
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%  
%  slides:
%
%  Rigid Dualizing Complexes and Perverse Coherent Sheaves
%
%  Amnon Yekutieli
%
%  written in AMSLaTex 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% %% ** class for notes **  
% \documentclass[10pt]{amsart}
% \usepackage{amscd}
% \usepackage{amsmath}
% \usepackage{amssymb}
% \usepackage[all]{xy}
% \setlength{\parskip}{1ex} 
% \setlength{\parindent}{0ex} 
% \usepackage{pst-all}
% \usepackage{graphicx}
% \newcommand{\negskip}[1]{}
% \newcommand{\hilite}[2]{{\it {#2}}}
% %% ** end class for notes **


%% ** class and  details for beamer **
%\documentclass[serif]{beamer}
\documentclass{beamer}
\setbeamertemplate{navigation symbols}{}
%\usetheme{Hannover}
\usefonttheme{serif}
\usepackage{amscd}
\usepackage{amsmath}
\usepackage{amssymb}
\setlength{\parskip}{1.5ex} 
\usepackage{pst-all}
\usepackage{graphicx}
\usepackage[all]{xy}
%
\usepackage[english]{babel}
\usepackage[T1]{fontenc}
\usepackage{times}
\newcommand{\negskip}[1]{\vspace{- #1 em}}
\newcommand{\hilite}[2]{\textcolor{#1}{#2}}
%
%\mode<presentation>{\setbeamercovered{transparent}} 
%
\useoutertheme{infolines}
%\usecolortheme{lily}
\usecolortheme{seahorse}
%
\title[Rigid \& Perverse]
{Rigid Dualizing Complexes and Perverse Coherent Sheaves}
%
\author[Amnon Yekutieli]{Amnon Yekutieli}
\institute[BGU]
{Department of Mathematics\\
Ben Gurion University\\[3em]
Notes available at 
{\blue http://www.math.bgu.ac.il/$\sim$amyekut/lectures}}
%
\date[]{\tiny Written 30 Oct 2007}
%\subject{Talks}
%
\pgfdeclareimage[height=0.5cm]{university-logo}{BGU_logo}
\logo{\pgfuseimage{university-logo}}
% ** end of details for beamer **


%% ** graphics **
\usepackage{pst-all}
\usepackage{graphicx}
\psset{unit=0.4cm}
\psset{linewidth=0.5pt}
%
\newrgbcolor{RU}{0.65 0.45 0.15}
\newrgbcolor{orange}{0.99 0.6 0}
\newcmykcolor{DB}{0.79 0.51 0 0.34}
\newcmykcolor{BG}{.8 .1 .5 .2}
%% ** end graphics **


%  ** new environments **
\numberwithin{equation}{section}
\newenvironment{cor}{\refstepcounter{equation} \medskip \par \noindent
        {\bf Corollary \theequation.} \em} {\medskip \par}
\newenvironment{prop}{\refstepcounter{equation} \medskip \par \noindent
        {\bf Proposition \theequation.} \em}{\medskip \par}
\newenvironment{thm}{\refstepcounter{equation} \medskip \par \noindent
        {\bf Theorem \theequation.} \em}{\medskip \par}
\newenvironment{lem}{\refstepcounter{equation} \medskip \par \noindent
        {\bf Lemma \theequation.} \em}{\medskip \par}
\newenvironment{dfn}{\refstepcounter{equation} \medskip \par \noindent
        {\bf Definition \theequation.}}{\medskip \par}
\newenvironment{rem}{\refstepcounter{equation} \medskip \par \noindent
        {\bf Remark \theequation.}\ }{\medskip \par}
\newenvironment{dig}{\refstepcounter{equation} \medskip \par \noindent
        {\bf Digression \theequation.}\ }{\medskip \par}
\newenvironment{exa}{\refstepcounter{equation} \medskip \par \noindent
        {\bf Example \theequation.}\ }{\medskip \par}
\newenvironment{prob}{\refstepcounter{equation} \medskip \par \noindent
        {\bf Problem \theequation.}\ }{\medskip \par}
\newenvironment{conj}{\refstepcounter{equation} \medskip \par \noindent
        {\bf Conjecture \theequation.}\ }{\medskip \par}
\newenvironment{que}{\refstepcounter{equation} \medskip \par \noindent
        {\bf Question \theequation.}\ }{\medskip \par}



