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%  Duality in noncommutative algebraic geometry
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%  Amnon Yekutieli
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\begin{document}

\begin{slide*}
%\mbox{}
%\vspace{2cm}

\begin{center}
{\large \bfseries Duality in Noncommutative Algebraic Geometry}

\vspace{1ex}
Amnon Yekutieli\\
Ben Gurion University, ISRAEL

\texttt{\small http://www.math.bgu.ac.il/$\sim$amyekut}



\vspace{1ex}
{\tiny written: 24 Jan 2006}
\end{center}

\newslide
\vspace{2ex}

This is a continuation of my talk ``Dualizing Complexes 
over Noncommutative Rings''. Here is the plan of the lecture:
\begin{enumerate}
\item[1.] Noncommutative Projective Schemes: Review
\item[2.] Grothendieck-Serre Duality for $\opn{Proj} A$
\item[3.] Quasi-Coherent Ringed Schemes
\item[4.] Dualizing Complexes on Noncommutative Ringed Schemes
\item[5.] Differential Quasi-Coherent Ringed Schemes
\item[6.] Perverse Coherent Sheaves
\item[7.] The Auslander Condition Revisited
\end{enumerate}

Most of the work is joint with James Zhang (UW, Seattle). 


\newslide
\section{Noncommutative Projective Schemes: Review}

Let $\K$ be a field and let $A$ be a noetherian connected graded 
$\K$-algebra. Recall that ``connected'' means that
$A = \boplus_{i \geq 0} A_i$,
with $A_0 = \K$ and each $A_i$ a finitely generated 
$\K$-module. 

Artin and Zhang \cite{AZ} define the {\em 
noncommutative projective spectrum} $\opn{Proj} A$ as follows.

Let $\cat{GrMod} A$ be the category of graded left $A$-modules.
In it there is the subcategory $\cat{Tors} A$ consisting of 
$\mfrak{m}$-torsion modules with respect to the 
augmentation ideal $\mfrak{m}$ of $A$. 

Consider the quotient abelian category
\[ \cat{Tails} A := \frac{\cat{GrMod} A}{\cat{Tors} A} , \]
also denoted by $\cat{QGr} A$, with projection functor $\pi$.
 The ``noncommutative space
$\opn{Proj} A$'' is the space such that ``the category of 
quasi-coherent sheaves on $\opn{Proj} A$ is $\cat{Tails} A$''.

Thus if we write $X := \opn{Proj} A$ then 
\[ \cat{QCoh} X = \cat{Tails} A \]
tautologically. 

The subcategory $\cat{Coh} X$ of 
``coherent sheaves'' is the image under $\pi$
of the finitely generated modules. 

\begin{rem}
$\opn{Proj} A$ is not a genuine space. It is only a metaphor, 
invented for heuristic purposes. 
\end{rem}

The category $\cat{QCoh} X$ is equipped with a ``global sections 
functor'' 
\[ \begin{aligned}
& \Gamma(X, -) := \opn{Hom}_{\cat{QCoh} X}(\mcal{O}_X, -) : \\
& \quad \quad \quad \quad \cat{QCoh} X \to \cat{Mod} \K ,
\end{aligned}  \]
where $\mcal{O}_X$ is the class of the graded module $A$. 

\newslide
There is 
also a Serre twist functor: if $\mcal{M} \in \cat{QCoh} X$
is the class of a graded module $M$, then $\mcal{M}(1)$ is the 
class of the shifted module $M(1)$.

Recall Serre's Theorem on projective schemes: if $X$ is a 
projective (commutative) scheme over $\K$ and $\mcal{L}$ is an 
ample invertible sheaf then the commutative graded algebra 
\[ A := \boplus_{i \geq 0} \Gamma(X, \mcal{L}^i) \]
is noetherian, and
$X \cong \opn{Proj} A$. 

The noncommutative definition of \cite{AZ} 
in effect takes Serre's Theorem and turns it around. 

\newslide
\section{Grothendieck-Serre Duality for $\opn{Proj} A$}

There is a version of Grothendieck-Serre Duality
for noncommutative projective schemes. Here we must 
assume that the algebra $A$ admits a balanced dualizing complex 
$R_A$ (see first lecture). 

