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\begin{document}

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

\begin{center}
{\large \bfseries Dualizing Complexes over Noncommutative Rings}

\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}

Here is the plan of my lecture:
\begin{enumerate}
\item[1.] Notation, and Review of Derived Categories
\item[2.] Dualizing Complexes
\item[3.] Existence of Dualizing Complexes
\item[4.] The Auslander Condition
\item[5.] Classification of Dualizing Complexes
\item[6.] Applications in Ring Theory
\end{enumerate}

There will be a second talk about the geometric aspects of 
noncommutative duality.

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


\newslide
\section{Notation, and Review of Derived Categories}

Let $A$ be a ring. We denote by $\cat{Mod} A$ the category of 
left $A$-modules.

The objects of the derived category $\msf{D}(\cat{Mod} A)$ are 
complexes of $A$-modules 
\[ M = \bigl( \cdots \to M^{-1} \to M^{0} \to M^{1} 
\to \cdots \bigr) . \]

Recall that a homomorphism of complexes
$\phi: M \to N$ is a quasi-isomorphism if 
$\mrm{H}^i(\phi) : \mrm{H}^i M \to \mrm{H}^i N$
is an isomorphism for all $i$.

The morphisms 
$\psi: M \to N$ in $\msf{D}(\cat{Mod} A)$ are of the form
$\psi = \phi_2^{-1} \circ \phi_1$ where 
$\phi_1: M \to L$ is a homomorphism of complexes and 
$\phi_2: N \to L$ is a quasi-isomorphism.

\newslide
There is a full embedding
\[ \cat{Mod} A \inj \msf{D}(\cat{Mod} A) \]
which is gotten by viewing a module $M$ as 
a complex concentrated in degree $0$. 

Of utmost importance for us is the derived functor
$\opn{RHom}$. Given complexes
$M, N \in \msf{D}(\cat{Mod} A)$
there is a complex
\[ \opn{RHom}_A(M, N) \in 
\msf{D}(\cat{Mod} \mbb{Z})  \]
depending functorially on $M$ and $N$. 

If $N$ happens to be an
$A$-bimodule then 
\[ \opn{RHom}_A(M, N) \in \msf{D}(\cat{Mod} A^{\mrm{op}}) , \]
where $A^{\mrm{op}}$ is the opposite ring. 

\newslide
There's a functorial isomorphism
\[ \mrm{H}^i \opn{RHom}_A(M, N) \cong 
\opn{Hom}_{\msf{D}(\cat{Mod} A)}(M, N[i]) \]
where $N[i]$ is the shifted complex.

If $M, N \in \cat{Mod} A$ then we recover the familiar Exts:
\[ \mrm{H}^i \opn{RHom}_A(M, N) = \opn{Ext}^i_A(M, N) . \]


\newslide
\section{Dualizing Complexes}

Dualizing complexes on (commutative) schemes were 
introduced by Grothendieck in the 1960's, in the book \cite{RD}. 

Let us recall the definition of a dualizing complex over a 
commutative noetherian ring $A$. It is a complex
$R \in \msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)$
such that the  contravariant functor
\[ \opn{RHom}_A(-, R) : 
\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)
\to \msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A) \]
is a duality (i.e.\ a contravariant equivalence). 
(I am omitting some details.) 

Here 
$\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)$ 
is the derived category of bounded complexes with
finitely generated cohomology modules. 

\newslide
\begin{exa}
Let $\K$ be a field. Then the complex $R := \K$ is a dualizing 
complex over $\K$. The duality $\opn{RHom}_{\K}(-, \K)$ extends the 
usual duality of linear algebra. 
\end{exa}

So far for the classical commutative picture. From now on 
$\K$ will be a field, and $A$ will be a noetherian, unital, 
associative $\K$-algebra (not necessarily commutative). 