%  ** new commands **
\newcommand{\ul}{\underline}
\newcommand{\ol}{\overline}
\newcommand{\iso}{\stackrel{\simeq}{\rightarrow}}
\newcommand{\inj}{\hookrightarrow}
\newcommand{\surj}{\twoheadrightarrow}
\newcommand{\xar}{\xrightarrow}
\newcommand{\opn}{\operatorname}
\newcommand{\opnt}[1]{\mathrm{#1}} % less space
\newcommand{\cat}[1]{\operatorname{\mathsf{#1}}}
\newcommand{\bdot}{{\textstyle \cdot}}
\newcommand{\rmitem}[1]{\item[\text{\textup{(#1)}}]}
\newcommand{\mfrak}[1]{\mathfrak{#1}}
\newcommand{\mcal}[1]{\mathcal{#1}}
\newcommand{\msf}[1]{\mathsf{#1}}
\newcommand{\mbf}[1]{\mathbf{#1}}
\newcommand{\mrm}[1]{\mathrm{#1}}
\newcommand{\mbb}[1]{\mathbb{#1}}
\newcommand{\gfrac}[2]{\genfrac{[}{]}{0pt}{}{#1}{#2}}
\newcommand{\tup}[1]{\textup{#1}}
\newcommand{\bsym}[1]{\boldsymbol{#1}}
\newcommand{\bwedge}{\bigwedge\nolimits}
\newcommand{\boplus}{\bigoplus\nolimits}
\newcommand{\bra}[1]{\langle #1 \rangle}
\newcommand{\abs}[1]{\lvert #1 \rvert}
\newcommand{\what}{\widehat}
\newcommand{\wtil}[1]{\widetilde{#1}}
\newcommand{\til}[1]{\tilde{#1}}
\renewcommand{\d}{\mathrm{d}}
\newcommand{\scrp}[1]{\scriptstyle{#1}}
\newcommand{\sect}[1]{\bigskip \begin{center}
{\textbf{#1}} \end{center}}
\newcommand{\head}[1]{\noindent \textbf{#1}}
\newcommand{\cmnt}[1]{\mbox{} \newline 
%\marginpar{\quad $\clubsuit$ \quad $\clubsuit$ \quad $\clubsuit$} 
\textsf{===[#1]===} \newline}
\DeclareMathSymbol{\mathbbk}{\mathord}{AMSb}{"7C}
\renewcommand{\k}{\mathbbk}
\newcommand{\K}{\mbb{K} \hspace{0.05em}}
\newcommand{\smfrac}[2]{\textstyle \frac{#1}{#2}}
\newcommand{\hatotimes}[1]{\, \what{\otimes}_{#1} \,}
%%


\begin{document}

% %% top matter for notes %%
% \mbox{}
% \vspace{2cm}
% 
% \begin{center}
% {\large \bfseries Rigid Dualizing Complexes and Perverse Coherent 
% Sheaves} 
% 
% \vspace{1ex}
% Lecture Notes\\
% 27 Oct 2007
% 
% \vspace{3ex}
% Amnon Yekutieli\\
% Ben Gurion University, ISRAEL
% 
% \texttt{\small http://www.math.bgu.ac.il/$\sim$amyekut/lectures}
% \end{center}
% 
% \vspace{2ex}
% %% ** end of top matter for notes **


\begin{frame}
\titlepage
\end{frame}
%
\begin{frame}
Here is the plan of my lecture: \pause

\begin{enumerate}
\item[1.] Background on Dualizing Complexes 
\item[2.] Rigid Complexes over Rings 
\item[3.] Rigid Dualizing Complexes over Rings 
\item[4.] Rigid Dualizing Complexes over Schemes 
\item[5.] Perverse Coherent Sheaves 
\item[6.] Cohen-Macaulay Complexes \pause
\end{enumerate}

\medskip
This talk is about joint work with James Zhang (Seattle). 
%\pause
\end{frame}


\section{1. Background on Dualizing Complexes}
\begin{frame}
\sect{1. Background on Dualizing Complexes}
\pause

Dualizing complexes over schemes were introduced by Grothendieck 
in the 1960's (see \cite{RD}), as a vast generalization of Serre 
duality. 
\pause

Suppose $X$ is a noetherian scheme. 
\pause

We denote by 
$\cat{Mod} \mcal{O}_X$ the category of sheaves of 
$\mcal{O}_X$-modules, and by 
$\msf{D}(\cat{Mod} \mcal{O}_X)$
its derived category. 
\pause

The full subcategory of bounded 
complexes with coherent cohomologies is
$\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X)$. It is 
equivalent to 
$\msf{D}^{\mrm{b}}(\cat{Coh} \mcal{O}_X)$.
\end{frame}

\begin{frame}
\begin{dfn} (Grothendieck \cite{RD}) \label{dfn1}
A \hilite{DB}{dualizing complex} on $X$ is a complex 
$\mcal{R} \in \msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X)$
satisfying the two conditions:
\begin{enumerate}
\rmitem{i} $\mcal{R}$ has finite injective dimension.
\rmitem{ii} The canonical morphism
$\mcal{O}_X \to \mrm{R} \mcal{H}om_{\mcal{O}_X}(\mcal{R}, \mcal{R})$ 
is an isomorphism.
\end{enumerate}
\end{dfn}
\pause


It follows that the functor
\[ \mcal{M} \mapsto \mrm{R} \mcal{H}om_{\mcal{O}_X}
(\mcal{M}, \mcal{R}) \]
is an auto-duality of 
$\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X)$.
\pause

When $X = \opn{Spec} A$ is affine, the complex 
$R := \mrm{R} \Gamma(X, \mcal{R}) \in
\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)$
is called a dualizing complex over $A$, and 
\[ M \mapsto \mrm{RHom}_A (M, R) \]
is an auto-duality of 
$\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)$.
\end{frame}


\begin{frame}
Suppose $\K$ is a  regular noetherian ring of 
finite Krull dimension, and $X$ is a finite type $\K$-scheme, with 
structural morphism  
$\pi : X \to \opn{Spec} \K$.
\pause

