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%  The derived Picard group and representations of quivers
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%  Amnon Yekutieli
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%  ** top matter **
\title{The Derived Picard Group and Representations of Quivers}
\author{Amnon Yekutieli}
\address{Department of Mathematics and Computer Science,
Ben Gurion University, 
Be'er Sheva 84105, ISRAEL}
\email{amyekut@math.bgu.ac.il \newline
\indent \textrm{\textit{WWW}}:\ 
http://www.math.bgu.ac.il/$\sim$amyekut}
\thanks{Complete version of the manuscript submitted to a journal}


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\begin{abstract}
The derived Picard group $\opn{DPic}(A)$ of a ring $A$ 
is the group of auto-equivalences of the derived category 
$\msf{D}^{\mrm{b}}(\cat{Mod} A)$
induced by tilting complexes. When $A$ is either local or 
commutative, $\opn{DPic}(A)$ is a product of the usual 
(noncommutative) Picard group $\opn{Pic}(A)$ and a cyclic group; 
in this sense $\opn{DPic}(A)$ has a noncommutative geometry content.

In this talk we consider the case where $A$ is the path algebra
of a finite quiver $\vec{\Delta}$ over a field $k$ (i.e. $A$ is a 
finite dimensional hereditary $k$-algebra). There is a natural action 
of $\opn{DPic}(A)$ on a certain infinite quiver. This action is 
faithful when $\vec{\Delta}$ is a tree, and otherwise a connected 
linear algebraic group may occur as a factor of $\opn{DPic}(A)$. At 
any rate we get an effective description of $\opn{DPic}(A)$; examples 
will be shown.

In the hereditary case $\opn{DPic}(A)$ coincides with the group of 
triangle auto-equivalences of the derived category of $A$-modules. 
This means that we can calculate the groups of auto-equivalences 
of various derived categories occurring, for example, in 
noncommutative geometry. (Joint work with J. Miyachi.)
\end{abstract}

\maketitle

%\begin{center}
%Lectures in Japan\\
%October 1999
%\end{center}

\setcounter{section}{-1}
\section{Introduction}

Let $k$ be a field, $A$ a $k$-algebra, and let 
$\cat{D}^{\mrm{b}}(\cat{Mod} A)$ be the derived 
category of $A$-modules.  The derived Picard 
group $\opn{DPic}_{k}(A)$ is the group of auto-equivalences 
of $\cat{D}^{\mrm{b}}(\cat{Mod} A)$ 
generated by tilting complexes. 

The group $\opn{DPic}_{k}(A)$ becomes interesting when $A$ is 
nonlocal and noncommutative. This takes us into the realm of 
noncommutative algebraic geometry.

Today I will talk about the case when $A$ is a finite dimensional 
$k$-algebra. For a hereditary algebra 
$A$ we can completely describe the structure of the group
$\opn{DPic}_{k}(A)$.
This is a survey of joint work with J. Miyachi. Full details 
are in the paper \cite{MY}. 



\section{Some background}

Suppose $k$ is a field and  $A$ is a (unital, associative) 
$k$-algebra. Let $\cat{Mod} A$ be the category of left $A$-modules. 
Recall that Morita Theory tells us what are all $k$-linear 
auto-equivalences 
$F : \cat{Mod} A \to \cat{Mod} A$. They are (up to isomorphism) 
$F M =  P \otimes_{A} M$, where $P$ is a $k$-central invertible 
bimodule (i.e.\ 
$A \cong P \otimes_{A} Q \cong Q \otimes_{A} P$
for some bimodule $Q$). We call the group generated by the 
invertible bimodules $P$ (with operation $P \otimes_{A} Q$) the 
(noncommutative) Picard group of $A$, denoted $\opn{Pic}_{k}(A)$.

\begin{rem}
If $A$ is commutative then $\opn{Pic}_{k}(A)$ contains the 
automorphism group $\opn{Aut}_{k}(A)$ and the usual (commutative) 
Picard group $\opn{Pic}_{A}(A)$ as subgroups.
\end{rem}