By \cite{YZ1, VdB} this is equivalent to $A$ 
satisfying the $\chi$ condition on both sides, and the functors 
$\Gamma_{\mfrak{m}}$, $\Gamma_{\mfrak{m}^{\mrm{op}}}$
having finite cohomological dimension.

Let $\mcal{R}_X$ be the class in 
$\msf{D}(\cat{QCoh} X)$ of $R_A[-1]$.

\begin{thm} \tup{(\cite{YZ1})}  \label{thm2.1}
There is a functorial isomorphism 
\[ \opn{RHom}_{\cat{QCoh} X}(\mcal{M}, \mcal{R}_X) \cong
\mrm{R} \Gamma(X, \mcal{M})^* \]
for $\mcal{M} \in \msf{D}^{\mrm{b}}(\cat{Coh} X)$.
\end{thm}

In the theorem $(-)^*$ denotes $\K$-linear dual. 

\newslide
This duality has a few generalizations. 
Let $X^{\mrm{op}} := \opn{Proj} A^{\mrm{op}}$ 
be the opposite noncommutative projective 
scheme, with projection functor
\[ \pi^{\mrm{op}} : \cat{GrMod} A^{\mrm{op}} \to 
\cat{QCoh} X^{\mrm{op}} . \]

Let
\[ \ul{\Gamma} : \cat{QCoh} X \to \cat{GrMod} A \]
be the functor
\[ \ul{\Gamma}\, \mcal{M}
:= \boplus_{i \geq 0} \Gamma(X, \mcal{M}(i)) . \]

\newslide
The next duality result appeared in the paper of Kapustin, 
Kuznetsov and Orlov (in a slightly weaker form). They used it to 
define vector bundles on $\opn{Proj} A$, in the study of 
noncommutative instantons. 


\begin{thm} \tup{(\cite{KKO})}
The functor
\[ \mcal{M} \mapsto 
\pi^{\mrm{op}} \opn{RHom}^{\mrm{gr}}_A(
\mrm{R} \ul{\Gamma}\, \mcal{M}, R_A[-1]) \]
is a duality
\[ \msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{QCoh} X) \to
\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{QCoh} X^{\mrm{op}}) . \]
\end{thm}

\newslide
Van den Bergh and de Naeghel \cite{NV} 
have recently proved yet another 
version of Serre Duality, this time in the bivariant formulation 
of Bondal-Kapranov \cite{BK}. 

Define a functor
\[ \mcal{R}_X \otimes^{\mrm{L}}_{\mcal{O}_X} - :
\msf{D}^-(\cat{QCoh} X) \to \msf{D}^-(\cat{QCoh} X)  \]
by
\[ \mcal{R}_X \otimes^{\mrm{L}}_{\mcal{O}_X} \mcal{M} :=
\pi  (R[-1] \otimes^{\mrm{L}}_{A} 
\mrm{R} \ul{\Gamma}\,  \mcal{M}) . \]

Let 
$\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{QCoh} X)_{\mrm{fpd}}$
be the category of complexes with finite projective dimension
(the compact objects in 
$\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{QCoh} X)$).

\newslide
\begin{thm} \tup{(\cite{NV})}
There is a functorial isomorphism
\[ \begin{aligned}
& \opn{Hom}_{\msf{D}(\cat{QCoh} X)}(\mcal{M}, \mcal{R}_X
\otimes^{\mrm{L}}_{\mcal{O}_X} \mcal{P}) \\
& \quad \cong
\opn{Hom}_{\msf{D}(\cat{QCoh} X)}(\mcal{P}, \mcal{M})^*
\end{aligned} \] 
for 
\[ \mcal{P} \in \msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{QCoh} 
X)_{\mrm{fpd}} \]
and
\[ \mcal{M} \in \msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{QCoh} X) . \]
\end{thm}

The reduces to Theorem \ref{thm2.1} when 
$\mcal{P} = \mcal{O}_X$. 