\newslide
We shall write $A^{\mrm{e}} := A \otimes_{\K} A^{\mrm{op}}$,
where $A^{\mrm{op}}$ is the opposite ring. So
$\cat{Mod} A^{\mrm{e}}$ is the category of $A$-bimodules. 

\begin{dfn} (\cite{Ye1}) \label{dfn1}
A complex $R \in \msf{D}^{\mrm{b}}(\cat{Mod} A^{\mrm{e}})$
is called {\em dualizing} if the functor
\[ \mrm{RHom}_A (-, R) :
\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A) \to 
\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A^{\mrm{op}}) \]
is a duality, with adjoint  
$\mrm{RHom}_{A^{\mrm{op}}} (-, R)$.
(Again I'm suppressing some details.)
\end{dfn}

\begin{exa}
The complex $R := A$ is a dualizing complex over $A$ iff $A$ is a 
Gorenstein ring (i.e.\ $A$ has finite injective dimension as left 
and right module over itself).
\end{exa}

\newslide
There is a graded version of dualizing complex. Suppose $A$ is a 
connected graded algebra, namely $A = \boplus_{i \geq 0} A_i$,
with $A_0 = \K$ and each $A_i$ a finitely generated 
$\K$-module. 

Consider the category $\cat{GrMod} A$ of 
graded left $A$-modules. Similarly to Definition \ref{dfn1} we may
define a graded dualizing 
complex $R \in \msf{D}^{\mrm{b}}(\cat{GrMod} A^{\mrm{e}})$.

The augmentation ideal of $A$ is denoted by $\mfrak{m}$, and the 
left (resp.\ right) $\mfrak{m}$-torsion functor is denoted by
$\Gamma_{\mfrak{m}}$ (resp.\ $\Gamma_{\mfrak{m}^{\mrm{op}}}$).

We let $A^* := \opn{Hom}_{\K}^{\mrm{gr}}(A, \K)$, the graded dual 
of $A$. 

\newslide
\begin{dfn} (\cite{Ye1})
Let $A$ be a connected graded $\K$-algebra. A
graded dualizing complex $R$ is called {\em balanced} if 
\[ \mrm{R} \Gamma_{\mfrak{m}} R \cong
\mrm{R} \Gamma_{\mfrak{m}^{\mrm{op}}} R \cong A^* \]
in $\msf{D}^{\mrm{b}}(\cat{GrMod} A^{\mrm{e}})$.
\end{dfn}

It is known that a balanced dualizing complex is unique up to 
isomorphism. 

\newslide
Again $A$ is any noetherian $\K$-algebra (not graded). 
Van den Bergh discovered the following condition on a 
dualizing complex $R$ that turns out to be extremely powerful. 

\begin{dfn} (\cite{VdB})
Let $R$ be a dualizing complex over $A$.
Suppose there is an isomorphism 
\[ \rho : R \iso \opn{RHom}_{A^{\mrm{e}}}(A, 
R \otimes_{\K} R) \]
in $\msf{D}(\cat{Mod} A^{\mrm{e}})$. Then $R$ is called a {\em 
rigid dualizing complex} and $\rho$ is a {\em rigidifying 
isomorphism}. 
\end{dfn}

\begin{thm} \tup{(\cite{VdB}, \cite{YZ1})}
A rigid dualizing complex 
$(R, \rho)$ is unique up to a unique isomorphism in 
$\msf{D}(\cat{Mod} A^{\mrm{e}})$. 
\end{thm}

\newslide
\begin{exa}
If $A$ is a commutative finitely generated $\K$-algebra, 
$X := \opn{Spec} A$ and $\pi : X \to \opn{Spec} \K$ is the 
structural morphism, then the dualizing complex 
$R := \mrm{R} \Gamma(X, \pi^! \K)$ from \cite{RD}
is rigid. 
\end{exa}

\begin{exa}
If $A$ is finite over $\K$ then the bimodule
$A^* := \opn{Hom}_{\K}(A, \K)$ is a rigid dualizing complex over 
$A$.
\end{exa}



\newslide
\section{Existence of Dualizing Complexes}
The question of existence of rigid dualizing complexes
is quite hard. The best existence criterion we know is due to 
Van den Bergh. 

\begin{thm}~\tup{(\cite{VdB})}
Suppose $A$ admits a nonnegative exhaustive
filtration $F = \{ F_i A \}_{i \in \mbb{Z}}$ such that the graded 
algebra  $\bar{A} := \opnt{gr}^F A$ is a connected graded,
commutative, finitely generated $\K$-algebra. Then $A$ has a rigid 
dualizing complex.
\end{thm}

\newslide
Here is an outline of the proof. Let 
\[ \tilde{A} := \boplus_i \, (F_i A) t^i \subset A[t] \]
be the Rees algebra, where $t$ is a central indeterminate of degree 
$1$. So $\bar{A} \cong \tilde{A} / (t)$ and 
$A \cong \tilde{A} / (t - 1)$. 