Now let us look at the bounded derived category 
$\cat{D}^{\mrm{b}}(\cat{Mod} A)$. 
We will not attempt to say too much about it now. All we need to know
are a few facts. The objects of $\cat{D}^{\mrm{b}}(\cat{Mod} A)$ 
are the bounded complexes of $A$-modules 
\[ M^{\bdot} = \left( \cdots \to 0 \to M^{p} \xrightarrow{\delta}
M^{p+1} \xrightarrow{\delta}
\cdots \to M^{q} \to 0 \to \cdots \right)  \]
where $\delta \circ \delta = 0$.
The category $\cat{Mod} A$ embeds in 
$\cat{D}^{\mrm{b}}(\cat{Mod} A)$ as the complexes concentrated in 
degree $0$. And $M[n]$ is the complex with $M$ in degree $-n$. If 
$T^{\bdot}$ is a complex of bimodules, then one can define the 
derived tensor product $T^{\bdot} \otimes^{\mrm{L}}_{A} M^{\bdot}$.
We call $T^{\bdot}$ a tilting complex if the functor
$F M^{\bdot} = T^{\bdot} \otimes^{\mrm{L}}_{A} M^{\bdot}$
is an auto-equivalence of $\cat{D}^{\mrm{b}}(\cat{Mod} A)$.
Tilting complexes were considered by Rickard and Keller.

\begin{exa}
If $P$ is an invertible bimodule then $P[n]$ is a tilting complex.
\end{exa}

\begin{dfn}
The derived Picard group $\opn{DPic}_{k}(A)$ is the group of 
isomorphism classes of tilting complexes, with operation 
$T^{\bdot} \otimes^{\mrm{L}}_{A} S^{\bdot}$.
\end{dfn}

One reason to study $\opn{DPic}_{k}(A)$ is:

\begin{thm}[Y.]
$\opn{DPic}_{k}(A)$ parameterizes the isomorphism classes of 
dualizing complexes over $A$.
\end{thm}

We note that $\opn{DPic}_{k}(A)$ contains two natural subgroups: a 
copy of $\mbb{Z}$ (represented by the complexes $A[n]$), 
and $\opn{Pic}_{k}(A)$.

\begin{thm}[Y., Zimmermann-Rouquier-...]
If $A$ is either local or commutative, then 
$\opn{DPic}_{k}(A) = \opn{Pic}_{k}(A) \times \mbb{Z}$.
\end{thm}

This however is not the general situation:

\begin{exa}[Y]
Consider the smallest nonlocal, noncommutative $k$-algebra
$A = \left[ \begin{matrix}
k & k \\
0 & k
\end{matrix} \right]$. The bimodule
$A^{*} := \opn{Hom}_{k}(A, k)$
turns out to be a tilting complex.
Let $\tau$ be the class of $A^{*}[-1]$ in $\opn{DPic}(A)$. Then 
one can show that 
$\tau \notin \opn{Pic}_{k}(A) \times \mbb{Z}$.
\end{exa}

This raises:

\begin{prob}
Study the structure of the group $\opn{DPic}_{k}(A)$ and its
relation to the geometry of the noncommutative ring $A$. 
\end{prob}

Let us write
$\opn{Out}^{\mrm{tr}}_{k}(\cat{D}^{\mrm{b}}(\cat{Mod} A))$
for the group of $k$-linear triangle auto-equivalences of 
$\cat{D}^{\mrm{b}}(\cat{Mod} A)$. The next problem is open: 

\begin{prob}
Is $\opn{DPic}_{k}(A) =
\opn{Out}^{\mrm{tr}}_{k}(\cat{D}^{\mrm{b}}(\cat{Mod} A))$?
\end{prob}

Recall that $A$ is hereditary if $\opn{gl.dim} A = 1$, or if
every left of right ideal is a projective module.

\begin{thm}[Miyachi and Y.]
If $A$ is hereditary then the answer to the previous problem 
is positive.
\end{thm}

\section{Quivers and Path Algebras}

We will concentrate on a finite dimensional $k$-algebra $A$. 
Let us assume that $k$ is algebraically closed, and that
$A$ is basic (i.e.\ $A / \mfrak{p} = k$ for every prime ideal). 
Since every $k$-algebra is Morita equivalent to a basic one
the latter is really no restriction. 