\newslide
\begin{dfn}
The noncommutative projective scheme 
$X$ is {\em smooth} if the category $\cat{Coh} X$ has finite 
global dimension; i.e.\ there is a number $d$ such that 
\[ \opn{Ext}^i_{\cat{Coh} X}(\mcal{M}, \mcal{N}) = 0 \]
for all objects $\mcal{M}, \mcal{N} \in \cat{Coh} X$
and all $i > d$.
\end{dfn}

If $X$ is smooth then every object of 
$\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{QCoh} X)$
has finite projective dimension; and the functor
$\mcal{R}_X \otimes^{\mrm{L}}_{\mcal{O}_X} -$
is an auto-duality of 
$\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{QCoh} X)$. Hence:

\begin{cor} \tup{(\cite{NV})}
If $X$ is smooth then the functor 
$\mcal{R}_X \otimes^{\mrm{L}}_{\mcal{O}_X} -$
is the Serre functor of 
$\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{QCoh} X)$.
\end{cor}



\newslide
\section{Quasi-Coherent Ringed Schemes}

In this section we look at another kind of noncommutative space. 
Throughout $\K$ is a base field. 

\begin{dfn} (\cite{YZ4})
Suppose $X$ is a $\K$-scheme, $\mcal{A}$ is a sheaf of 
$\K$-algebras on $X$, and there is 
ring homomorphism $\mcal{O}_X \to \mcal{A}$ making $\mcal{A}$ 
into a quasi-coherent $\mcal{O}_X$-module on both sides.
We call $(X, \mcal{A})$
a {\em quasi-coherent ringed scheme over $\K$}. 
\end{dfn}


\newslide
Such ringed schemes are abundant; some prototypical examples are:
\begin{enumerate}
\item $X$ is any $\K$-scheme and $\mcal{A} = \mcal{O}_X$.
\item $X = \opn{Spec} \K$ and $\mcal{A}$ is any $\K$-algebra.
\item $X$ is smooth, $\opn{char} \K = 0$ and 
$\mcal{A} := \mcal{D}_X$, the sheaf of differential operators.
\item $X$ is any $\K$-scheme and
$\mcal{A}$ is any quasi-coherent $\mcal{O}_X$-algebra.
\end{enumerate} 

\newslide
Given a ringed scheme $(X, \mcal{A})$ its opposite 
ringed scheme is $(X, \mcal{A}^{\mrm{op}})$, and the product 
is a quasi-coherent ringed scheme which we denote by 
$(X^2, \mcal{A}^{\mrm{e}})$. The definition of the product
is pretty obvious. See Figure \ref{fig1}.

It is not hard to show uniqueness of
$(X^2, \mcal{A}^{\mrm{e}})$. 
Surprisingly existence is not 
automatic! In \cite{YZ4} we proved that the product exists iff
certain Ore conditions are satisfied. (There are counterexamples.)

\newslide
\begin{figure}
\begin{center}
\begin{pspicture}(-5,-5)(12,12)
\psframe[fillstyle=solid,linecolor=lightgray,fillcolor=lightgray]
(0,0)(10,10) 
\psline[linewidth=8pt,linecolor=lightgray]{-}(0,-3)(10,-3)
\psline[linewidth=8pt,linecolor=lightgray]{-}(-3,0)(-3,10)
\pcline{->}(-0.1,4.5)(-2.4,4.5)
\bput(0.5){$\mrm{p}_{2}$}
\pcline{->}(5,-0.1)(5,-2.4)
\bput(0.5){$\mrm{p}_{1}$}
\psframe[framearc=0.2,fillstyle=vlines,linecolor=blue,
hatchcolor=blue,hatchwidth=0.2](1,1)(9,9) 
\psframe[framearc=0.5,fillstyle=vlines,linecolor=blue,
hatchcolor=blue,hatchwidth=0.2](1,-3.5)(9,-2.5) 
\psframe[framearc=0.5,fillstyle=vlines,linecolor=blue,
hatchcolor=blue,hatchwidth=0.2](-3.5,1)(-2.5,9) 
\rput(-1,-3){$X$}
\rput(3,-4){$\mcal{A}$}
\rput(-3,11){$X$}
\rput(-4.5,2.5){$\mcal{A}^{\mrm{op}}$}
\rput(5,11){$X^2$}
\rput(9.5,7){$\mcal{A}^{\mrm{e}}$}
\end{pspicture}
\end{center}
\caption{The product of $(X, \mcal{A})$ and 
$(X, \mcal{A}^{\mrm{op}})$.} 
\label{fig1}
\end{figure}




\newslide
\section{Dualizing Complexes on Noncommutative Ringed Schemes}

Let $(X, \mcal{A})$ be a noetherian quasi-coherent ringed 
scheme. By noetherian I mean that $X$ is 
noetherian, and for any affine open set $U$ the ring
$A := \Gamma(U, \mcal{A})$ is noetherian. 

