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% {Duality in Noncommutative Algebra and Geometry}
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\begin{document}

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\mbox{}
\vspace{2cm}

\begin{center}
{\large \bfseries Duality in Noncommutative Algebra and Geometry} 

\vspace{1ex}
Lecture Notes\\
9 June 2008

\vspace{3ex}
Amnon Yekutieli\\
Ben Gurion University, ISRAEL

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

\vspace{2ex}
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% \begin{frame}
% \titlepage
% \end{frame}

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

\begin{enumerate}
\item[1.] Recalling Duality
\item[2.] The Derived Category
\item[3.] Duality in Commutative Algebraic Geometry
\item[4.] Duality in Noncommutative Algebra
\item[5.] Applications in Ring Theory
\item[6.] Noncommutative Algebraic Geometry
\end{enumerate}
%\end{frame}


\section{Recalling Duality}
%\begin{frame}
%\sect{1. Recalling Duality}
%\pause

Duality is one of the fundamental concepts in mathematics.
%\pause

The most basic duality is that of linear algebra. We take a vector 
space $V$ over a field $\mbb{K}$ and assign to it
\negskip{1}
\[ V^{*} = \opn{Hom}_{\mbb{K}}(V, \mbb{K}) . \]

\negskip{1}
If $V$ is finite dimensional then $V \cong V^{**}$. 
%\pause

This can be generalized in many ways. For instance we can make $V$ 
infinite, and then often we impose a topology to retain 
reflexivity (e.g.\ a Banach space). 
%\pause

Now change the ring of coefficients: instead of the field 
$\mbb{K}$ take a commutative ring $A$, and instead of a vector 
space $V$ we consider an $A$-module $M$. Immediately we run into 
difficulties. 
%\end{frame}

%\begin{frame}
\begin{exa} \label{exa1}
Let us look at $A := \mbb{Z}$, the ring of integers. A
$\mbb{Z}$-module is just an abelian group. 
%\pause

There are two distinct dualities for 
finitely generated  abelian groups. For a finite group $H$ the dual is
\[ H^{*} = \opn{Hom}_{\mbb{Z}}(H, 
\mbb{Q} / \mbb{Z}) , \]
whereas for a free group $G$ the dual is
\[ G^{*} = \opn{Hom}_{\mbb{Z}}(G, \mbb{Z}) . \]
\negskip{2}
%\pause

(We want to stay within finitely generated groups, so the 
Pontryagin dual \linebreak
$\opn{Hom}_{\mbb{Z}}(G, \mbb{R} / \mbb{Z})$ is ruled out.) 
%\pause

We know that $G^{* *} \cong G$ and  $H^{* *} \cong H$.
\end{exa}
%\pause

Is it possible to unite these two dualities into one? What 
about more complicated rings?
%\end{frame}


\section{The Derived Category}
%\begin{frame}
%\sect{2. The Derived Category}
%\pause

This is where the \hilite{RU}{derived category} enters. The idea of 
Grothendieck and Verdier (see \cite{RD}) was to work with 
\hilite{DB}{complexes of modules}. 
%\pause

A complex of $A$-modules consists of 
\[ M = \left( \cdots \to M^{i - 1} \xrightarrow{\d^{i - 1}}
M^{i} \xrightarrow{\d^{i}} M^{i + 1} \to \cdots \right) \]
where the $M^{i}$ are $A$-modules, the $\d^{i}$ are $A$-linear maps,
and $\d^{i} \circ \d^{i - 1} = 0$. 
%\pause

The module $M^i$ is called the degree $i$ component of $M$.  
%\pause

The derived category 
$\msf{D}(\cat{Mod} A)$ has complexes as objects. Morphisms in 
$\msf{D}(\cat{Mod} A)$ are more complicated to explain.
%\end{frame} 