Then, according to \cite{RD}, there is a special dualizing complex
on $X$, namely the \hilite{DB}{Grothendieck dualizing complex} 
$\mcal{R}_X := \pi^! \K$.
\pause

The proof of existence of this complex, and its properties, is very 
difficult. 
\pause

In this lecture I will explain an alternative approach to 
Grothendieck duality.
\pause

For other approaches see the papers in the references, mainly by Joseph  
Lipman and his coauthors.
\end{frame}




\section{2. Rigid Complexes over Rings}
\begin{frame}
\sect{2. Rigid Complexes over Rings}
\pause

Suppose $A$ is a commutative ring, and $B$ is a commutative 
$A$-algebra. 
\pause

In \cite{YZ4} we constructed a functor
\[ \opn{Sq}_{B / A} : \msf{D}(\cat{Mod} B) \to
\msf{D}(\cat{Mod} B) , \]
called the \hilite{DB}{squaring operation}.
\pause

When $B$ is flat over $A$ one has
\[ \opn{Sq}_{B / A} M =
\mrm{RHom}_{B \otimes_{A} B}(B, 
M \otimes^{\mrm{L}}_{A} M) \]
for $M \in \msf{D}(\cat{Mod} B)$. 
\pause

But in general one has to use DG 
algebras to define $\opn{Sq}_{B / A} M$. 
\end{frame}

\begin{frame}
The functor $\opn{Sq}_{B / A}$ is quadratic, in the following sense.
Given a morphism $\phi : M \to N$ in $\msf{D}(\cat{Mod} B)$, and an 
element $b \in B$, one has
\[ \opn{Sq}_{B / A}(b \phi) = b^2 \opn{Sq}_{B / A}(\phi) \]
in
\[ \opn{Hom}_{\msf{D}(\cat{Mod} B)}(\opn{Sq}_{B / A} M,
\opn{Sq}_{B / A} N) . \]
\pause

\begin{dfn}
Let $B$ be a noetherian $A$-algebra, and let
$M$ be a complex in 
$\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} B)$ that has finite flat 
dimension over $A$. Assume
\[ \rho : M \iso \opn{Sq}_{B / A} M \]
is an isomorphism in $\msf{D}(\cat{Mod} B)$. Then the pair
$(M, \rho)$ is called a \hilite{DB}{rigid complex over $B$ relative to 
$A$}.
\end{dfn}
\end{frame}

\begin{frame}
\begin{dfn} \label{dfn2}
Say $(M, \rho)$ and $(N, \sigma)$ are rigid complexes over $B$ 
relative to $A$. A morphism $\phi : M \to N$ in
$\msf{D}(\cat{Mod} B)$ is called a \hilite{DB}{rigid morphism relative to $A$}
if the diagram
\[ \begin{CD}
M @>{\rho}>> \opn{Sq}_{B / A} M \\
@V{\phi}VV @VV{\opn{Sq}_{B / A} (\phi)}V \\
N  @>{\sigma}>> \opn{Sq}_{B / A} N
\end{CD} \]\
is commutative. 
\end{dfn}
\end{frame}




\section{3. Rigid Dualizing Complexes over Rings}
\begin{frame}
\sect{3. Rigid Dualizing Complexes over Rings}
\pause

 From now on $\K$ denotes a fixed noetherian regular ring of finite Krull 
dimension (e.g.\ a field or the ring of integers). 
\pause

Let $A$ be a noetherian $\K$-algebra. The next definition is due to 
Michel Van den Bergh \cite{VdB}. 
\pause

\begin{dfn}
A \hilite{DB}{rigid dualizing complex} over $A$ relative to $\K$ is a 
rigid complex $(R, \rho)$, such that $R$ is a dualizing 
complex. 
\end{dfn}
\pause

Note that the only rigid automorphism of
a rigid dualizing complex $(R, \rho)$ is the identity 
$\bsym{1}_R$. Indeed, any automorphism 
$\phi$ of $R$ has to be of the form 
$\phi = a \bsym{1}_R$ for some invertible element $a \in A$. If 
$\phi$ is rigid then $a^2 = a$, and hence $a = 1$. 
\end{frame}

\begin{frame}
Recall that an $A$-algebra $B$ is called 
\hilite{DB}{essentially finite type} if it is a localization of some 
finitely generated $A$-algebra. 
\pause

The next theorems are taken from \cite{YZ5}
\pause

\begin{thm} \label{thm4}
\hilite{BG}{Let $A$ be an essentially finite type $\K$-algebra.
Then $A$ has a rigid dualizing complex 
$(R_A, \rho_A)$, which is unique up to a unique rigid isomorphism.}
\end{thm}
\pause