Assume the 
product $(X^2, \mcal{A}^{\mrm{e}})$ exists and is noetherian too. 

%\newslide
Let 
$\mcal{R} \in 
\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{A}^{\mrm{e}})$
be some complex. We may define a functor
\[ \mrm{D} : \msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{A})
\to \msf{D}(\cat{Mod} \mcal{A}^{\mrm{op}}) \]
as follows:
\[ \mrm{D} \mcal{M} :=
\mrm{R} \opnt{p}_{2 *}\, 
\mrm{R} \mcal{H}om_{\opnt{p}_{1}^{-1} \mcal{A}}
(\opnt{p}_{1}^{-1} \mcal{M}, \mcal{R}) . \]


\newslide
\begin{figure} 
\begin{center}
\begin{pspicture}(-5,-5)(12,12)
\psframe[fillstyle=solid,linecolor=lightgray,fillcolor=lightgray]
(0,0)(10,10) 
\psline[linewidth=8pt,linecolor=lightgray]{-}(0,-3)(10,-3)
\psline[linewidth=8pt,linecolor=lightgray]{-}(-3,0)(-3,10)
\pcline{->}(-0.1,4.5)(-2.5,4.5)
\bput(0.5){$\mrm{p}_{2}$}
\pcline{->}(5,-0.1)(5,-2.5)
\bput(0.5){$\mrm{p}_{1}$}
\pscustom[fillstyle=vlines,linecolor=red,
hatchcolor=red,hatchwidth=0.3,hatchangle=0]
{\pscurve(2,2)(2,6)(5,9)(8,5)(5,1)(2,2)}
\rput(2,7){$\mcal{R}$}
\psellipse[fillstyle=vlines,linecolor=brtgreen,hatchcolor=brtgreen,
hatchwidth=0.2](8,-3)(2,0.6)
\rput(8,-4.5){$\mcal{M}$}
\psellipse[fillstyle=vlines,linecolor=brtgreen,hatchcolor=brtgreen,
hatchwidth=0.2](-3,5)(0.5,3)
\rput(-5,7){$\mrm{D} \mcal{M}$}
\psframe[framearc=0.5,fillstyle=vlines,
linecolor=brtgreen,hatchcolor=brtgreen,hatchwidth=0.2](6,0)(10,10)
\rput(11.4,8){$\mrm{p}_{1}^{-1} \mcal{M}$}
\end{pspicture}
\end{center}
\caption{The duality functor $\mrm{D}$.}
\label{fig2}
\end{figure}

This is a contravariant Fourier-Mukai transform. 
%See Figure \ref{fig2}.

\newslide
There is an opposite version of this functor:
\[ \mrm{D}^{\mrm{op}} : 
\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{A}^{\mrm{op}})
\to \msf{D}(\cat{Mod} \mcal{A}) . \]

\begin{dfn}
A complex 
$\mcal{R} \in 
\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{A}^{\mrm{e}})$
is called a {\em dualizing complex} if the adjunction morphisms
$\bsym{1} \to \mrm{D}^{\mrm{op}} \mrm{D}$ and
$\bsym{1} \to \mrm{D} \mrm{D}^{\mrm{op}}$
are both isomorphisms. (I am suppressing some details.)
\end{dfn}

The definition I just gave allows for all kinds of exotic 
dualizing complexes. 

\begin{exa}
Say $X$ is an elliptic curve 
and take $\mcal{A} := \mcal{O}_X$. Then the product is 
\[ (X^2, \mcal{A}^{\mrm{e}}) = (X^2, \mcal{O}_{X^2}) . \] 
It turns out that the Poincar\'e bundle 
$\mcal{R} \in 
\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{A}^{\mrm{e}})$
is a dualizing complex over $(X, \mcal{O}_X)$ 
in the noncommutative sense. 
\end{exa}

\newslide
We are interested in dualizing complexes $\mcal{R}$ that behave 
similarly to Grothen\-dieck's dualizing complex $\pi^! \K$. 
Hence the definition below.