The geometry of $A$ is described by a quiver $\vec{\Delta}$. Here 
is a nonorthodox definition of $\vec{\Delta}$. The set 
of vertices of $\vec{\Delta}$ is $\opn{Spec} A$, namely the set of 
prime ideals. Let $\mfrak{r}$ be the Jacobson radical of $A$, and 
for each $\mfrak{p}_{i} \in \opn{Spec} A$ let 
$e_{i} \in A / \mfrak{r}$ be the corresponding central idempotent. 
So
\[ \mfrak{r} / \mfrak{r}^{2} = 
\bigoplus_{i,j} e_{i} (\mfrak{r} / \mfrak{r}^{2}) e_{j} 
\cong \bigoplus_{i,j} \frac{\mfrak{p}_{i} \cap \mfrak{p}_{j} }
{\mfrak{p}_{i} \mfrak{p}_{j}} \]
as $A$-$A$-bimodules. Define
$d_{i,j} := \opn{dim}_{k}
\frac{\mfrak{p}_{i} \cap \mfrak{p}_{j} } 
{\mfrak{p}_{i} \mfrak{p}_{j}}$.
Then there are $d_{i,j}$ arrows $\mfrak{p}_{i} \to \mfrak{p}_{j}$.
The connected components of $\vec{\Delta}$ are called cliques, and 
they control \"{O}re localization. 

\begin{exa}
Take 
$A = \left[ \begin{matrix}
k & k \\
0 & k
\end{matrix} \right]$.
The prime ideals are 
$ \mfrak{p}_{1} = \left[ \begin{matrix}
0 & k \\
0 & k
\end{matrix} \right]$ 
and
$\mfrak{p}_{2} = \left[ \begin{matrix}
k & k \\
0 & 0
\end{matrix} \right]$. Clearly 
$\frac{\mfrak{p}_{1} \cap \mfrak{p}_{2} }
{\mfrak{p}_{1} \mfrak{p}_{2}}
=  \left[ \begin{matrix}
0 & k \\
0 & 0
\end{matrix} \right]$.
So the quiver $\vec{\Delta}$ is
$\UseTips
\begin{xy} 
            (0,0)*+@{*}="1"*+!U{\mfrak{p}_{1}},"1"
\ar@{->}    (10,0)*+@{*}="2"*+!U{\mfrak{p}_{2}},"2" 
\end{xy}$,
a Dynkin quiver of type $A_{2}$.
\end{exa}

On the other hand, given a finite quiver $\vec{\Delta}$ we can 
construct an algebra, called the path algebra $k \vec{\Delta}$.
A path in $\vec{\Delta}$ is either a trivial path $e_{x}$ for each 
vertex $x$, or a sequence of arrows 
$\alpha_{1} \alpha_{2} \cdots \alpha_{m}$, such that 
$\alpha_{i+1}$ starts where $\alpha_{i}$ ends:
\[ \UseTips
\begin{xy} 
                        (0,0)*+@{*}="1"*+!U{x_{1}},"1"
\ar@{->}^{\alpha_{1}}   "1";(10,0)*+@{*}="2"*+!U{x_{2}},"2"
\ar@{->}^{\alpha_{2}}   "2";(20,0)*+@{*}="3"*+!U{x_{3}},"3" 
\ar@{->}^{\alpha_{3}}   "3";(30,0)*+@{*}="4"*+!U{x_{4}},"4"
\end{xy} \]
The paths form a basis of $k \vec{\Delta}$ as vector space, and 
multiplication of paths is by concatenation if possible, or $0$.
 
\begin{exa}
Take the quiver 
$\vec{\Delta} = \UseTips
\begin{xy} 
            (0,0)*+@{*}="1"*+!U{x_{1}},"1"
\ar@{->}^{\alpha}    (10,0)*+@{*}="2"*+!U{x_{2}},"2" 
\end{xy}$. 
Then 
$e_{x_{1}}, e_{x_{2}}$ are orthogonal idempotents;
$e_{x_{1}} \alpha = \alpha = \alpha e_{x_{2}}$;
and
$\alpha e_{x_{1}} = \alpha^{2} = e_{x_{2}} \alpha = 0$. 
Thus
$k \vec{\Delta} \cong 
\left[ \begin{matrix}
k & k \\
0 & k
\end{matrix} \right]$, with
\[ e_{x_{1}} =  
\left[ \begin{matrix}
1 & 0 \\
0 & 0
\end{matrix} \right],\ 
e_{x_{2}} =  
\left[ \begin{matrix}
0 & 0 \\
0 & 1
\end{matrix} \right],\ 
\alpha =  
\left[ \begin{matrix}
0 & 1 \\
0 & 0
\end{matrix} \right] . \]
\end{exa}

Here is a classical structure theorem for finite dimensional 
algebras. 

\begin{thm}[Gabriel]
Let $A$ be a basic finite dimensional $k$-algebra with quiver 
$\vec{\Delta}$. Then
\begin{enumerate}
\item $A \cong k \vec{\Delta} / I$ for an ideal 
$I \subset \mfrak{a}^{2}$,
where $\mfrak{a} \subset k \vec{\Delta}$ is the ideal generated by 
the arrows.  
\item $I = 0$ iff $A$ is hereditary.
\end{enumerate} 
\end{thm}