Since $\bar{A}$ is commutative it follows that $\tilde{A}$
satisfies the $\chi$ condition of \cite{AZ}. 
This implies that the local duality functor
$\tilde{M} \mapsto (\mrm{R} \Gamma_{\tilde{\mfrak{m}}}\, 
\tilde{M})^*$
is represented by a balanced dualizing complex
$\tilde{R}$ over $\tilde{A}$. Then 
\[ R_A := A \otimes_{\tilde{A}} \tilde{R}[-1] 
\otimes_{\tilde{A}} A \]
is a rigid dualizing complex over $A$.

One should think of the filtration $F$ as a 
``compactification of $\opn{Spec} A$''. Indeed if $A$ is 
commutative then $\opn{Proj} \tilde{A}$ is a projective 
$\K$-scheme, $\{ t = 0 \}$ is an ample divisor, and its complement 
is isomorphic to $\opn{Spec} A$.

\newslide
In practice often an algebra $A$ comes equipped with a
filtration $G$ that satisfies the conditions of the next 
definition, but is not connected (i.e.\ $\opnt{gr}^G A$ is not a 
connected graded $\K$-algebra).

\begin{dfn}
A nonnegative exhaustive
filtration $G = \{ G_i A \}_{i \in \mbb{Z}}$ such that 
$\opnt{gr}^G A$ is finite over its center $\mrm{Z}(\opnt{gr}^G A)$, 
and $\mrm{Z}(\opnt{gr}^G A)$ is a finitely generated $\K$-algebra,
is called a {\em differential filtration of finite type}.  

If $A$ admits such a filtration then it is called a 
{\em differential $\K$-algebra of finite type}. 
\end{dfn}

\newslide
We call the next result the ``Theorem on the Two Filtrations''. 
A slightly weaker result appeared in \cite{MS}.

\begin{thm}~\tup{(\cite{YZ5})} 
Assume the ring $A$ has a differential filtration of finite type 
$G$. Then there exists a differential filtration of finite type 
$F$ on $A$ such that the graded algebra  $\opnt{gr}^F A$ 
is connected and commutative.
\end{thm}

\newslide
The prototypical example is:

\begin{exa}
Let $\opn{char} \K = 0$. Consider the first Weyl algebra
\[ A := \K \bra{x, y} / (yx - xy - 1) . \]
It is of course isomorphic to the ring of differential operators 
$\mcal{D}(\mbf{A}^1)$ on the affine line
$\mbf{A}^1 = \opn{Spec} \K[x]$,
via $y \mapsto \frac{\partial}{\partial x}$. 

The first filtration of $A$ is the filtration $G$ by order of 
operator, namely $\opn{deg}^G(x) = 0$ and
$\opn{deg}^G(y) = 1$. The filtration $G$ has the benefit of 
localizing to a filtration of the sheaf of differential operators
$\mcal{D}_{\mbf{A}^1}$. However 
$\opnt{gr}_0^G A = \K[\bar{x}]$, 
so $\opnt{gr}^G A$ is not connected. 

