in Def.~5.2. It is easy to see that a morphism $F$ of reduced schemes $X,Y$ is completely determined by the set-theoretic map $f:X\lra Y$. \titlec{\S 9.4.}{Coherent Sheaves on Analytic Schemes} If $(X,\cO_X)$ is an analytic scheme, a sheaf $\cS$ of $\cO_X$-modules is said to be {\it coherent} if it satisfies the same properties as those already stated when $X$ is a manifold: \smallskip \noindent{$(9.10)$}~~$\cS$ is locally finitely generated over $\cO_X\,$; \hfill\break \noindent{$(9.10')$}~for any open set $U\subset X$ and any sections $G_1\ld G_q\in\cS(U)$, the\break \phantom{$(9.10')$~}relation sheaf $\cR(G_1\ld G_q)\subset \cO_{X\restriction U}^{\oplus q}$ is locally finitely generated. \smallskip \noindent Locally, we have $\cO_{X\restriction A_\lambda}=\cO_{\Omega_\lambda}/\cJ_\lambda$, so if $i_\lambda:A_\lambda\to\Omega_\lambda$ is the injection, the direct image $\cS_\lambda=(i_\lambda)_\star(\cS_{\restriction A_\lambda})$ is a module over $\cO_{\Omega_\lambda}$ such that $\cJ_\lambda.\cS_\lambda=0$. It is clear that $\cS_{\restriction\Omega_\lambda}$ is coherent if and only if $\cS_\lambda$ is coherent as a module over $\cO_{\Omega_\lambda}$. It follows immediately that the Oka theorem and its consequences 3.16--20 are still valid over analytic schemes. \titlec{\S 9.5.}{Subschemes} Let $X$ be an analytic scheme and $\cG$ a coherent sheaf of ideals in $\cO_X$. The image of $\cG$ in $\cO_{X,\red}$ is a coherent sheaf of ideals, and its zero set $Y$ is clearly an analytic subset of $X_\red$. We can make $Y$ into a scheme by introducing the structure sheaf $$\cO_Y=(\cO_X/\cG)_{\restriction Y},\leqno(9.11)$$ and we have a scheme morphism $F:(Y,\cO_Y)\lra(X,\cO_X)$ such that $f$ is the inclusion and $F^\star:f^{-1}\cO_X\lra\cO_Y$ the obvious map of $\cO_{X\restriction Y}$ onto its quotient $\cO_Y$. The scheme $(Y,\cO_Y)$ will be denoted $V(\cG)$. When the analytic set $Y$ is given, the structure sheaf of $V(\cG)$ depends of course on the choice of the equations of $Y$ in the ideal $\cG\,$; in general $\cO_Y$ has nilpotent elements. \titlec{\S 9.6.}{Inverse Images of Coherent Sheaves} Let $F:(X,\cO_X)\lra(Y,\cO_Y)$ be a morphism of analytic schemes and $\cS$ a coherent sheaf over~$Y$. The sheaf theoretic inverse image $f^{-1}\cS$, whose stalks are $(f^{-1}\cS)_x=\cS_{f(x)}$, is a sheaf of modules over $f^{-1}\cO_Y$. We define the {\it analytic inverse image} $F^\star\cS$ by $$F^\star\cS=\cO_X\otimes_{f^{-1}\cO_Y}f^{-1}\cS,~~~~ (F^\star\cS)_x=\cO_{X,x}\otimes_{\cO_{Y,f(x)}}\cS_{f(x)}.\leqno(9.12)$$ Here the tensor product is taken with respect to the comorphism $F^\star:f^{-1}\cO_Y\to\cO_X$, which yields a ring morphism $\cO_{Y,f(x)}\to\cO_{X,x}$. If $\cS$ is given over $U\subset Y$ by a local presentation $$\cO_{Y\restriction U}^{\oplus p}\buildo A\over\lra\cO_{Y\restriction U}^{\oplus q}\lra \cS_{\restriction U}\lra 0$$ where $A$ is a $(q\times p)$-matrix with coefficients in $\cO_Y(U)$, our definition shows that $F^\star\cS$ is a coherent sheaf over $\cO_X$, given over $f^{-1}(U)$ by the local presentation $$\cO_{X\restriction f^{-1}(U)}^{\oplus p}\buildo{F^\star A}\over{\larex 24 } \cO_{X\restriction f^{-1}(U)}^{\oplus q}\lra F^\star\cS_{\restriction f^{-1}(U)} \lra 0.