Recall that a ring homomorphism $f^* : A \to B$ is called 
\hilite{RU}{finite} if $B$ is a finitely generated $A$-module.
\pause

\begin{thm} \label{thm5}
\hilite{BG}{Let $A$ and $B$ be essentially finite type $\K$-algebras, and let
$f^* : A \to B$ be a \hilite{RU}{finite} homomorphism. Then the complex 
$\opn{RHom}_A(B, R_A)$ 
has an induced rigidifying isomorphism, and there 
is a unique rigid isomorphism
\[ \opn{RHom}_A(B, R_A) \cong R_B . \]}
\end{thm}
\end{frame}

\begin{frame}
We say that $B$ is \hilite{RU}{essentially smooth}
of relative dimension $n$ over $A$ if it is essentially 
finite type, formally smooth, and the rank of the projective 
$B$-module $\Omega^1_{B / A}$ is $n$. When $n=0$ we say $B$ is 
\hilite{DB}{essentially \'etale}.
\pause

\begin{exa}
If $A'$ is a localization of $A$ then $A \to A'$ is essentially 
\'etale. If $B = A[t_1, \ldots, t_n]$ is a polynomial algebra then 
$A \to B$ is  essentially smooth of relative dimension $n$.
\end{exa}
\pause

\begin{thm} \label{thm6}
\hilite{BG}{Let $A$ and $B$ be essentially finite type $\K$-algebras, and let
$f^* : A \to B$ be an \hilite{RU}{essentially smooth} homomorphism 
of relative dimension $n$. Then the complex 
$\Omega^n_{B/A}[n] \otimes_A R_A$
has an induced rigidifying isomorphism, and there 
is a unique rigid isomorphism
\[ \Omega^n_{B/A}[n] \otimes_A R_A \cong R_B . \]}
\end{thm}
\end{frame}

\begin{frame}
Taking $n = 0$ we get an important corollary:
\pause

\begin{cor} \label{cor1}
\hilite{BG}{Given an essentially \'etale  homomorphism $f^* : A \to B$, 
there is a unique rigid isomorphism
\[ B \otimes_A R_A \cong R_B . \]}
\end{cor}
\end{frame}





\section{4. Rigid Dualizing Complexes over Schemes}
\begin{frame}
\sect{4. Rigid Dualizing Complexes over Schemes}
\pause

\begin{dfn}
Let $X$ be a finite type separated $\K$-scheme. 
A \hilite{DB}{rigid dualizing complex over $X$} (relative to $\K$) is a 
the data $( \mcal{R}, \bsym{\rho})$, where:
\pause
\begin{enumerate}
\item $\mcal{R} \in \msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X)$ 
is a dualizing complex on $X$.
\item $\bsym{\rho} = \{ \rho_U \}$ is a collection 
of rigidifying isomorphisms, indexed 
by the affine open sets of $X$. Namely, for any affine open set $U$,
$\rho_U$ is a rigidifying isomorphism for the dualizing complex
$\mrm{R} \Gamma(U, \mcal{R})$ over the $\K$-algebra 
$A := \Gamma(U, \mcal{O}_X)$. 
\end{enumerate}
\pause

The condition is:
\begin{enumerate}
\rmitem{\dag} For any inclusion $V \subset U$ of affine open sets, 
with $A := \Gamma(U, \mcal{O}_X)$ and $B := \Gamma(V, \mcal{O}_X)$,
the canonical isomorphism
\[ B \otimes_A \mrm{R} \Gamma(U, \mcal{R}) \cong
\mrm{R} \Gamma(V, \mcal{R}) \]
is rigid, with respect to the 
rigidifying isomorphisms $\rho_U$ and $\rho_V$.
\end{enumerate}
\end{dfn}
\end{frame}


\begin{frame}
\only<1>{
\begin{figure}
\includegraphics[scale = 0.8, 
%bb = 150 500 595 730,
bb = 150 0 500 510,
%00pt 900pt 240pt 1000pt,
clip = true]{fig2.ps}
\caption{cap!} %\label{fig1}
\end{figure}}

\only<2>{\begin{figure}
\includegraphics[scale = 0.8, 
%bb = 150 500 595 730,
bb = 150 0 500 510,
%00pt 900pt 240pt 1000pt,
clip = true]{fig2_a.ps}
\caption{cap!} %\label{fig1}
\end{figure}}

\only<3>{\begin{figure}
\includegraphics[scale = 0.8, 
%bb = 150 500 595 730,
bb = 150 0 500 510,
%00pt 900pt 240pt 1000pt,
clip = true]{fig2_b.ps}
\caption{cap!} %\label{fig1}
\end{figure}}

\only<4>{\begin{figure}
\includegraphics[scale = 0.8, 
%bb = 150 500 595 730,
bb = 150 0 500 510,
%00pt 900pt 240pt 1000pt,
clip = true]{fig2_c.ps}
\caption{cap!} %\label{fig1}
\end{figure}}
\end{frame}



\begin{frame}
We would like to prove \hilite{DB}{existence and uniqueness of a rigid 
dualizing complex on $X$}. 
\pause

Consider an affine open set $U \subset X$, and let
$A := \Gamma(U, \mcal{O}_X)$. According to Theorem \ref{thm4}
there exists a rigid dualizing complex 
$(R_A, \rho_A)$ over $A$. Let us denote by 
$\mcal{R}_U$ the corresponding complex of sheaves on $U$, which is 
of course a dualizing complex over $U$. And let's write 
$\rho_U := \rho_A$. 
\pause