\begin{dfn}
A {\em rigid dualizing complex over $(X, \mcal{A})$} is a pair
$(\mcal{R}, \bsym{\rho})$, where:
\begin{enumerate}
\item $\mcal{R} \in 
\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{A}^{\mrm{e}})$ 
is a dualizing complex supported on the diagonal in $X^2$.
(See Figure \ref{fig3}.)
\item $\bsym{\rho} = \{ \rho_U \}$ is a collection
of rigidifying isomorphisms. Namely
for any affine open set $U \subset X$, 
letting $A := \Gamma(U, \mcal{A})$ and 
\[ R := \mrm{R} \Gamma(U^2, \mcal{R}) \in
\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} A^{\mrm{e}}) , \] 
the pair $(R, \rho_U)$ is a rigid dualizing complex over $A$.
\item The collection $\bsym{\rho}$ satisfies 
a compatibility condition for inclusions of affine open 
sets. 
\end{enumerate}
\end{dfn}


\newslide
\begin{figure}
\begin{center}
\begin{pspicture}(-5,-5)(12,12)
\psframe[fillstyle=solid,linecolor=lightgray,fillcolor=lightgray]
(0,0)(10,10) 
\psline[linewidth=8pt,linecolor=lightgray]{-}(0,-3)(10,-3)
\psline[linewidth=8pt,linecolor=lightgray]{-}(-3,0)(-3,10)
\pcline{->}(-0.1,4.5)(-2.4,4.5)
%\bput(0.5){$\mrm{p}_{2}$}
\pcline{->}(5,-0.1)(5,-2.4)
%\bput(0.5){$\mrm{p}_{1}$}
\psframe[framearc=0.1,fillstyle=none,linecolor=blue,
linewidth=0.15](4,4)(8,8) 
\rput(3.3,7){$U^2$} 
%\psframe[framearc=0.1,fillstyle=none,linecolor=blue,
%linewidth=0.15](4,-3.5)(8,-2.5) 
\psellipse[fillstyle=none,linecolor=blue,
linewidth=0.15](6,-3)(2,0.6)
\rput(6,-4){$U$} 
%\psframe[framearc=0.1,fillstyle=none,linecolor=blue,
%linewidth=0.15](-3.5,4)(-2.5,8)
\psellipse[fillstyle=none,linecolor=blue,
linewidth=0.15](-3,6)(0.6,2)
\rput(-4,6){$U$} 
\rput(9.5,8.2){$\mcal{R}$}
%\rput(9,7){$\mcal{R}_{\mcal{A}|_U}$}
\pscustom[fillstyle=vlines,linecolor=red,
hatchcolor=red,hatchwidth=0.1,hatchangle=0,hatchsep=0.1]
{\pspolygon[linecolor=red,linearc=0.2]
(0,0.4)(9.6,10)(10,9.6)(0.4,0)}
%
%\pscurve[linewidth=8pt,linecolor=red]
%{cc-cc}(0,0)(5,5)(10,10) 
\end{pspicture}
\end{center}
\caption{A dualizing complex $\mcal{R}$ supported on the 
diagonal.}
\label{fig3}
\end{figure}

%See Figure \ref{fig3} for an illustration.

\newslide
\begin{exa}
Consider a separated finite type $\K$-scheme $X$ with structural 
morphism $\pi : X \to \opn{Spec} \K$.

If one looks carefully at the variance properties of 
the dualizing complex $\mcal{R} := \pi^! \K$ 
that are worked out in  \cite{RD}, one sees that this is in fact a 
rigid dualizing over $(X, \mcal{O}_X)$. 
\end{exa}


\newslide
\section{Differential Quasi-Coherent Ringed Schemes}

All quasi-coherent ringed schemes $(X, \mcal{A})$
that ``occur naturally'' are of the following kind.

\begin{dfn}
A {\em differential quasi-coherent ringed scheme of finite type 
over $\K$} is a quasi-coherent ringed scheme $(X, \mcal{A})$
such that:
\begin{enumerate}
\item $X$ is a finite type $\K$-scheme.
\item There is an exhaustive nonnegative filtration 
$G = \{ G_i \mcal{A} \}_{i \in \mbb{Z}}$ 
on the sheaf of rings $\mcal{A}$, such 
that $\opnt{gr}^{G} \mcal{A}$ is a coherent module over its center
$\mrm{Z}(\opnt{gr}^{G} \mcal{A})$, and  
$\mrm{Z}(\opnt{gr}^{G} \mcal{A})$ is a 
quasi-coherent $\mcal{O}_X$-algebra of finite type. 
\end{enumerate}
\end{dfn}

\newslide
\begin{exa}
If $X$ is smooth in characteristic $0$ and 
$\mcal{A} = \mcal{D}_X$ is the ring of differential operators,
then we can take $G$ to be the order filtration on $\mcal{A}$.
\end{exa}

\begin{exa}
If $\mcal{A}$ is a coherent 
$\mcal{O}_X$-algebra (e.g.\ an Azumaya algebra) then we can take
$G$ to be the trivial filtration.
\end{exa}

\newslide
Let $(X, \mcal{A})$ be a separated differential quasi-coherent 
ringed scheme of finite type over $\K$. 