\leqno(9.13)$$ \titlec{\S 9.7.}{Products of Analytic Schemes} Let $(X,\cO_X)$ and $(Y,\cO_Y)$ be analytic schemes, and let $(A_\lambda,\cO_{\Omega_\lambda}/\cJ_\lambda)$,\break $(B_\mu,\cO_{\Omega'_\mu}/\cJ'_\mu)$ be local models of $X$, $Y$, respectively. The {\it product scheme} $(X\times Y,\cO_{X\times Y})$ is obtained by gluing the open patches $$\Big(A_\lambda\times B_\mu~,~\cO_{\Omega_\lambda\times\Omega'_\mu}\big/ \big({\rm pr}_1^{-1}\cJ_\lambda+{\rm pr}_2^{-1}\cJ'_\mu\big) \cO_{\Omega_\lambda\times\Omega'_\mu}\Big).\leqno(9.14)$$ In other words, if $A_\lambda$, $B_\mu$ are the subschemes of $\Omega_\lambda$, $\Omega'_\mu$ defined by the equations $g_{\lambda,j}(x)=0$, $g'_{\mu,k}(y)=0$, where $(g_{\lambda,j})$ and $(g'_{\mu,k})$ are generators of $\cJ_\lambda$ and $\cJ'_\mu$ respectively, then $A_\lambda\times B_\mu$ is equipped with the structure sheaf $\cO_{\Omega_\lambda\times\Omega'_\mu}\big/\big( g_{\lambda,j}(x),g'_{\mu,k}(y)\big)$. Now, let $\cS$ be a coherent sheaf over $\cO_X$ and let $\cS'$ be a coherent sheaf over $\cO_Y$. The (analytic) {\it external tensor product} $\cS\stimes\cS'$ is defined to be $$\cS\stimes\cS'={\rm pr}_1^\star\cS\otimes_{\cO_{X\times Y}} {\rm pr}_2^\star\cS'.\leqno(9.15)$$ If we go back to the definition of the inverse image, we see that the stalks of $\cS\stimes\cS'$ are given by $$(\cS\stimes\cS')_{(x,y)}=\cO_{X\times Y,(x,y)}\otimes_ {\cO_{X,x}\otimes\cO_{Y,y}}(\cS_x\otimes_\bbbc\cS'_y)~,\leqno(9.15')$$ in particular $(\cS\stimes\cS')_{(x,y)}$ {\it does not coincide} with the sheaf theoretic tensor product $\cS_x\otimes\cS'_y$ which is merely a module over $\cO_{X,x}\otimes\cO_{Y,y}$. If $\cS$ and $\cS'$ are given by local presentations $$\cO_{X\restriction U}^{\oplus p}\buildo A\over\lra\cO_{X\restriction U}^{\oplus q}\lra \cS_{\restriction U}\lra 0,~~~~ \cO_{Y\restriction U'}^{p'}\buildo B\over\lra\cO_{Y\restriction U'}^{q'} \lra\cS'_{\restriction U'}\lra 0,$$ then $\cS\stimes\cS'$ is the coherent sheaf given by $$\cO_{X\times Y\restriction U\times U'}^{pq'\oplus qp'} \buildo{(A(x)\otimes\Id,\Id\otimes B(y))}\over{\larex 84 } \cO_{X\times Y\restriction U\times U'}^{qq'}\lra (\cS\stimes\cS')_{\restriction U\times U'}\lra 0.$$ \titlec{\S 9.8.}{Zariski Embedding Dimension} If $x$ is a point of an analytic scheme $(X,\cO_X)$, the {\it Zariski embedding dimension} of the germ $(X,x)$ is the smallest integer $N$ such that $(X,x)$ can be embedded in $\bbbc^N$, i.e.\ such that there exists a patch of $X$ near $x$ isomorphic to $(A,\cO_\Omega/\cJ)$ where $\Omega$ is an open subset of $\bbbc^N$. This dimension is denoted $$\hbox{\rm embdim}(X,x)=~\hbox{\rm smallest~such}~~N.