The second filtration of $A$ is the filtration
$F$ in which $\opn{deg}^F(x) = \opn{deg}^F(y) = 1$. 
Here $\opnt{gr}^F A$ is a polynomial algebra in the variables
$\bar{x}, \bar{y}$, both of degree $1$, so it is connected.
\end{exa}

\newslide
More examples of differential $\K$-algebras of finite type are:

\begin{exa}
The ring $\mcal{D}(X)$ of differential operators 
on a smooth affine variety $X$ in characteristic $0$. 
The rigid dualizing complex is 
$\mcal{D}(X)[2n]$ where $n := \opn{dim} X$.
\end{exa}

\begin{exa}
The universal enveloping algebra $\mrm{U}(\mfrak{g})$
of a finite dimensional Lie algebra $\mfrak{g}$. The rigid 
dualizing complex is 
$\mrm{U}(\mfrak{g}) \otimes (\bwedge^n \mfrak{g})[n]$
 where $n := \opn{dim} \mfrak{g}$.
\end{exa}

\begin{exa}
Generalizing the previous two examples, the universal enveloping 
algebroid $\mrm{U}_C(L)$, where $C$ is a f.g.\ commutative 
$\K$-algebra and $L$ is a f.g.\ Lie algebroid over $C$.
\end{exa}

\newslide
\begin{exa}
Any quotient ring $A / I$ or any matrix ring
$\mrm{M}_n(A)$ of a differential $\K$-algebra of finite type $A$.
\end{exa}

By combining Van den Bergh's existence result with the Theorem on 
the Two Filtrations, and some more work, we get:

\newslide
\begin{thm} \tup{(\cite{YZ5})}
Let $A$ be a differential $\K$-algebra of finite type.
\begin{enumerate}
\item $A$ has a rigid dualizing complex $R_A$, which is unique up 
to a unique rigid isomorphism.
\item Suppose $A'$ is a localization of $A$ such that each bimodule
$\mrm{H}^i R_A$ is evenly localizable to $A'$. Then $A'$ has a 
rigid dualizing complex $R_{A'}$, and there is a unique rigid 
localization morphism  
\[ \mrm{q}_{A / A'} : R_A \to R_{A'} . \]
\item Suppose $A \to B$ is a finite centralizing homomorphism.
Then $B$ has a rigid dualizing complex $R_B$, and there is a 
unique rigid trace morphism 
\[ \opn{Tr}_{B/A} : R_B \to R_A . \]
\end{enumerate}
\end{thm}

\newslide
``Evenly localizable'' is a variant of the Ore condition. 
Part (2) basically says that 
\[ R_{A'} \cong A' \otimes_A R_A \otimes_A A'  \]
in $\msf{D}(\cat{Mod} {A'}^{\, \mrm{e}})$. 

And part (3) says that 
\[ R_B \cong \opn{RHom}_A(B, R_A) \cong 
\opn{RHom}_{A^{\mrm{op}}}(B, R_A) \]
$\msf{D}(\cat{Mod} A^{\mrm{e}})$.

\newslide
\begin{rem}
I wish to amplify the significance of part (3) of the theorem. 
Suppose $B = A / I$ and 
$M \in \msf{D}(\cat{Mod} A^{\mrm{e}})$. Then 
$\opn{Ext}^i_A(B, M)$ is a
$B \otimes_{\K} A^{\mrm{op}}$ -module, but usually it is {\em not} 
a $B \otimes_{\K} B^{\mrm{op}}$ -module, i.e.\
\[ \opn{Ext}^i_A(B, M) \cdot I \neq 0 . \]

The existence of the rigid trace implies, among other things, 
that $\opn{Ext}^i_A(B, R_A)$ is indeed a 
$B \otimes_{\K} B^{\mrm{op}}$ -module.
\end{rem}

Applications of this theorem to ring theory will be discussed in 
Section 6. The geometric significance of part (2) will be 
explained in the second lecture. 

\newslide
\section{The Auslander Condition}

We continue with the hypothesis that $A$ is a noetherian algebra 
over a field $\K$. 

\begin{dfn} (\cite{Ye2}, \cite{YZ1})
Let $R$ be a dualizing complex over $A$. We say $R$ is {\em 
Auslander} if the two conditions below hold.
\begin{enumerate}
\rmitem{i} For any finitely generated $A$-module $M$, any 
integers $p > q$ and any $A^{\mrm{op}}$-submodule
$N \subset \opn{Ext}^p_{A}(M, R)$ one has
$\opn{Ext}^q_{A^{\mrm{op}}}(M, R) = 0$.
\rmitem{ii} The same after exchanging $A$ and $A^{\mrm{op}}$.
\end{enumerate}
\end{dfn}

\newslide
This is a generalization of the classical notion of 
{\em Auslander-Gorenstein} ring. Indeed, a $\K$-algebra $A$ is
called Auslander-Gorenstein precisely if it is Gorenstein, and
the dualizing complex
$R := A$ is Auslander in the sense of the definition above.

