Name Juan Elias title On the Grauert-Riemenschneider vanishing theorem Abstracts ================================================================================= \documentclass[12pt]{amsart} \textwidth = 15truecm \textheight = 20.8truecm \hoffset = -1.4truecm \sloppy %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % theorem environments %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{defn0}{Definition}[section] \newtheorem{prop0}[defn0]{Proposition} \newtheorem{thm0}[defn0]{Theorem} \newtheorem{lemma0}[defn0]{Lemma} \newtheorem{corollary0}[defn0]{Corollary} \newtheorem{example0}[defn0]{Example} \newtheorem{remark0}[defn0]{Remark} \newtheorem{conjecture0}[defn0]{Conjecture} \newenvironment{definition}{\bigskip \begin{defn0}}{\end{defn0}} \newenvironment{proposition}{\bigskip \begin{prop0}}{\end{prop0}} \newenvironment{theorem}{\bigskip \begin{thm0}}{\end{thm0}} \newenvironment{lemma}{\bigskip \begin{lemma0}}{\end{lemma0}} \newenvironment{corollary}{\bigskip \begin{corollary0}}{\end{corollary0}} \newenvironment{example}{\bigskip \begin{example0}\rm}{\end{example0}} \newenvironment{remark}{\bigskip \begin{remark0}\rm}{\end{remark0}} \newenvironment{conjecture}{\bigskip \begin{conjecture0}}{\end{conjecture0}} \newcommand{\defref}[1]{Definition~\ref{#1}} \newcommand{\propref}[1]{Proposition~\ref{#1}} \newcommand{\thmref}[1]{Theorem~\ref{#1}} \newcommand{\lemref}[1]{Lemma~\ref{#1}} \newcommand{\corref}[1]{Corollary~\ref{#1}} \newcommand{\exref}[1]{Example~\ref{#1}} \newcommand{\secref}[1]{Section~\ref{#1}} \newcommand{\remref}[1]{Remark~\ref{#1}} \newcommand{\conjref}[1]{Conjecture~\ref{#1}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% local definitions for this paper %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\max{{\bf m}} %% maximal ideal \def\res{{\bf k}} %% residual field \def\rees{{\mathcal R}} %% Rees algebra \def\vv{{\mathcal V}^{2}} %% Vallavrega-Valla cond. \def\brees{\overline{{\mathcal R}}} %% quotient of Rees algebra \def\bG{\overline{G}} %% quotient of G \def\HiRIM{H^i_{\mathcal M}(\rees(I))} \def\HiRInM{H^i_{\mathcal M^{(n)}}(\rees(I^n))} \def\LCR#1{H^{#1}_{\mathcal M}(\rees(I))} \def\LC#1#2#3{H^{#1}_{#2}(#3)} \def\PI{\;{{\bf Proj}\; }(\mathcal R(I))} \def\OX{{\mathcal O}_{X}} \def\OXx{{\mathcal O}_{(X,x)}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title[Vanishing theorem]{{\bf On the Grauert-Riemenschneider vanishing theorem}} \author[Juan Elias]{Juan Elias } \address{Departament d'\`{A}lgebra i Geometria \newline \indent Facultat de Matem\`{a}tiques \newline \indent Universitat de Barcelona \newline \indent Gran Via 585, 08007 Barcelona, Spain} \email{{\tt elias@mat.ub.es}} \maketitle \baselineskip 15pt Let $R$ be a Cohen-Macaulay local ring essentially of finite type over $\mathbb C$ and let $X=\PI$ be the blowing up of ${\bf Spec}(R)$ along the closed sub-scheme defined by an ideal $I$ of $R$. Let $Y$ be the closed fiber of $\pi: X \longrightarrow {\bf Spec}(R)$. Sancho de Salas prove that if $X$ is a Cohen-Macaulay scheme then \medskip \noindent {\bf SdS-1.} There exists an integer $n_0$ such that the associated graded ring $gr_{I^n}(R)$ is Cohen-Macaulay for all $n \ge n_0$ if and only if $H^{i}_Y(X,\OX)=0$ for $i < dim(R)$. \medskip In the proof of this result Sancho de Salas introduced a new exact sequence relating some local cohomology groups of $R$ and sheaf cohomology groups of $X$. We use here an algebraic version of this sequence due to Karen Smith, see \cite{Lip94}, page 150. >From {\bf SdS-1} Sancho de Salas deduce the following version of Grauert-Riemenschneider vanishing theorem, \cite{GR70}, Satz 2,3, \cite{HO74}, Proposition 2.2, \medskip \noindent {\bf SdS-2.