\leqno(9.16)$$ Consider the maximal ideal $\gm_{X,x}\subset\cO_{X,x}$ of functions which vanish at point~$x$. If $(X,x)$ is embedded in $(\Omega,x)=(\bbbc^N,0)$, then $\gm_{X,x}/\gm_{X,x}^2$ is generated by $z_1\ld z_N$, so $d=\dim\gm_{X,x}/\gm_{X,x}^2\le N$. Let $s_1\ld s_d$ be germs in $\gm_{\Omega,x}$ which yield a basis of $\gm_{X,x}/\gm_{X,x}^2\simeq\gm_{\Omega,x}/(\gm^2_{\Omega,x}+\cJ_x)$. We can write $$z_j=\sum_{1\le k\le d}c_{jk}s_k+u_j+f_j,~~~c_{jk}\in\bbbc,~~u_j\in \gm^2_{\Omega,x},~~f_j\in\cJ_x,~~1\le j\le n.$$ Then we find $dz_j=\sum c_{jk}\,ds_k(x)+df_j(x)$, so that the rank of the system of diffe\-rentials $\big(df_j(x)\big)$ is at least $N-d$. Assume for example that $df_1(x)\ld df_{N-d}(x)$ are linearly independant$\,$. By the implicit function theorem, the equations $f_1=\ldots=f_{N-d}=0$ define a germ of smooth subvariety $(Z,x)\subset(\Omega,x)$ of dimension $d$ which contains $(X,x)$. We have $\cO_Z=\cO_\Omega/(f_1\ld f_{N-d})$ in a neighborhood of $x$, thus $$\cO_X=\cO_\Omega/\cJ\simeq\cO_Z/\cJ'~~~~\hbox{\rm where}~~ \cJ'=\cJ/(f_1\ld f_{N-d}).$$ This shows that $(X,x)$ can be imbedded in $\bbbc^d$, and we get $$\hbox{\rm embdim}(X,x)=\dim\gm_{X,x}/\gm_{X,x}^2.\leqno(9.17)$$ \begstat{(9.18) Remark} \rm For a given dimension $n=\dim(X,x)$, the embedding dimension $d$ can be arbitrarily large. Consider for example the curve $\Gamma\subset\bbbc^N$ parametrized by $\bbbc\ni t\longmapsto(t^N,t^{N+1}\ld t^{2N-1})$. Then $\cO_{\Gamma,0}$ is the ring of convergent series in $\bbbc\{t\}$ which have no terms $t,t^2\ld t^{N-1}$, and $\gm_{\Gamma,0}/\gm_{\Gamma,0}^2$ admits precisely $z_1=\smash{t^N} \ld z_N=\smash{t^{2N-1}}$ as a basis. Therefore $n=1$ but $d=N$ can be as large as we want. \endstat \titleb{\S 10.}{Bimeromorphic maps, Modifications and Blow-ups} It is a very frequent situation in analytic or algebraic geometry that two complex spaces have isomorphic dense open subsets but are nevertheless different along some analytic subset. These ideas are made precise by the notions of modification and bimeromorphic map. This will also lead us to generalize the notion of meromorphic function to maps between analytic schemes. If $(X,\cO_X)$ is an analytic scheme, $\cM_X$ denotes the sheaf of meromorphic functions on~$X$, defined at the beginning of \S~6.2. \begstat{(10.1) Definition} Let $(X,\cO_X)$, $(Y,\cO_Y)$ be analytic schemes. An analytic morphism $F:X\to Y$ is said to be a modification if $F$ is proper and if there exists a nowhere dense closed analytic subset $B\subset Y$ such that the restriction $F:X\ssm F^{-1}(B)\to Y\ssm B$ is an isomorphism. \endstat \begstat{(10.2) Definition} If $F:X\to Y$ is a modification, then the comorphism $F^\star:f^\star\cO_Y\to\cO_X$ induces an isomorphism $F^\star:f^\star\cM_Y\to\cM_X$ for the sheaves of meromorphic functions on $X$ and~$Y$. \endstat \begproof{} Let $v=g/h$ be a section of $\cM_Y$ on a small open set $\Omega$ where $u$ is actually given as a quotient of functions $g,h\in\cO_Y(\Omega)$. Then $F^\star u=(g\circ F)/(h\circ F)$ is a section of $\cM_X$ on $F^{-1}(\Omega)$, for $h\circ F$ cannot vanish identically on any open subset $W$ of~$F^{-1}(\Omega)$ (otherwise $h$ would vanish on the open subset $F(W\ssm F^{-1}(B))$ of $\Omega\ssm B$). Thus the extension of the comorphism to sheaves of meromorphic functions is well defined. Our claim is that this is an isomorphism. The injectivity of $F^\star$ is clear: $F^\star u=0$ implies $g\circ F=0$, which implies $g=0$ on $\Omega\ssm B$ and thus $g=0$ on $\Omega$ because $B$ is nowhere dense. In order to prove surjectivity, we need only show that every section $u\in\cO_X(F^{-1}(\Omega))$ is in the image of $\cM_Y(\Omega)$ by~$F^\star$. For this, we may shrink $\Omega$ into a relatively compact subset $\Omega'\compact\Omega$ and thus assume that $u$ is bounded (here we use the properness of $F$ through the fact that $F^{-1}(\Omega')$ is relatively compact in $F^{-1}(\Omega))$. Then $v=u\circ F^{-1}$ defines a bounded holomorphic function on~$\Omega\ssm B$. By Th.~7.2, it follows that $v$ is weakly holomorphic for the reduced structure of~$Y$. Our claim now follows from the following Lemma.\qed \endproof \begstat{(10.3) Lemma} If $(X,\cO_X)$ is an analytic scheme, then every holomorphic function $v$ in the complement of a nowhere dense analytic subset $B\subset Y$ which is weakly holomorphic on~$X_\red$ is meromorphic on~$X$. \endstat \begproof{} It is enough to argue with the germ of $v$ at any point $x\in Y$, and thus we may suppose that $(Y,\cO_Y)=(A,\cO_\Omega/\cI)$ is embedded in~$\bbbc^N$. Because $v$ is weakly holomorphic, we can write $v=g/h$ in $Y_\red$, for some germs of holomorphic functions $g,h$. Let $\wt g$ and $\wt h$ be extensions of $g$, $h$ to $\cO_{\Omega,x}$. Then there is a neighborhood $U$ of $x$ such that $\wt g-v\wt h$ is a nilpotent section of $cO_\Omega(U\ssm B)$ which is in $\cI$ on \endproof \begstat{(10.4) Definition} A meromorphic map $F:X\merto Y$ is a scheme morphism $F:X\ssm A\to Y$ defined in the complement of a nowhere dense analytic subset $A\subset X$, such that the closure of the graph of $F$ in $X\times Y$ is an analytic subset $($for the reduced complex space structure of $X\times Y)$. \endstat \titleb{\S 11.}{Exercises}\begpet \titled{11.1.} Let $\cA$ be a sheaf on a topological space~$X$. If the sheaf space $\wt\cA$ is Hausdorff, show that $\cA$ satisfies the following {\it unique continuation principle}: any two sections $s,s'\in\cA(U)$ on a connected open set $U$ which coincide on some non empty open subset~$V\subset U$ must coincide identically on~$U$. Show that the converse holds if $X$ is Hausdorff and locally connected. \titled{11.2.} Let $\cA$ be a sheaf of abelian groups on~$X$ and let $s\in\cA(X)$. The support of~$s$, denoted $\Supp\,s$, is defined to be $\{x\in X\,;\,s(x)\ne 0\}$. Show that $\Supp\,s$ is a closed subset of~$X$. The support of $\cA$ is defined to be $\Supp\,\cA=\{x\in X\,;\,\cA_x\ne 0\}$. Show that $\Supp\,\cA$ is not necessarily closed: if $\Omega$ is an open set in $X$, consider the sheaf $\cA$ such that $\cA(U)$ is the set of continuous functions $f\in\cC(U)$ which vanish on a neighborhood of $U\cap(X\ssm\Omega)$. \titled{11.3.