Derived categories

From Talbot

Get PDF | Get log | Regenerate PDF | Export Source

---

\documentclass{article}
\usepackage{amsmath,amsfonts,amssymb,epsfig,color}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\iso}{\cong}
\parskip1em
\parindent0em
\begin{document}
 
\underline{Derived categories of $A_\infty$-categories}
 
This talk introduces derived categories. To prevent boredom, I want to
stay away from a too classical viewpoint, and emphasize the intrinsic
nature of exact triangles on the $A_\infty$ level. Much of this is due to
Kontsevich. A basic reference is [Keller, lectures on $A_\infty$-algebras
and modules]. It would still be good if the speaker is acquainted with
classical derived categories, or even better with [Bondal-Kapranov, framed
triangulated categories]. Most of the material can be found in some form in
[Seidel, Fukaya categories and Picard-Lefschetz theory, Chapter I].
 
\underline{Plan}
 
{\it $A_\infty$-categories.} Reminder of $A_\infty$-categories,
$A_\infty$-functors, and the invertibility of quasi-equivalences. Take the
standard category ${\mathcal T}_2$ with two objects and one isomorphism in
either direction. Then, two objects in an arbitrary $A_\infty$-category
$\mathcal A$ are said to be isomorphic if there is an $A_\infty$-functor
${\mathcal T}_2 \rightarrow {\mathcal A}$ mapping the two model objects to
them.
 
{\it Exact triangles.} Generalizing the previous example, there is a
special $A_\infty$-category with three objects, the Kontsevich triangle
category ${\mathcal T}_3$, described in [Seidel op. cit., Section 3g]. One
can think of this as an $A_\infty$-deformation of $\Lambda(\C) \semidirect
\Z/3$, where $\Z/3$ acts on $\C$ by rotation. In any case, exact triangles
in $\mathcal A$ are defined as images of the standard triangle under
$A_\infty$-functors ${\mathcal T}_3 \rightarrow \mathcal A$. One should
then prove some standard properties: the fact that all $A_\infty$-functors
preserve exact triangles is obvious from this point of view; however, one
has to prove that an exact triangle induces long exact sequences of $Hom$
spaces on the left and right; and more challengingly, if one changes the
arrows in an exact triangle to cohomologous ones, the resulting triangle
is still exact. Some of this may be postponed to later, when we have
twisted complexes and such.
 
{\it Triangulated $A_\infty$-categories.} An $A_\infty$-category $\A$ is
called triangulated if it is closed under up and down shifts, and if every
morphism can be completed to an exact triangle. The first example are
categories of $A_\infty$-modules, and mapping cones. If the speaker is
feeling ambitious, this would be the moment to introduce an
$A_\infty$-version of the notion of strong generators, and to prove that
an arbitrary triangulated $A_\infty$-category admitting suitable colimits,
and having a strong generator, is quasi-equivalent to the category of
modules over the endomorphism ring of that generator. However, I don't
know a reference for this (the corresponding result for dg categories is
in [Keller, math.KT/0601185], and I assume one could derive the
$A_\infty$-case from that if necessary; however, it may be too much
trouble to define what one actually means by admitting suitable colimits).
 
{\it Twisted complexes.} For $A_\infty$-categories, this was first
introduced by [Kontsevich, Homological algebra of mirror symmetry], even
if his definition is a little clumsy. One should define them, mention that
the resulting $A_\infty$-categories of twisted complexes are triangulated,
define the notion of generators and of the triangulated envelope, which
then leads to the definition of derived category of an
$A_\infty$-category. As usual, we also need a nice example. I feel that
the directed $A_n$ quivers give a nice example, since there one can see
various indecomposable objects in the derived category as twisted
complexes (this is an old story, going back to
[Bernstein-Gelfand-Ponomarev] on reflection functors).
 
\underline{Notes}
 
Kontsevich's $\mathcal T_3$ has a generalization $\mathcal T_n$, which is
in fact related to the derived category of the $A_n$ quiver. There is a
nice explanation of the octahedral axiom in this context [Kontsevich,
unpublished]. These $A_\infty$-categories can also be viewed as
deformations of $\Lambda(\C) \rtimes \Z/n$, where $\Z/n$ acts on $\C$ by
$\exp(2\pi i/n)$.
 
Another topic which may or may not fit into this lecture is idempotent
completion (split-closure) and the associated notion of split-generator.
However, since that is not particularly deep, we may fit it in anywhere
later when it's needed.
 
\underline{Dependencies}
 
This relies on: the basic talk on $A_\infty$-structures.
 
\end{document}