Auslander-Gorenstein rings were studied by Gabber, Levasseur and 
Bj\"{o}rk, especially in the context of $\mcal{D}$-modules. 

However 
the Gorenstein condition is very restrictive (recall that unlike 
the commutative situation, a noncommutative noetherian
ring $A$ is seldom a quotient of a ``nice'' noetherian ring). 

\newslide
On the other hand, Auslander dualizing complexes are relatively 
easy to find:

\begin{thm} \tup{(\cite{YZ5})}
Suppose $A$ is a differential $\K$-algebra of finite type. Then 
its rigid dualizing complex $R_A$ is Auslander.
\end{thm}

Applications of the theorem to ring theory will be discussed in 
Section 6. The geometric significance (the relation with 
perverse t-structures) will be explained in the second lecture. 

\newslide
\section{Classification of Dualizing Complexes}

In the commutative case the dualizing complexes are classified by 
the Picard group. Namely, given two dualizing complexes $R, R'$ 
over a commutative noetherian ring $A$, one has
\[ R' \cong L[n] \otimes_A R \] 
for some invertible $A$-module $L$ and 
some integer $n$. See \cite{RD}.

The noncommutative picture is much more complicated. Again let
$A$ be a noetherian algebra over a field $\K$. 
A {\em two-sided tilting complex} over $A$ is a complex
$P \in \msf{D}^{\mrm{b}}(\cat{Mod} A^{\mrm{e}})$
such there exists some 
$Q \in \msf{D}^{\mrm{b}}(\cat{Mod} A^{\mrm{e}})$
and isomorphisms
\[ P \otimes^{\mrm{L}}_{A} Q \cong
Q \otimes^{\mrm{L}}_{A} P \cong A \]
in $\msf{D}(\cat{Mod} A^{\mrm{e}})$.

\newslide
\begin{dfn} (\cite{Ye3})
The derived Picard group of $A$ is the group
\[ \opn{DPic}(A) :=
\frac{ \{ \text{two-sided tilting complexes over } A \}}
{\text{isomorphism}} . \] 
\end{dfn}

The derived Picard group classifies dualizing complexes in the 
following sense:

\begin{thm} \tup{(\cite{Ye3})}
Assume $A$ has at least one dualizing complex. Then the action
of $\opn{DPic}(A)$ on the set 
\[ \frac{ \{ \text{\rm dualizing complexes over } A \}}
{\text{\rm isomorphism}} , \] 
given by 
$(P, R) \mapsto P \otimes^{\mrm{L}}_{A} R$,
is transitive with trivial stabilizers.
\end{thm}

\newslide
The group $\opn{DPic}(A)$ always contains the subgroup 
$\opn{Pic}(A) \times \mbb{Z}$, where $\opn{Pic}(A)$ is the 
noncommutative Picard group of $A$ (consisting of invertible 
bimodules), and $\mbb{Z}$ is generated by the shift $\sigma$.

However when $A$ is neither commutative nor local, often 
$\opn{DPic}(A)$ is bigger than $\opn{Pic}(A) \times \mbb{Z}$.

\newslide
\begin{exa}
Let 
$A := \left[ \begin{smallmatrix} \K & \K \\ 0 & \K 
\end{smallmatrix} \right]$,
the algebra of upper triangular $2 \times 2$ matrices 
over $\K$. The rigid dualizing complex $R_A = A^*$ turns out to be 
a two-sided tilting complex. 

In fact the functor
$R_A \otimes^{\mrm{L}}_{A} -$ is the Serre functor of 
$\msf{D}^{\mrm{b}}_{\mrm{f}}(\cat{Mod} A)$, in the sense of 
\cite{BK}. 