The fact that $(X, \mcal{A})$ is 
differential implies that the product 
$(X^2, \mcal{A}^{\mrm{e}})$ exists. 
It is not
hard to show that the product itself is a 
differential quasi-coherent ringed scheme of finite type.

For any affine open set $U \subset X$ the ring 
$A := \Gamma(U, \mcal{A})$ is a differential $\K$-algebra of 
finite type. As explained in the first lecture, 
Van den Bergh's existence criterion implies that $A$ 
has a rigid dualizing complex $R_A$. 

Furthermore, we proved in \cite{YZ3} that $R_A$ 
is supported on the diagonal in $U^2$. 

\newslide
The fact that $R_A$ is supported on the diagonal implies that it 
sheafifies to a complex 
\[ \mcal{R}_{\mcal{A}|_U} \in
\msf{D}(\cat{Mod} \mcal{A}^{\mrm{e}}|_{U^2}) , \]
which is a dualizing complex over the affine ringed scheme
$(U, \mcal{A}|_U)$. 

%\newslide
Because of the uniqueness of rigid dualizing complexes we obtain 
canonical isomorphisms 
\[ \mcal{R}_{\mcal{A}|_U}|_{U^2 \cap V^2} \cong 
\mcal{R}_{\mcal{A}|_{V}}|_{U^2 \cap V^2} \]
in 
$\msf{D}(\cat{Mod} \mcal{A}^{\mrm{e}}|_{U^2 \cap V^2})$
for any two affine open sets $U$ and $V$.
See Figure \ref{fig4}

\newslide
\begin{figure}
\begin{center}
\begin{pspicture}(-5,-5)(12,12)
\psframe[fillstyle=solid,linecolor=lightgray,fillcolor=lightgray]
(0,0)(10,10) 
\psline[linewidth=8pt,linecolor=lightgray]{-}(0,-3)(10,-3)
\psline[linewidth=8pt,linecolor=lightgray]{-}(-3,0)(-3,10)
\pcline{->}(-0.1,4.5)(-2,4.5)
%\bput(0.5){$\mrm{p}_{2}$}
\pcline{->}(5,-0.1)(5,-2)
%\bput(0.5){$\mrm{p}_{1}$}
%
\psframe[framearc=0.1,fillstyle=none,linecolor=blue,
linewidth=0.15](4,4)(8,8) 
%\rput(3.3,7){$U^2$} 
%\psframe[framearc=0.1,fillstyle=none,linecolor=blue,
%linewidth=0.15](4,-3.5)(8,-2.5) 
\psellipse[fillstyle=none,linecolor=blue,
linewidth=0.15](6,-3)(2,0.6)
\rput(7,-4.5){$U$} 
%\psframe[framearc=0.1,fillstyle=none,linecolor=blue,
%linewidth=0.15](-3.5,4)(-2.5,8)
\psellipse[fillstyle=none,linecolor=blue,
linewidth=0.15](-3,6)(0.6,2)
\rput(-4.5,7){$U$} 
%
\psframe[framearc=0.1,fillstyle=none,linecolor=drkblue,
linewidth=0.15](2,2)(6,6) 
%\rput(1.3,4){$V^2$} 
%\psframe[framearc=0.1,fillstyle=none,linecolor=drkblue,
%linewidth=0.15](2,-3.7)(6,-2.3)
\psellipse[fillstyle=none,linecolor=drkblue,
linewidth=0.15](4,-3)(2,0.8) 
\rput(3,-4.5){$V$} 
%\psframe[framearc=0.1,fillstyle=none,linecolor=drkblue,
%linewidth=0.15](-3.7,2)(-2.3,6)
\psellipse[fillstyle=none,linecolor=drkblue,
linewidth=0.15](-3,4)(0.8,2)
\rput(-4.5,3){$V$} 
%
\rput(9.1,8){$\mcal{R}_{\mcal{A}|_U}$}
\pscustom[fillstyle=vlines,linecolor=red,
hatchcolor=red,hatchwidth=0.1,hatchangle=0,hatchsep=0.15]
{\pspolygon[linecolor=red,linearc=0.2]
(4,4.4)(7.6,8)(8,7.6)(4.4,4)}
%
\rput(3.5,1.4){$\mcal{R}_{\mcal{A}|_V}$}
\pscustom[fillstyle=vlines,linecolor=orange,
hatchcolor=orange,hatchwidth=0.1,hatchangle=45,hatchsep=0.15]
{\pspolygon[linecolor=red,linearc=0.2]
(2,2.5)(5.5,6)(6,5.5)(2.5,2)}
%
\rput(5.2,4.7){\rnode{A}{}}
\rput(12,3){\rnode{B}{\psframebox{
\parbox{6em}{canonical\\ isomorphism}}}}
\ncline{-}{A}{B}
\end{pspicture}
\end{center}
\caption{Gluing data for a rigid complex}
\label{fig4}
\end{figure}