Now suppose $V \subset U$ is another affine open set, with 
$B :=  \Gamma(V, \mcal{O}_X)$ and rigid dualizing complex 
$(R_B, \rho_B)$. According to Corollary \ref{cor1} there is a 
unique isomorphism
\begin{equation} \label{eqn1}
\phi_{V/U} : \mcal{R}_U |_V \iso \mcal{R}_V
\end{equation}
in $\msf{D}(\cat{Mod} \mcal{O}_V)$ which respects rigidity. 
\pause

Therefore given an affine open set $W \subset V$, 
these isomorphisms satisfy
\[ \phi_{W / V} \circ \phi_{V/U} = \phi_{W / U} . \]
\end{frame}

\begin{frame}
The next step would be to try to glue the affine dualizing complexes
$\mcal{R}_U$ to a global complex 
$\mcal{R}_X \in \msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X)$.
\pause

But here we encounter a genuine problem: \hilite{DB}{usually objects in derived 
categories cannot be glued!}
\pause

Grothendieck's solution in the commutative case, in 
\cite{RD}, was to use Cousin complexes. 
This solution can be used in our setup too, but it has a 
disadvantage: we are forced to leave the derived category, and then 
return to it. 
\pause

We propose an alternative solution: \hilite{RU}{perverse coherent sheaves}. 
\pause

\begin{rem}
For noncommutative ringed schemes one is forced to use perverse 
coherent sheaves, since Cousin complexes are ill-behaved. See 
\cite{YZ3}.
\end{rem}
\end{frame}




\section{5. Perverse Coherent Sheaves}
\begin{frame}
\sect{5. Perverse Coherent Sheaves}
\pause

The notion of t-structures and perverse sheaves were introduced by 
Beilinson, Bernstein and Deligne \cite{BBD} around 1980. This was in 
the context of intersection cohomology on singular spaces. For such a 
space $X$ they were interested in t-structures on subcategories of
$\msf{D}(\cat{Mod} \K_X)$, where $\K_X$ is a constant sheaf of 
rings on $X$. 
\pause

Perverse coherent sheaves came into the scene only very recently, 
independently in the work of Bezrukavnikov (after Deligne) 
\cite{Bz}, Bridgeland \cite{Br}, Kashiwara \cite{Ka} and our paper 
\cite{YZ3}. 
\end{frame}

\begin{frame}
Let me recall what is a t-structure on a triangulated category 
$\cat{D}$. It consists of the datum of two full subcategories
$\cat{D}^{\leq 0}$ and $\cat{D}^{\geq 0}$ satisfying the axioms 
below, where $\cat{D}^{\leq n} := \cat{D}^{\leq 0}[-n]$
and
$\cat{D}^{\geq n} := \cat{D}^{\geq 0}[-n]$.
\pause
\begin{enumerate}
\rmitem{i} $\cat{D}^{\leq -1} \subset \cat{D}^{\leq 0}$
and
$\cat{D}^{\geq 1} \subset \cat{D}^{\geq 0}$.
%
\rmitem{ii} $\opn{Hom}_{\cat{D}}(M, N) = 0$ for
$M \in \cat{D}^{\leq 0}$ and $N \in \cat{D}^{\geq 1}$.
%
\rmitem{iii} For any $M \in \cat{D}$ there is a distinguished
triangle
\[ M' \to M \to M'' \to M'[1] \]
in $\cat{D}$ with $M'\in \cat{D}^{\leq 0}$ and
$M'' \in \cat{D}^{\geq 1}$.
\end{enumerate}
\pause

When these conditions are satisfied one defines
the \hilite{DB}{heart} of $\cat{D}$ to be the full subcategory
$\cat{D}^{0} := \cat{D}^{\leq 0} \cap \cat{D}^{\geq 0}$.
This is an abelian category. 
\end{frame}

\begin{frame}
Given a scheme $X$, the derived category
$\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X)$
has the \hilite{DB}{standard t-structure}, in which 
\[ \begin{aligned}
& \msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X)^{\leq 0}
:= \\
& \quad \{ \mcal{M} \in 
\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X) \mid
\mrm{H}^i \mcal{M} = 0 \text{ for all } i > 0 \} , \\
& \msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X)^{\geq 0}
:= \\ 
& \quad \{ \mcal{M} \in 
\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X) \mid
\mrm{H}^i \mcal{M} = 0 \text{ for all } i < 0 \} .
\end{aligned} \]
\pause
The heart 
$\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X)^{0}$
is equivalent to the category $\cat{Coh} \mcal{O}_X$ of coherent 
sheaves.
\pause

Other t-structures will be referred to as \hilite{DB}{perverse 
t-structures}.
\end{frame}

\begin{frame}
Here is an observation.
Suppose $A$ is an essentially finite type $\K$-algebra, where as 
before $\K$ is a finite dimensional regular noetherian ring. Let 
$R_A$ be the rigid dualizing complex of $A$. 
\pause