%\begin{frame}
For any $i$, the \hilite{DB}{ $i$-th cohomology} of a complex $M$ is
the
module
\[ \mrm{H}^i M := \opn{Ker}(\d^{i}) / \opn{Im}(\d^{i - 1}) . \]
%\pause
This is a functor 
\[ \mrm{H}^i : \msf{D}(\cat{Mod} A) \to \cat{Mod} A . \]
%\pause

The category of modules $\cat{Mod} A$ embeds as a full 
subcategory into $\msf{D}(\cat{Mod} A)$, as the complexes 
concentrated in degree $0$:
\[ \left( \cdots \to 0 \to M^{0} \to 0 \to \cdots \right) . \]
%\pause

There is a \hilite{RU}{ shift} operation on $\msf{D}(\cat{Mod} A)$.
Given a complex $M$ and an integer $k$, the shifted complex 
$M[k]$ has $M^{k+i}$ as its degree $i$ component.
%\end{frame}

%\begin{frame}
Given two complexes $M$ and $N$, we get a new complex 
\[ \opn{RHom}_{A}(M, N) \in \msf{D}(\cat{Mod} A) . \]
%\pause
The operation
$\opn{RHom}_{A}(-,-)$ is called the \hilite{DB}{ right derived Hom
functor}.
%\pause

The complex $\opn{RHom}_{A}(M, N)$ is related to the familiar Ext
modules. Indeed, 
\[ \mrm{H}^{i}  \opn{RHom}_{A}(M, N) = 
\opn{Ext}^{i}_{A}(M, N) . \]
%\end{frame}

%\begin{frame}
Now suppose that for a certain complex 
$R \in \msf{D}(\cat{Mod} A)$, the functor 
\[ M \mapsto M^{*} :=  \opn{RHom}_{A}(M, R) \]
is a duality, in the sense that 
$M \cong M^{* *}$.
%\pause

Then we call $R$ a \hilite{BG}{ dualizing complex}. 
%\pause

\begin{rem}
There are certain \hilite{DB}{ finiteness conditions} that we must
impose on
$R$ and $M$ in order to make things really work. These are analogous
to the finiteness already appearing in linear algebra. 
%\pause

I will ignore such conditions (as well as other technicalities)
in this lecture.
\end{rem}
%\end{frame}

%\begin{frame}
\begin{exa} \label{exa2}
Getting back to the ring $\mbb{Z}$, and the setup of Example
\ref{exa1}, one can show (without much difficulty) that the module
$R := \mbb{Z}$ is a dualizing complex. 
%\pause

The two dualities, for finite groups and free groups, are included in 
the duality $ \opn{RHom}_{\mbb{Z}}(-, \mbb{Z})$. 
%\pause

Indeed, for a free
group $G$ we have
\[ G^{*} = \opn{Ext}^{0}_{\mbb{Z}}(G, \mbb{Z}); \]
and for a finite group $H$:
\[ H^{*} = \opn{Ext}^{1}_{\mbb{Z}}(H, \mbb{Z}) . \]
\end{exa}
%\pause

Generally speaking, most commutative rings encountered in algebra 
will admit a dualizing complex. 
%\end{frame}



\section{Duality in Commutative Algebraic Geometry} 
\label{sec3}
%\begin{frame}
%\sect{3. Duality in Commutative Algebraic Geometry}
%\pause

Grothendieck's duality theory has geometric aspects to it, when we
pass from commutative rings to \hilite{DB}{ algebraic varieties}. 
%\pause

It has local features (such as local duality and residues). It also
has global features -- indeed, it is a vast generalization of Serre
duality, which talks about vector bundles on smooth projective
varieties. 
%\pause

Moreover, Grothendieck's duality is a relative theory, in the
sense that it deals with maps between varieties. 
%\pause

Applications of Grothendieck duality in algebraic geometry are 
numerous: moduli problems, resolution of singularities, 
enumerative geometry\ldots.
%\end{frame}