\newslide
We would like to glue the affine dualizing complexes 
$\mcal{R}_{\mcal{A}|_U}$ into a global complex
$\mcal{R}_{\mcal{A}} \in \msf{D}(\cat{Mod} \mcal{A}^{\mrm{e}})$. 

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

Grothendieck's solution in the commutative case, in 
\cite{RD}, was to use Cousin complexes. However, as explained 
in \cite{YZ2}, this solution seldom applies in the noncommutative 
context. 

The main discovery in \cite{YZ4} is that {\em 
perverse coherent sheaves can be used instead of Cousin 
complexes to glue dualizing complexes}. 



\newslide
\section{Perverse Coherent Sheaves}

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 $X$. 

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{YZ4}. 

\newslide
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
$\cat{D}^{\geq n} := \cat{D}^{\geq 0}[-n]$.
\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}

When these conditions are satisfied one defines
the {\em 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. 

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

Other t-structures will be referred to as {\em perverse 
t-structures}.

\newslide
Here is an observation.
Suppose the ring $A$ has a rigid dualizing complex $R_A$. Then 
the duality $\mrm{D} := \mrm{RHom}_{A}(-, R_A)$ 
gives rise to a perverse 
t-structure 
\[ \begin{aligned}
& {}^{p}\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)^{\leq 0} := \\
& \qquad 
\{ M \mid \mrm{H}^i \mrm{D} M = 0 \text{ for all } i < 0 \}, \\
& {}^{p}\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)^{\geq 0} := \\
& \qquad
\{ M \mid \mrm{H}^i \mrm{D} M = 0 \text{ for all } i > 0 \} .
\end{aligned} \]
%\[ \begin{aligned}
%{}^{p}\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)^{\leq 0} := 
%\{ M \mid \mrm{H}^i \mrm{D} M = 0 \text{ for all } i < 0 \}, \\
%{}^{p}\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)^{\geq 0} := 
%\{ M \mid \mrm{H}^i \mrm{D} M = 0 \text{ for all } i > 0 \} .
%\end{aligned} \]
We call it the {\em rigid perverse t-structure}. 
The heart is denoted by 
${}^{p}\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)^{0}$.

\newslide
\begin{thm} \tup{(\cite{YZ4})} \label{thm0.1}
Let $(X, \mcal{A})$ be a separated differential
noetherian quasi-coherent ringed $\K$-scheme of finite type. 
Define
\[ \begin{aligned}
& {}^{p}\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{A})^{\star} 
:= \{ \mcal{M} \in  
\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{A}) \mid \\
& \qquad \mrm{R} \Gamma(U, \mcal{M}) 
\in {}^{p}\msf{D}^{\mrm{b}}_{\mrm{f}}
\bigl( \cat{Mod} \Gamma(U, \mcal{A}) \bigr)^{\star} \\
& \qquad \text{ for all affine open sets } U \} .
\end{aligned} \]
%\[ \begin{aligned}
%{}^{p}\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{A})^{\star} 
%:= \{ \mcal{M} \in  
%\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{A}) \mid\, & 
%\mrm{R} \Gamma(U, \mcal{M}) 
%\in {}^{p}\msf{D}^{\mrm{b}}_{\mrm{f}}
%\bigl( \cat{Mod} \Gamma(U, \mcal{A}) \bigr)^{\star} \\
%& \quad \text{ for all affine open sets } U \} .
%\end{aligned} \]
Then: 
\begin{enumerate}
\rmitem{1} The pair 
\[ \bigl( 
{}^{p}\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{A})^{\leq 0}, 
{}^{p}\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{A})^{\geq 0} 
\bigr) \]
is a t-structure on 
$\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{A})$.
\rmitem{2} The assignment 
$V \mapsto 
{}^{p}\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{A}|_{V})^{0}$,
for $V \subset X$ open, is a stack of abelian categories on $X$.
\end{enumerate} 
\end{thm}