Here the group $\opn{Pic}(A)$ is trivial, and
$\opn{DPic}(A) \cong \mbb{Z}$, generated by the class $\nu$ 
of $R_A$. The shift satisfies $\sigma = \nu^3$.
Thus
\[ \opn{Pic}(A) \times \mbb{Z} \subsetneqq \opn{DPic}(A) . \]

The relation $\sigma = \nu^3$ says that $A$ has ``Calabi-Yau 
dimension $\frac{1}{3}$'', in the terminology of Kontsevich. 
See \cite{MY} for details.
\end{exa}


\newslide
\section{Applications in Ring Theory}

Here are a few applications of the theory of dualizing 
complexes.

\subsection{Left vs.\ Right Gorenstein}
In \cite{Jo1} J\"{o}rgensen used balanced dualizing complexes to 
prove that a connected graded algebra $A$ is left Gorenstein iff 
it is right Gorenstein.

\newslide
\subsection{Free Resolutions}
J\"{o}rgensen \cite{Jo2} used balanced dualizing complexes 
(implicitly) to establish a noncommutative version of 
Castelnuovo-Mumford regularity. 

In \cite{Jo3} he proceeded to show 
that if $A$ is a Koszul connected graded algebra with balanced 
dualizing complex, then any finitely generated $A$-module $M$,
possibly after truncating low degrees, will 
admit a linear free resolution.
 
\newslide
\subsection{Duals of Verma Modules}
Consider the universal enveloping algebra 
$A := \mrm{U}(\mfrak{g})$ of a 
finite dimensional Lie algebra $\mfrak{g}$. In \cite{Ye4} we
described the structure of the rigid dualizing complex of
$A$ (this had 
been conjectured by Van den Bergh). 

As a consequence, and using 
the functoriality of rigid dualizing complexes (the rigid trace) 
we extended results of Duflo, Brown and Levasseur \cite{BL} 
regarding the Ext duals of Verma modules. 

\newslide
\subsection{Multiplicities of Injectives}
In \cite{YZ4} we obtained several results regarding multiplicities 
of indecomposable injectives in the minimal injective resolution 
of a ring $A$. 

These results extend work of previous authors
(see Barou and Malliavin \cite{BM}, Brown and Levasseur \cite{BL}).


Of particular interest is the case $A = \mrm{U}(\mfrak{g})$, 
the universal enveloping algebra of a 
finite dimensional Lie algebra $\mfrak{g}$. 
Earlier papers on this topic tended to rely
on localization; and this restricted their scope to 
solvable Lie algebras. 

Since Auslander rigid dualizing complexes
were used in \cite{YZ4}, we
were able to obtain similar results for any Lie algebra 
(solvable or not). 

\newslide
\subsection{Homological Transcendence Degree}
In the paper \cite{YZ6} we introduced a new notion 
of transcendence degree for division rings,
called the {\em homological transcendence degree}, and denoted by
$\opn{Htr} D$. 

This invariant 
seems to be better-behaved than other noncommutative invariants 
meant to generalize the commutative transcendence degree.

For instance, if $D$ is the total ring of fractions of an 
Artin-Schelter regular algebra $A$ of global dimension $n$, then
$\opn{Htr} D = n$. 

This, and some other good properties of the
homological transcendence degree, 
were established with the aid of Auslander rigid dualizing complexes.

\newslide
\subsection{Catenarity}
Recall that a noetherian 
ring $A$ is called catenary if given two prime 
ideals $\mfrak{p} \subset \mfrak{q}$, any saturated chain of prime 
ideals
\[ \mfrak{p} = \mfrak{p}_0 \subset \mfrak{p}_1 \subset \cdots
\subset \mfrak{q} \]
has the same length. 

It is known that if $A$ is commutative and 
admits some dualizing complex then it is catenary (see \cite{RD}).

In \cite{YZ1} we proved that some rings of quantum type 
are catenary. This was extended by Goodearl-Zhang \cite{GZ} to  
the case of the quantized coordinate rings $\mcal{O}_q(G)$.
 
\end{slide*}


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\end{document}