} If $X$ is non singular then $gr_{I^n}(R)$ is Cohen-Macaulay for all large values of $n$. \medskip Lipman in \cite{Lip94}, Theorem 4.3, prove that {\bf SdS-1} holds for all Cohen-Macaulay local ring $R$ without the assumption to be essentially of finite type over $\mathbb C$. Cutkovsky, \cite{cut90}, part III, and Huckaba and Huneke, \cite{HH99}, Theorem 3.12, give examples showing that {\bf SdS-2} result does not holds for a $d-$dimensional Cohen-Macaulay ring $R$, with $d\ge 3$ and such that $X$ is a normal scheme instead a non-singular scheme. Huneke prove in \cite{MR96m:13001}, Exercises 5.12, 5.13, an algebraic version of Grauert-Riemenschneider vanishing theorem, see also \cite{HH99}, Corollary 3.8, \medskip \noindent {\bf aGR.} Let $R$ be a $2$-dimensional Cohen-Macaulay local ring, and let $I$ be a normal $\max-$primary ideal. Then $gr_{I^n}(R)$ is Cohen-Macaulay for all large values of $n$. \medskip This result has been generalized by Huckaba and Marley in \cite{HM00}, Corollary 3.9. Valla in \cite{Val76} proved that if $R$ is Cohen-Macaulay and $I$ is a complete intersection ideal then $gr_{I^n}(R)$ and $\rees(I^n)$ are Cohen-Macaulay for all $n \ge 1$. \medskip We generalize {\bf SdS-1} to schemes $X=\PI$ with bounded projective Cohen-Macaulay deviation. As a corollary of this result we propose an explanation of Cutkovsky's and Huckaba-Huneke's examples, in particular these examples shows that the generalization of {\bf SdS-1} is sharp. We also improve {\bf aGR} showing that we only need to assume that $gr_{I}(R)$ is a positive depth ring for getting $gr_{I^n}(R)$ Cohen-Macaulay for all large values of $n$. In particular we obtain as corollary {\bf aGR}, \cite{HH99}, Corollary 3.8, see also \cite{HM00}, Corollary 3.5 and Corollary 3.9. Finally, we show that the improvement of {\bf aGR} cannot be generalized to higher dimensions by considering Huckaba-Huneke's examples. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip %\bibliography{biblatex,tight} %\bibliographystyle{kkalpha} %\bibliographystyle{amsplain} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \begin{thebibliography}{1} \bibitem{cut90} S.D. Cutkosky, \emph{A new characterization of rational surface singularities}, Inv. Math. \textbf{102} (1990), 157--177. \bibitem{GR70} H.~Grauert and O.~Riemenschneider, \emph{Verschwindungss\"atze f\"ur analytische kohomologiegruppen auf komplexen r\"aumen}, Inv. Math. \textbf{11} (1970), 263--292. \bibitem{HO74} R.~Hartshorne and A.~Ogus, \emph{On the factoriality of local rings of small embedding codimension}, Comm. in Algebra \textbf{1(5)} (1974), 415--437. \bibitem{HH99} S.~Huckaba and C.~Huneke, \emph{Normal ideals in regular rings}, J. Reine Angew. Math. \textbf{510} (1999), 63--82. \bibitem{HM00} S.~Huckaba and T.~Marley, \emph{On associated graded rings of normal ideals}, Preprint (2000). \bibitem{MR96m:13001} C.~Huneke, \emph{Tight closure and its applications}, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996, With an appendix by Melvin Hochster. \bibitem{Lip94} J.~Lipman, \emph{Cohen-{M}acaulayness in graded rings}, Mathematical Research Letters \textbf{1} (1994), 149--157. \bibitem{Val76} G.~Valla, \emph{Certain graded algebras are {C}ohen-{M}acaulay}, J. of Algebra \textbf{42} (1976), 537--548. \end{thebibliography} \end{document} \end ================================================================================== -- Juan Elias Dept. Àlgebra i Geometria Facultat de Matemàtiques Universitat de Barcelona Gran Via 585 BARCELONA 08007 SPAIN http://www.mat.ub.es/~elias/ Fax 34-934021601 Telf 34-934021604