\newslide
The last piece in the puzzle is the fact 
that the affine dualizing complexes 
$\mcal{R}_{\mcal{A}|_U}$ are perverse bimodules, namely they lie 
in
${}^{p}\msf{D}^{\mrm{b}}_{\mrm{f}}
(\cat{Mod} \mcal{A}^{\mrm{e}}|_{U^2})^{0}$.
Therefore the gluing data (arising from uniqueness of rigid 
dualizing complexes) becomes effective. 

We thus obtain:

\begin{thm} \tup{(\cite{YZ4})} \label{thm0.2}
Let $(X, \mcal{A})$ be a separated differential 
quasi-coherent ringed $\K$-scheme of finite type.
Then there exists a rigid dualizing complex 
$(\mcal{R}_{\mcal{A}}, \bsym{\rho})$ over $\mcal{A}$. 
It is unique up to a unique isomorphism in 
$\msf{D}^{\mrm{b}}_{\mrm{c}}(\cat{Mod} \mcal{A}^{\mrm{e}})$.
\end{thm}


\newslide
\section{The Auslander Condition Revisited}

Let $A$ be a differential algebra of finite type over $\K$.
In the first lecture I stated that the rigid dualizing complex 
$R_A$ of $A$ satisfies the Auslander condition. 

\begin{dfn}
The canonical dimension $\opn{Cdim} M$ of an $A$-module $M$
is defined by
\[ \begin{aligned}
&\opn{Cdim} M := \\
& \quad - \inf\, \{ q \mid \opn{Ext}^q_A(M, R_A) \neq 0 \}
\in {\mbb{Z}} \cup \{ -\infty \} 
\end{aligned} \]
for a finitely generated $A$-module $M$, and by 
\[ \begin{aligned}
& \opn{Cdim} M := \\
& \quad \opn{sup}\, \{ \opn{Cdim} M' \mid M' \subset M 
\text{ is finitely generated} \} 
\end{aligned} \]
in general. 
\end{dfn}

\newslide
The Auslander condition implies that $\opn{Cdim}$ is a ``nice'' 
dimension function.

Here is an alternative characterization of the rigid perverse 
t-structure on 
$\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)$,
which resembles the original definition in 
\cite{BBD}. 

For a module $M$ and any integer $i$ define 
$\Gamma_{\msf{M}_i} M$ to be the biggest submodule of 
$M$ with $\opn{Cdim} \leq i$. This is a functor 
\[ \Gamma_{\msf{M}_i} : \cat{Mod} A \to \cat{Mod} A , \] 
and we denote by $\mrm{H}^j_{\msf{M}_i}$ its $j$-th right 
derived functor. Thus $\mrm{H}^j_{\msf{M}_i} M$
is the ``$i$-th cohomology of $M$ with supports in 
$\msf{M}_i$''. 

\newslide
\begin{thm}~\tup{(\cite{YZ3})} \label{thm0.4}
Let $A$ be a differential $\K$-algebra of finite type and 
$M \in \msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)$. 
\begin{enumerate}
\item $M \in 
{}^{p}\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)^{\leq 0}$
iff $\opn{Cdim} \mrm{H}^j M < i$ for all integers $i, j$ such that 
$j > -i$.
\item $M \in 
{}^{p}\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)^{\geq 0}$
iff $\mrm{H}^j_{\msf{M}_i} M = 0$ for all integers $i, j$ such that 
$j < -i$.
\end{enumerate}
\end{thm}


\end{slide*}


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\end{thebibliography}



\end{document}