%\begin{frame}
I'll now try to give a taste of how Grothendieck
duality relates to familiar notions from complex differential
geometry. 
%\pause

\begin{exa}
Consider the ring $A = \mbb{C}[t]$ of 
polynomials over the field $\mbb{C}$. 
%\pause

Like in Example \ref{exa2}, the module $A$ is a dualizing complex.
However it is not the ``correct'' dualizing complex. 
%\pause

We shall view $A$ as the ring of algebraic $\mbb{C}$-valued functions 
on the affine line $\mbf{A}^{1}(\mbb{C})$. Consider the 
module of algebraic differential $1$-forms
$\Omega^{1}_{A}$,
which is a free $A$-module of rank $1$. 
%\pause

The ``correct'' dualizing complex is the shifted
complex
\[ R := \Omega^{1}_{A}[1]  , \]

\negskip{1}
i.e.\ the complex which has the
module $\Omega^{1}_{A}$ in degree $-1$, and the zero module in all
other degrees.
\end{exa}
%\end{frame}

%\begin{frame}
The reason is that we can integrate meromorphic $1$-forms. 
%\pause

Indeed, suppose $f(t) \d t$ is a meromorphic 
differential form on the projective line 
\[ \mbf{P}^{1}(\mbb{C}) = \mbf{A}^{1}(\mbb{C}) \cup \{ \infty \} , \]
and $x \in \mbf{P}^{1}(\mbb{C})$ is a point. 
%\pause

By contour integration we get the residue
\[ \opn{Res}_{x} f(t) \mrm{d} t := 
\frac{1}{2 \pi i} \oint_{x} f(t) \mrm{d} t \in \mbb{C} . \]
%\pause

An indication that the residue has geometric significance is the 
Residue Theorem: 
\[ \sum_{x \in \mbf{P}^{1}(\mbb{C})} 
\opn{Res}_{x} f(t) \mrm{d} t = 0 . \]
%\end{frame}

%\begin{frame}
The residue $\opn{Res}_{x} f(t) \d t$ above can be defined on any 
algebraic curve (over any field $\mbb{K}$), and there is a Residue 
Theorem. 
%\pause

This residue map was used by Serre to build his duality, and
to prove the Riemann-Roch Theorem for curves \cite{Se}. 
%\pause

Grothendieck expanded this idea enormously by  in the book
Residues and Duality \cite{RD}. 
%\pause

Indeed, Grothendieck showed how to define residues, in an abstract
form, on any algebraic variety $X$. These residues were
used to construct his global duality. 
%\pause

See the work of Lipman and others \cite{Li, LS, Ye2} for more
on this direction. 
%\end{frame}



\section{Duality in Noncommutative Algebra}
%\begin{frame}
%\sect{4. Duality in Noncommutative Algebra}
%\pause

Consider a noncommutative ring $A$. What is to prevent us from
seeking 
a duality of the form
$\opn{RHom}_{A}(-, R)$?
Nothing really, as long as we take $R$ to be a complex of 
\hilite{DB}{ $A$-bimodules}. 
%\pause

Suppose $A$ is a unital associative algebra over a field $\K$. 
Recall that an $A$-bimodule $M$ has two commuting actions by $A$:
left multiplication and right multiplication.
%\pause 

It is convenient to view the bimodule $M$ as a left module over the
algebra
$A \otimes_{\K} A^{\mrm{op}}$. 
Here $A^{\mrm{op}}$ is the opposite algebra, where the order of
multiplication is reversed. 
%\pause

Thus for $m \in M$ and
\[ a_1 \otimes a_2 \in A \otimes_{\K} A^{\mrm{op}} \]
we have
\[ (a_1 \otimes a_2) \cdot m = a_1 \cdot m \cdot a_2 \in M . \]
%\end{frame}

%\begin{frame}
Let us denote by $\cat{Mod} A$ the category of left $A$-modules. 
Then $\cat{Mod} A^{\mrm{op}}$ is really the category of right
$A$-modules. 
%\pause