Then the duality $\mrm{D} := \mrm{RHom}_{A}(-, R_A)$ 
gives rise to a perverse 
t-structure 
\[ \begin{aligned}
& {}^{\mrm{p}}\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)^{\leq 0} := \\
& \qquad 
\{ M \mid \mrm{H}^i \mrm{D} M = 0 \text{ for all } i < 0 \}, \\
& {}^{\mrm{p}}\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)^{\geq 0} := \\
& \qquad
\{ M \mid \mrm{H}^i \mrm{D} M = 0 \text{ for all } i > 0 \} .
\end{aligned} \]
on 
$\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)$.
\pause

We call it the \hilite{DB}{rigid perverse t-structure}. 
The heart is denoted by 
${}^{\mrm{p}}\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)^{0}$.
\pause

The next theorem was proved in \cite{YZ3}.
\end{frame}

\begin{frame}
\begin{thm} \label{thm0.1}
\hilite{BG}{Let $X$ be a finite type $\K$-scheme.
Let $\star$ denote either $\leq 0$, $\geq 0$ or $0$. }

\pause
\hilite{BG}{Define 
${}^{\mrm{p}}\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} 
\mcal{O}_X)^{\star}$
to be the class of complexes 
$\mcal{M} \in  
\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X)$
such that 
\[ \mrm{R} \Gamma(U, \mcal{M}) 
\in {}^{\mrm{p}}\msf{D}^{\mrm{b}}_{\mrm{f}}
\bigl( \cat{Mod} A  \bigr)^{\star} \]
for any affine open set $U$, with $A := \Gamma(U, \mcal{O}_X)$.}
\pause

\hilite{BG}{Then}: 
\begin{enumerate}
\rmitem{1} \hilite{BG}{The pair 
\[ \bigl( 
{}^{\mrm{p}}\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X)^{\leq 0}, 
{}^{\mrm{p}}\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X)^{\geq 0} 
\bigr) \]
is a t-structure on 
$\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X)$.}
\pause
%
\rmitem{2} \hilite{BG}{The assignment 
$V \mapsto 
{}^{\mrm{p}}\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_{V})^{0}$,
for $V \subset X$ open, is a stack of abelian categories on $X$.}
\end{enumerate} 
\end{thm}
\end{frame}

\begin{frame}
Part (2) says that the objects of 
${}^{\mrm{p}}\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X)^{0}$, 
which we call \hilite{DB}{perverse coherent sheaves}, can be glued. They
behave like sheaves, and hence the name.
\pause

For any affine open set $U = \opn{Spec} A$, the dualizing complex $\mcal{R}_U$
(the sheafification of the rigid dualizing complex $R_A$) is clearly a perverse
coherent sheaf on $U$.
\pause

Thus we can use the isomorphisms 
\[ \phi_{U, V} : \mcal{R}_U|_{U \cap V} \iso 
\mcal{R}_V|_{U \cap V} , \]
deduced from equation (\ref{eqn1}), to
glue the affine dualizing complexes.
\end{frame}


\begin{frame}
\only<1>{
\begin{figure}
\includegraphics[scale = 0.8, 
%bb = 150 500 595 730,
bb = 150 0 550 510,
%00pt 900pt 240pt 1000pt,
clip = true]{fig3.ps}
\caption{cap!} %\label{fig1}
\end{figure}}

\only<2>{\begin{figure}
\includegraphics[scale = 0.8, 
%bb = 150 500 595 730,
bb = 150 0 550 510,
%00pt 900pt 240pt 1000pt,
clip = true]{fig3_a.ps}
\caption{cap!} %\label{fig1}
\end{figure}}

\only<3>{\begin{figure}
\includegraphics[scale = 0.8, 
%bb = 150 500 595 730,
bb = 150 0 550 510,
%00pt 900pt 240pt 1000pt,
clip = true]{fig3_b.ps}
\caption{cap!} %\label{fig1}
\end{figure}}

\end{frame}


\begin{frame}
In this way we obtain:
\pause

\begin{thm}
\hilite{BG}{Let $X$ be a finite type $\K$-scheme.
There exists a rigid dualizing complex 
$(\mcal{R}_X, \bsym{\rho}_X)$
over $X$ relative to $\K$, and it is unique up to a unique rigid 
isomorphism.}
\end{thm}
\end{frame}

\begin{frame}
Along the same lines, using Theorems \ref{thm5} and \ref{thm6},
we can also prove:
\pause

\begin{thm}
\hilite{BG}{Let $X$ and $Y$ be finite type $\K$-schemes,
and let $f : X \to Y$ be a morphism.}
\pause
\begin{enumerate}
\item \hilite{BG}{If $f$ is finite, then there is a unique isomorphism
\[ \mrm{R} f_* \mcal{R}_X \cong
\mrm{R} \mcal{H}om_{\mcal{O}_Y}(f_* \mcal{O}_X, \mcal{R}_Y) \]
which respects rigidity.} 
\pause
%
\item \hilite{BG}{If $f$ is smooth of relative dimension $n$, 
then there is a unique isomorphism
\[ \Omega^n_{X / Y}[n] \otimes_{\mcal{O}_X} f^* \mcal{R}_Y
\cong \mcal{R}_X  \]
which respects rigidity.}
\end{enumerate}
\end{thm}
\end{frame}

\begin{frame}
Proper morphisms and residues are treated in \cite{Ye6}.
\pause

\begin{rem}
Recently I discovered a totally new proof of the duality theorem for proper
morphisms, which uses perverse sheaves only, avoiding residue calculations. If
correct, the new proof will make the paper \cite{Ye6} much shorter. 
\end{rem}
\pause

\begin{rem}
I think all the results here work also for essentially finite type $\K$-schemes.
\end{rem}
\end{frame}