Some remarks: 

\begin{rem}
According to the definitions of Cuntz-Quillen, a finite 
dimensional algebra $A$ is smooth iff it is hereditary. 
\end{rem}

\begin{rem}
The surjection $\vec{\Delta} \surj A$ should be viewed as a 
noncommutative analog of the Cohen theorem for commutative rings:
$k[[t_{1}, \ldots, t_{n}]] \surj A$.
Indeed the arrows in $k \vec{\Delta}$ are lifts of a basis of 
$\mfrak{r} / \mfrak{r}^{2}$. 
\end{rem}

\begin{rem}
Beilinson showed that the derived category 
$\cat{D}^{\mrm{b}}(\cat{Coh} \mbf{P}^{1})$
of coherent sheaves on the projective line $\mbf{P}^{1}$ is 
equivalent to 
$\cat{D}^{\mrm{b}}(\cat{mod} k \vec{\Delta}_{2})$, 
for a certain quiver $\vec{\Delta}_{2}$. According to 
Kontsevich-Rosenberg one has
$\cat{D}^{\mrm{b}}(\cat{Coh} \mbf{NP}^{n}) \approx
\cat{D}^{\mrm{b}}(\cat{mod} k \vec{\Delta}_{n+1})$,
where $\mbf{NP}^{n}$ is their noncommutative projective $n$-space,
so 
$\opn{Out}^{\mrm{tr}}_{k}(\cat{D}^{\mrm{b}}(\cat{Coh} \mbf{NP}^{n}))
\cong \opn{DPic}_{k}(k \vec{\Delta}_{n+1})$. 
\end{rem}


\section{The Auslander-Reiten Quiver}

From now on $A = k \vec{\Delta}$ and the quiver $\vec{\Delta}$ is 
connected. An $A$-module is also called a representation of  
$\vec{\Delta}$. As customary we write $\cat{mod} A$ for 
the category of finitely generated $A$-modules (=finite dimensional 
representations). 

One of the important invariants of the  algebra 
$A$ is the set of isomorphism classes of indecomposable modules. If 
this set is finite $A$ is said to have finite representation type. 

\begin{thm}[Gabriel]
$A$ has finite representation type iff $\vec{\Delta}$ is a Dynkin 
quiver of type $A_{n}, D_{n}, E_{6},E_{7}$ or $E_{8}$. 
\end{thm}

The Auslander-Reiten quiver
$\vec{\Gamma}(\cat{mod} A)$ is defined as follows. 
Its vertices are the isomorphism classes of indecomposable 
$A$-modules, and there are $d$ arrows $x \to y$,
where $d$ is the dimension of the space of irreducible 
homomorphisms $\opn{Irr}(M_{x}, M_{y})$ (this is like 
$\mfrak{r} / \mfrak{r}^{2}$). 

Happel showed how to define the Auslander-Reiten quiver
$\vec{\Gamma} = \vec{\Gamma}(\cat{D}^{\mrm{b}}(\cat{mod} A))$. 
This is an infinite quiver. Usually it is very complicated, but 
it contains a nice subquiver denoted $\vec{\mbb{Z}} \vec{\Delta}$
that is easy to describe ($\vec{\mbb{Z}} \vec{\Delta}$ was 
introduced by Riedtmann). When $\vec{\Delta}$ is a Dynkin quiver, 
then in fact 
$\vec{\Gamma}(\cat{D}^{\mrm{b}}(\cat{mod} A)) 
= \vec{\mbb{Z}} \vec{\Delta}$.