Given a complex of bimodules $R$, we actually have
two duality functors:
\[  \opn{RHom}_{A}(-, R) : \msf{D}(\cat{Mod} A) \to
\msf{D}(\cat{Mod} A^{\mrm{op}}) \]
and
\[  \opn{RHom}_{A^{\mrm{op}}}(-, R) : \msf{D}(\cat{Mod} A^{\mrm{op}}) 
\to \msf{D}(\cat{Mod} A) . \]
%\pause

\negskip{1}
If these two functors are equivalences, then $R$ is called a 
\hilite{DB}{dualizing complex over $A$}. 
This definition goes back to my paper \cite{Ye1} from 1990. 
%\pause

It turns out that many interesting noncommutative rings admit 
dualizing complexes. A very important existence criterion was
discovered by Van den Bergh \cite{VdB1} in 1995. 
%\end{frame}

%\begin{frame}
It should be noted that many of the methods of commutative 
algebra and geometry do not apply in the noncommutative setting
(e.g.\ localization). 
%\pause

However the theory of duality works extremely
well over noncommutative rings, and hence it has become one of
the standard homological tools of ring theory. 
%\pause

One of the first observations when passing to noncommutative rings is
that the notion of ``correct'' dualizing complex becomes sharper. 
%\pause

By considerations of \hilite{RU}{ local cohomology} (for graded rings)
or the 
\hilite{DB}{ rigidity equation} of Van den Bergh
\[ R \cong  \opn{RHom}_{A \otimes_{\mbb{K}} A^{\mrm{op}}}
(A, R \otimes_{\mbb{K}} R) , \]

\negskip{1}
a particular dualizing complex is chosen. 
This is the \hilite{RU}{balanced},
respectively \hilite{DB}{ rigid}, dualizing complex. See 
\cite{Ye1, VdB1, YZ2}.
%\pause

The rigid dualizing complex enjoys several functorial properties,
similar to the commutative situation. As we shall see 
this has nice consequences. 
%\end{frame}



\section{Applications in Ring Theory}
%\begin{frame}
%\sect{5. Applications in Ring Theory}
%\pause
 
Let us look at some applications. 
%\pause

\begin{exa}
Let $\mfrak{g}$ be an $n$-dimensional semisimple Lie algebra over
$\K$, and let $A = \mrm{U}(\mfrak{g})$ be its universal enveloping
algebra. Van den Bergh \cite{VdB2} proved a Poincare-type duality
formula between Hochschild homology and cohomology of $A$. 
%\pause

This formula relies on the properties of the rigid
dualizing complex $R$ of $A$, which in this case is just 
$R = A[n]$. 
\end{exa}
%\end{frame}

%\begin{frame}
\begin{exa}
Suppose $\mfrak{b}$ is an $n$-dimensional Lie algebra over $\mbb{K}$.
%\pause

In \cite{Ye3} I proved that the rigid dualizing complex of
$\mrm{U}(\mfrak{b})$ is the complex $M[n]$, where 
$M$ is the twisted bimodule
\begin{equation} \label{eqn1}
M := \mrm{U}(\mfrak{b}) \otimes_{\mbb{K}}
\bigwedge\nolimits^{n} \mfrak{b} .
\end{equation}
%\pause

\negskip{2}
When $\mfrak{b}$ is not semisimple, $M$ could be a
nontrivial bimodule (cf.\ previous example). 
%\pause

Now suppose $\mfrak{g}$ is a simple Lie algebra. Duflo \cite{Du} and 
Brown-Levasseur \cite{BL} obtained important structural results on 
Verma modules over $\mrm{U}(\mfrak{g})$ and their duals. 
%\pause

By looking at the Borel subalgebra 
$\mfrak{b} \subset \mfrak{g}$, and using formula (\ref{eqn1}),
together with the functorial properties of rigid dualizing
complexes, I got a new proof of these results, as well as some 
generalizations. 
\end{exa}