\section{6. Cohen-Macaulay Complexes}
\begin{frame}
\sect{6. Cohen-Macaulay Complexes}
\pause

As before $X$ is a finite type $\K$-scheme. Let $\mcal{R}_X$ be the 
rigid dualizing complex of $X$. 
\pause

Given a point 
$x \in X$ let $\bsym{k}(x)$ be its residue field. We denote by
$\opn{dim}_{\K}(x)$ the unique integer $i$ such that
\[ \opn{Ext}^{-i}_{\mcal{O}_{X, x}} \bigl( \bsym{k}(x), 
\mcal{R}_{X, x} \bigr) \neq 0 . \]
\pause

Then the function 
\[ \opn{dim}_{\K} : X \to \mbb{Z} \]
is a dimension function, i.e.\ 
%\negskip{1}
\[ \opn{dim}_{\K}(y)  =  \opn{dim}_{\K}(x) - 1 \]
when $y$ is an immediate specialization of $x$. 

\mbox{}
\end{frame}


\begin{frame}
\begin{exa}
Take $\K := \mbb{Z}$, the ring of integers, and 
$X := \mbf{A}^1_{\K} = \opn{Spec} \K[t]$, the affine line. 
Consider the following points in $X$: $x_0$ is the generic point;
$x_1$ is the prime ideal $(t)$; and $x_2$ is the maximal ideal 
$(t, 2)$. Then 
\[ \opn{dim}_{\K}(x_i) = 1 - i. \]
\end{exa}
%\pause
\end{frame}

\begin{frame}
\begin{figure}
\includegraphics[scale = 1, %bb = 150 500 595 730,
bb = 150 0 595 730,
%00pt 900pt 240pt 1000pt,
clip = true]{fig4.ps}
%\caption{} \label{fig1}
\end{figure}
\end{frame}

\begin{frame}
Recall from \cite{RD} that a complex
$\mcal{M} \in \msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X)$
is called \hilite{DB}{Cohen-Macaulay} if for every point $x$ the local 
cohomologies $\mrm{H}^i_x \mcal{M}$ all vanish except for
$i = - \opn{dim}_{\K}(x)$.
\pause

Here is another result from \cite{YZ3}.
\end{frame}

\begin{frame}
\begin{thm} \label{thm0.3}
\hilite{BG}{Let $X$ be a finite type scheme over $\K$, let 
$\mcal{R}_X$ be the rigid dualizing complex of $X$, and 
let $\mrm{D}$ be the duality functor
$\mrm{R} \mcal{H}om_{\mcal{O}_X}(-, \mcal{R}_X)$.}
\pause

\hilite{BG}{Then the following conditions are equivalent for 
$\mcal{M} \in
\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X)$.}
\pause

\begin{enumerate}
\rmitem{i} \hilite{BG}{$\mcal{M}$ is a perverse coherent sheaf
\tup{(}for the rigid perverse t-structure\tup{)}.}
\pause
\rmitem{ii} \hilite{BG}{$\mrm{D} \mcal{M}$ is a coherent sheaf, i.e.\ 
$\mrm{H}^i \mrm{D} \mcal{M} = 0$ for all $i \neq 0$.}
\pause
\rmitem{iii} \hilite{BG}{$\mcal{M}$ is a Cohen-Macaulay complex.}
\end{enumerate} 
\end{thm}
\pause

In particular this implies the Cohen-Macaulay complexes form an 
abelian subcategory of 
$\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{O}_X)$, 
a fact that seems to have eluded Grothendieck.
\pause