\begin{exa}
Take 
$\vec{\Delta} = 
\UseTips
\begin{xy} 
            (0,0)*+@{*}="1"*+!U{1},"1"
\ar@{->}   "1";(10,0)*+@{*}="2"*+!U{2},"2"
\ar@{->}   "2";(20,0)*+@{*}="3"*+!U{3},"3" 
\end{xy}$. 
Then $\vec{\Gamma}(\cat{D}^{\mrm{b}}(\cat{mod} A)) 
= \vec{\mbb{Z}} \vec{\Delta}$ 
is
\[ \UseTips
\begin{xy} 
            (0,0)*+@{*}="01"*+!U{\scrp{(0,1)}},"01"
\ar@{->}    "01";"01"+(10,10)*+@{*}="02"*++!R{\scrp{(0,2)}},"02" 
\ar@{->}    "02";"02"+(10,10)*+@{*}="03"*+!D{\scrp{(0,3)}},"03" 
%
\ar@{->}    "02";"01"+(20,0)*+@{*}="11"*+!U{\scrp{(1,1)}},"11" 
\ar@{->}    "03";"02"+(20,0)*+@{*}="12"*++!R{\scrp{(1,2)}},"12" 
\ar@{->}    "11";"12" 
\ar@{->}    "12";"12"+(10,10)*+@{*}="13"*+!D{},"13" 
%
\ar@{->}    "12";"11"+(20,0)*+@{*}="21"*+!U{\scrp{(2,1)}},"21" 
\ar@{->}    "13";"12"+(20,0)*+@{*}="22"*++!R{},"22" 
\ar@{->}    "21";"22" 
\ar@{->}    "22";"22"+(10,10)*+@{*}="23"*+!D{},"23" 
%
\ar@{->}    "22";"21"+(20,0)*+@{*}="31"*++!L{\cdots},"31" 
\ar@{->}    "23";"22"+(20,0)*+@{*}="32"*++!L{\cdots},"32" 
\ar@{->}    "31";"32"
\ar@{->}    "32";"32"+(10,10)*+@{*}="33"*++!L{\cdots},"33" 
%
\ar@{->}    "02"-(20,0)*+@{*}="-12"*++!R{\cdots},"-12";"01"
\ar@{->}    "03"-(20,0)*+@{*}="-13"*++!R{\cdots},"-13";"02"
\ar@{->}    "01"-(20,0)*+@{*}="-11"*++!R{\cdots},"-11";"-12"
\ar@{->}    "-12";"-13"
\end{xy} \]
The vertices in $\vec{\Gamma}(\cat{mod} A)$ are labeled.
\end{exa}

The group $\opn{DPic}_{k}(A)$ acts naturally on the category
$\cat{D}^{\mrm{b}}(\cat{mod} A)$, and hence on the quiver 
$\vec{\Gamma}(\cat{D}^{\mrm{b}}(\cat{mod} A))$
(by quiver automorphisms). The element 
$\tau \in \opn{DPic}_{k}(A)$, represented by the tilting complex
$A^{*}[-1]$, is called the translation. In the example above it is a 
shift by $1$ to the left. Using this action we proved the 
following theorems. Let
$\opn{Aut}(\vec{\mbb{Z}} \vec{\Delta})$ denote the group of quiver 
automorphisms. 

\begin{thm}[Miyachi and Y.]
Suppose $A$ has finite representation type. Then 
\[ \opn{DPic}_{k}(A) \cong
\opn{Aut}(\vec{\mbb{Z}} \vec{\Delta})^{\bra{\tau}} . \]
\end{thm}

\begin{exa}
Let $\sigma \in \opn{DPic}_{k}(A)$ be represented by $A[1]$.
If $\vec{\Delta}$ is of Dynkin type $A_{n}$ then 
$\opn{DPic}_{k}(A)$ is abelian, generated by $\tau$ and $\sigma$, 
with one relation
$\tau^{n+1} = \sigma^{-2}$.
\end{exa}

We denote by
$\opn{Aut}((\vec{\mbb{Z}} \vec{\Delta})_{0}; d)$
the group of permutations of the vertex set 
$(\vec{\mbb{Z}} \vec{\Delta})_{0}$
that preserve the arrow-multiplicity $d$.

\begin{thm}[Miyachi and Y.]
If $A$ has infinite representation type then there is an 
isomorphism of groups
\[ \opn{DPic}_{k}(A) \cong
\bigl( \opn{Aut}((\vec{\mbb{Z}} \vec{\Delta})_{0}; d)^{\bra{\tau}}
\ltimes \opn{Pic}^{0}_{k}(A) \bigr) \times \mbb{Z} . \]
$\opn{Pic}^{0}_{k}(A)$ is a connected linear algebraic group, and 
is trivial when $\vec{\Delta}$ is a tree.
\end{thm}

To conclude,

\begin{prob}
Can this analysis of $\opn{DPic}_{k}(A)$ be carried out for other 
finite dimensional algebras $A$? We note that that the quiver 
$\vec{\Gamma}(\cat{D}^{\mrm{b}}(\cat{mod} A))$
exists whenever $\opn{gl.dim} A < \infty$.
\end{prob}


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