%\hyperlink{sec6}{\beamergotobutton{skip}}
%\end{frame}

%\begin{frame}
\begin{exa} 
Suppose $G$ is a semisimple Lie group over $\mbb{C}$. Let
$\mcal{O}_q(G)$ be its quantized coordinate ring. 
Goodearl and Zhang \cite{GZ} obtained results on the structure of 
$\mcal{O}_q(G)$, using rigid dualizing complexes. 
\end{exa}
%\pause

\begin{exa}
The notion of Cohen-Macaulay module is important in commutative
algebra, and it has several equivalent definitions. It appears that
the only useful definition in the noncommutative context is the one
based on the rigid dualizing complex (cf.\ \cite{YZ2}). 
This was used, for instance, in the paper of Etingof and Ginzburg on
symplectic reflection algebras \cite{EG}. 
\end{exa}
%\pause

\begin{exa}
Suppose $D$ is a division algebra over a field $\K$. Zhang and I
\cite{YZ3} introduced the homological transcendence degree
$\opn{Htr} D$. This is a numerical invariant of $D$, which, when $D$
is commutative, is the usual transcendence degree. 
%\pause

The definition itself is quite simple minded. But the
important properties of this invariant rely on the theory of rigid
dualizing complexes.
\end{exa}
%\end{frame}




\section{Noncommutative Algebraic Geometry}
%\begin{frame} [label=sec6]
%\sect{6. Noncommutative Algebraic Geometry}
%\pause

By \hilite{RU}{noncommutative algebraic geometry} people usually
refer to the confluence of ring theory and algebraic geometry, with an
emphasis on homological methods. This is a new and developing area of
research. 
%\pause

Initial work in this area was done by Artin, Van den Bergh, Tate,
Zhang, Stafford and others (the so called ``Artin school''), in the
1980's. See the papers \cite{ATV1, AZ, SV} and their
references.
%\pause

Independently, the ``Moscow school'', consisting of Manin, Bondal,
Orlov, Kapranov and others, worked on related problems. See 
\cite{BK, BO1, BO2}. 
%\end{frame}

%\begin{frame}
The main object of study of the Artin school was 
\hilite{DB}{ noncommutative projective varieties}.
%\pause

Suppose $A$ is a noncommutative graded algebra over a field. 
Artin and Zhang \cite{AZ} showed that a meaningful geometric object,
namely the projective spectrum $\opn{Proj} A$, can be constructed by
``reversing'' Serre's theorem about coherent sheaves on projective
varieties. 
%\pause

In this way $A$ becomes the ``homogeneous coordinate ring''
of the ``projective variety'' $\opn{Proj} A$.
%\pause

This only works well if the algebra $A$ satisfies a 
condition called $\chi$. This condition is satisfied if $A$ is 
commutative, but also in many other interesting cases. 
%\pause

In \cite{YZ1} Zhang and I we proved that if the graded algebra $A$ 
has a balanced dualizing complex, then condition $\chi$ holds, and 
moreover $\opn{Proj} A$ has Serre duality. The converse was proved 
by Van den Bergh \cite{VdB1}.
%\end{frame}

%\begin{frame}
The main idea of the Moscow school was the \hilite{DB}{ Serre
functor}. This
is a very powerful abstraction of Serre's original duality. 
%\pause

\begin{exa}
Suppose $X$ is a smooth projective algebraic variety of dimension
$n$. 
%\pause

Let $\Omega^n_X$
be the sheaf of degree $n$ differential forms. 
%\pause

It is a line bundle on
$X$, sometimes called the \hilite{RU}{dualizing sheaf} or the 
\hilite{DB}{canonical bundle}. 
%\pause

Let us denote by $\msf{D}(X)$ the derived category of coherent
sheaves on $X$. 
%\pause

The rigid dualizing complex of $X$ is the complex
\[ \Omega^n_X[n] \in \msf{D}(X) . \]
%\pause