\bigskip
\begin{center}
{\em \bfseries \red - END -}
\end{center}
\end{frame}




\begin{thebibliography}{VdB1}
\bibitem[AJL]{AJL} L.\ Alonso, A.\ Jerem\'{\i}as and J.\ Lipman,
        ``Studies in Duality on Noetherian Formal Schemes 
        and Non-Noetherian Ordinary Schemes,'' Contemp.\ Math.\ 
        \textbf{244}, Amer.\ Math.\ Soc., 1999.    
\bibitem[AK]{AK} A.\ Altman and S.\ Kleiman, ``Introduction to
    Grothendieck Duality,'' Lecture Notes in Math.\ \textbf{20}, 
    Springer, 1970.
\bibitem[BBD]{BBD} A.A.\ Beilinson, J. Bernstein and P. Deligne,
    Faisceaux perverse, in ``Analyse et Topologie sur les Espaces 
    Singulieres,'' Ast\'erisque \textbf{100}, Soc.\ Math.\ France, 
    1982, 5-171.
\bibitem[Br]{Br} T. Bridgeland, Flops and derived categories,
        Invent.\ Math.\ \textbf{147} (2002), no. 3, 613-632. 
\bibitem[Bz]{Bz} R. Bezrukavnikov, 
        Perverse coherent sheaves (after Deligne),
        eprint math.AG/0005152 at http://arXiv.org.
\bibitem[Co]{Co} B. Conrad, ``Grothendieck Duality and Base Change,''
    Lecture Notes in Math.\ \textbf{1750}, Springer, 2000. 
\bibitem[HS]{HS} R. H\"ubl and P. Sastry, 
    Regular differential forms and relative duality,
    Amer.\ J.\ Math.\ {\bf 115} (1993),  no.\ 4, 749-787.
\bibitem[HK]{HK} R. H\"ubl and E. Kunz, 
    Regular differential forms and duality for projective morphisms,
    J. Reine Angew.\ Math.\ {\bf 410} (1990), 84-108. 

\bibitem[Ka]{Ka} M.\ Kashiwara, T-structures on the derived categories of
	holonomic D-modules and coherent O-modules, 
	Moscow Mathematical Journal, {\bf 4} (2004), no.\ 4, 847-868.
\bibitem[KS]{KS} M. Kashiwara and P. Schapira, 
    ``Sheaves on Manifolds,'' Springer, 1990.

\bibitem[Li]{Li} J.\ Lipman, Dualizing sheaves, differentials and
    residues on algebraic varieties, Ast\'{e}risque \textbf{117} (1984).

\bibitem[LNS]{LNS} J. Lipman, S. Nayak and P. Sastry,
    ``Variance and Duality for Cousin Complexes on Formal Schemes'', 
    Contemp.\ Math.\ {\bf 375}, AMS 2005. 

\bibitem[Ne]{Ne} A. Neeman, The Grothendieck duality theorem via 
    Bousfield's techniques and Brown representability, Jour.\ 
    Amer.\ Math.\ Soc.\ \textbf{9} (1996), 205-236.
\bibitem[RD]{RD} R. Hartshorne, ``Residues and Duality,''
        Lecture Notes in Math.\ {\bf 20},
        Springer-Verlag, Berlin, 1966.
\bibitem[VdB]{VdB} M.\ Van den Bergh, Existence theorems for 
    dualizing complexes over non-commutative graded and filtered 
    rings, J. Algebra \textbf{195} (1997), no. 2, 662-679. 

\bibitem[Ye1]{Ye1} A.\ Yekutieli,
    Dualizing complexes over noncommutative graded algebras,
    J.\ Algebra {\bf 153} (1992), 41-84.
\bibitem[Ye2]{Ye2} A.\ Yekutieli,
    ``An Explicit Construction of the
    Grothendieck Residue Complex'' (with an appendix by P.\ Sastry),
    Ast\'{e}risque {\bf 208} (1992).
\bibitem[Ye3]{Ye3} A.\ Yekutieli,
    Smooth formal embeddings and the residue complex,
    Canadian J.\ Math.\ \textbf{50} (1998), 863-896.
\bibitem[Ye4]{Ye4} A.\ Yekutieli, Residues and differential
    operators on schemes, Duke Math. J. \textbf{95} (1998), 305-341.
\bibitem[Ye5]{Ye5} A.\ Yekutieli, Dualizing complexes, Morita
    equivalence and the derived Picard group of a ring, 
    J. London Math. Soc. (2) \textbf{60} (1999), no. 3, 723-746.
\bibitem[Ye6]{Ye6} A.\ Yekutieli,
    Rigidity, Residues, and Grothendieck Duality for Schemes,
    in preparation.

    
\bibitem[YZ1]{YZ1} A.\ Yekutieli and J.J.\ Zhang,
        Rings with Auslander dualizing complexes,
        J.\ Algebra \textbf{213} (1999), no. 1, 1-51.
\bibitem[YZ2]{YZ2} A.\ Yekutieli and J.J.\ Zhang, 
        Rigid Dualizing Complexes and Perverse Modules over
        Differential Algebras, 
        Compositio Math.\ {\bf 141} (2005), 620-654.
\bibitem[YZ3]{YZ3} A.\ Yekutieli and J.J.\ Zhang, 
        Dualizing Complexes and Perverse Sheaves on 
        Noncommutative Ringed Schemes, 
        Selecta Math.\ {\bf 12} (2006), 137-177.
\bibitem[YZ4]{YZ4} A.\ Yekutieli and J.J.\ Zhang, 
    Rigid Complexes via DG Algebras, 
    to appear in Trans.\ Amer.\ Math.\ Soc. Eprint math.AC/0603733 at 
    http://arXiv.org.     
\bibitem[YZ5]{YZ5} A.\ Yekutieli and J.J.\ Zhang, 
    Rigid Dualizing Complexes over Commutative Rings,
    to appear in Algebr.\ Represent.\ Theory,
    eprint math.AG/0601654 http://arXiv.org.
\end{thebibliography}


\end{document}