} Let $\cA$ be a sheaf of rings on a topological space~$X$ and let $\cF$, $\cG$ be sheaves of $\cA$-modules. We define a presheaf $\cH=\cHom_\cA(\cF,\cG)$ such that $\cH(U)$ is the module of all sheaf-homomorphisms $\cF_{\restriction U}\to\cG_{\restriction U}$ which are $\cA$-linear.\smallskip \item{a)} Show that $\cHom_\cA(\cF,\cG)$ is a sheaf and that there is a canonical homomorphism $\varphi_x:\cHom_\cA(\cF,\cG)_x\lra \hom_{\cA_x}(\cF_x,\cG_x)$ for every $x\in X$. \smallskip \item{b)} If $\cF$ is locally finitely generated, then $\varphi_x$ is injective, and if $\cF$ has local finite presentations as in (3.12), then $\varphi_x$ is bijective. \smallskip \item{c)} Suppose that $\cA$ is a coherent sheaf of rings and that $\cF$, $\cG$ are coherent modules over~$\cA$. Then $\cHom_\cA(\cF,\cG)$ is a coherent $\cA$-module.\newline {\it Hint}\/: observe that the result is true if $\cF=\cA^{\oplus p}$ and use a local presentation of $\cF$ to get the conclusion. \titled{11.4.} Let $f:X\to Y$ be a continuous map of topological spaces. Given sheaves of abelian groups $\cA$ on $X$ and $\cB$ on~$Y$, show that there is a natural isomorphism $$\hom_X(f^{-1}\cB,\cA)=\hom_Y(\cB,f_\star\cA).$$ {\it Hint}\/: use the natural morphisms (2.17). \titled{11.5.} Show that the sheaf of polynomials over $\bbbc^n$ is a coherent sheaf of rings (with either the ordinary topology or the Zariski topology on $\bbbc^n$). Extend this result to the case of regular algebraic functions on an algebraic variety.\newline {\it Hint}\/: check that the proof of the Oka theorem still applies. \titled{11.6.} Let $P$ be a non zero polynomial on~$\bbbc^n$. If $P$ is irreducible in $\bbbc[z_1\ld z_n]$, show that the hypersurface $H=\smash{P^{-1}(0)}$ is globally irreducible as an analytic set. In general, show that the irreducible components of $H$ are in a one-to-one correspondence with the irreducible factors of~$P$.\newline {\it Hint}\/: for the first part, take coordinates such that $P(0\ld 0,z_n)$ has degree equal to $P\,$; if $H$ splits in two components $H_1$, $H_2$, then $P$ can be written as a product $P_1P_2$ where the roots of $P_j(z',z_n)$ correspond to points in~$H_j$. \titled{11.7.} Prove the following facts:\smallskip \item{a)} For every algebraic variety $A$ of pure dimension $p$ in~$\bbbc^n$, there are coordinates $z'=(z_1\ld z_p)$, $z''=(z_{p+1}\ld z_n)$ such that $\pi:A\to\smash{\bbbc^p}$, $z\mapsto z''$ is proper with finite fibers, and such that $A$ is entirely contained in a cone $$|z''|\le C(|z'|+1).$$ {\it Hint}\/: imitate the proof of Cor.~4.11. \smallskip \item{b)} Conversely if an analytic set $A$ of pure dimension $p$ in $\bbbc^n$ is contained in a cone $|z''|\le C(|z'|+1)$, then $A$ is algebraic.\newline {\it Hint}\/: first apply $(5.9)$ to conclude that the projection $\pi: A\to\bbbc^p$ is finite. Then repeat the arguments used in the final part of the proof of Th.