\negskip{1}
And the Serre functor $S$ of $\msf{D}(X)$ is 
\[ S \mcal{M} = \Omega^n_X[n] \otimes_{\mcal{O}_X} \mcal{M} \]
for $\mcal{M} \in \msf{D}(X)$.
\end{exa}
%\end{frame}

%\begin{frame}
A big push to noncommutative algebraic geometry came with the
realization that it is useful for some problems in mathematical
physics. See Examples below.
%\pause

With the progress on the Homological Mirror Symmetry Conjecture of
Kontsevich (see \cite{Ko}, \cite{KS}), it became clear that all the
previously disparate ideas on noncommutative algebraic geometry are
actually pretty tightly linked. 
%\pause

There are even suggestions now 
that there is a close connection to noncommutative geometry in the
sense of Connes and Quillen (e.g.\ cyclic homology).
%\end{frame}

%\begin{frame}
To finish, here are several examples that illustrate recent work. 
%\hyperlink{more_exa}{\beamergotobutton{skip}}
%\pause

\begin{exa}
This first example shows how the distinction between commutative
algebraic geometry and noncommutative ring theory can blur. 
%\pause

Let $A$ be the ring of $n \times n$ upper triangular matrices over
a field $\K$. 
%\pause

The rigid dualizing complex of $A$ is the bimodule
\[ A^* = \opn{Hom}_{\K}(A, \K) . \]
%\pause

\negskip{2}
A calculation by Kreines and myself \cite{Ye4} showed that 
\[ \underset{n+1}{\underbrace{ A^* \otimes^{\mrm{L}}_{A} \cdots
\otimes^{\mrm{L}}_{A} A^* }} \cong A[n-1] . \]
Here $\otimes^{\mrm{L}}_{A}$ is the derived tensor product. 
\end{exa}
%\end{frame}

%\begin{frame}
Now the Serre functor $S$ of $\msf{D}(\cat{Mod} A)$ is
\[ S M = A^* \otimes^{\mrm{L}}_{A} M   \]
for $M \in \msf{D}(\cat{Mod} A)$.
%\pause
 
Let us denote by $T$ the shift functor, so that
\[ T M = M[1] . \]
We see that 
\[ S^{n+1} = T^{n-1} . \]
%\pause

\negskip{2}
According to Kontsevich, this says that ``$A$ is a
Calabi-Yau manifold'' of dimension $\frac{n-1}{n+1}$.
%\end{frame}

%\begin{frame}[label=more_exa]
\begin{exa}
Kapustin, Kuznetsov and Orlov \cite{KKO} studied noncommutative
instantons (this is mathematical physics). Their classification
turned out to involve noncommutative projective surfaces. 
A significant part of the paper is devoted to
expanding and applying the noncommutative Serre duality. 
\end{exa}
%\pause

\begin{exa}
Zhang and I studied rigid dualizing complexes on certain
noncommutative spaces. This work involved the notion of perverse
coherent  sheaves. See \cite{YZ3}, \cite{YZ5}. 
%\pause

The idea that dualizing complexes should be related to perverse
coherent sheaves was realized
independently by Kashiwara [Ka]. 
\end{exa}
%\end{frame}

%\begin{frame}[label=end]
\begin{exa}
More recently, Zhang and I showed how to obtain a large part of the
original Grothendieck duality theory (namely the commutative picture)
very efficiently, using Van den Bergh's rigid dualizing complexes,
plus the theory of perverse coherent sheaves. See \cite{YZ6},
\cite{YZ7}.
\end{exa}
%\pause

Finally: 

\begin{exa}
In the paper [AKO] by Auroux, Katzarkov and Orlov, the authors prove 
the homological mirror symmetry conjecture for weighted
projective spaces and their noncommutative deformations.
(Again, mathematical physics.)
Their work uses the duality for noncommutative projective
schemes. 
\end{exa}
%\pause

\begin{center}
{\red $\sim$ \ END \ $\sim$}
\end{center}
%\end{frame}



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\end{thebibliography}


\end{document}