~4.19. \item{c)} Deduce from a), b) that an algebraic set in $\bbbc^n$ is irreducible if and only if it is irreducible as an analytic set. \titled{11.8.} Let $\Gamma:f(x,y)=0$ be a germ of analytic curve in $\bbbc^2$ through~$(0,0)$ and let $(\Gamma_j,0)$ be the irreducible components of $(\Gamma,0)$. Suppose that $f(0,y)\not\equiv 0$. Show that the roots $y$ of $f(x,y)=0$ corresponding to points of $\Gamma$ near $0$ are given by {\it Puiseux expansions} of the form $y=g_j(x^{1/q_j})$, where $g_j\in\cO_{\bbbc,0}$ and where $q_j$ is the sheet number of the projection $\Gamma_j\to\bbbc$, $(x,y)\mapsto x$.\newline {\it Hint}\/: special case of the parametrization obtained in~(4.27). \titled{11.9.} The goal of this exercise is to prove the existence and the analyticity of the {\it tangent cone} to an arbitrary analytic germ $(A,0)$ in~$\bbbc^n$. Suppose that $A$ is defined by holomorphic equations $f_1=\ldots=f_N=0$ in a ball $\Omega=B(0,r)$. Then the (set theoretic) tangent cone to $A$ at $0$ is the set $C(A,0)$ of all limits of sequences $t_\nu^{-1}z_\nu$ with $z_\nu\in A$ and $\bbbc^\star\ni t_\nu$ converging to~$0$.\smallskip \item{a)} Let $E$ be the set of points $(z,t)\in\Omega\times\bbbc^\star$ such that $z\in t^{-1}A$. Show that the closure $\smash{\ovl E}$ in $\Omega\times\bbbc$ is analytic.\newline {\it Hint}\/: observe that $E=A\ssm(\Omega\times\{0\})$ where $A=\{f_j(tz)=0\}$ and apply Cor.~5.4.\smallskip \item{b)} Show that $C(A,0)$ is a conic set and that $\ovl E\cap(\Omega\times \{0\})=C(A,0)\times\{0\}$ and conclude. Infer from this that $C(A,0)$ is an algebraic subset of~$\bbbc^n$. \titled{11.10.} Give a new proof of Theorem~5.5 based on the coherence of ideal sheaves and on the strong noetherian property. \titled{11.11.} Let $X$ be an analytic space and let $A$, $B$ be analytic subsets of pure dimensions. Show that $\codim_X(A\cap B)\le\codim_X A+\codim_X B$ if $A$ or $B$ is a local complete intersection, but that the equality does not necessarily hold in general.\newline {\it Hint}\/: see Remark~(6.5). \titled{11.12.} Let $\Gamma$ be the curve in $\bbbc^3$ parametrized by $\bbbc\ni t\longmapsto(x,y,z)=(t^3,t^4,t^5)$. Show that the ideal sheaf $\cI_\Gamma$ is generated by the polynomials $xz-y^2$, $x^3-yz$ and $x^2y-z^2$, and that the germ $(\Gamma,0)$ is not a (sheaf theoretic) complete intersection.\newline {\it Hint}\/: $\Gamma$ is smooth except at the origin. Let $f(x,y,z)= \sum a_{\alpha\beta\gamma}x^\alpha y^\beta z^\gamma$ be a convergent power series near~$0$. Show that $f\in\cI_{\Gamma,0}$ if and only if all weighted homogeneous components $f_k=\sum_{3\alpha+4\beta+5\gamma=k} a_{\alpha\beta\gamma}x^\alpha y^\beta z^\gamma$ are in~$\cI_{\Gamma,0}$. By means of suitable substitutions, reduce the proof to the case when $f=f_k$ is homogeneous with all non zero monomials satisfying $\alpha\le 2$, $\beta\le 1$, $\gamma\le 1$; then check that there is at most one such monomial in each weighted degree${}\le 15$ the product of a power of $x$ by a homogeneous polynomial of weighted degree~$\le 8$. \endpet \end