Subject: Re: A_infty structures and Hochschild cohomology
Date: Fri, 8 Mar 2002 08:39:12 +0000 (GMT Standard Time)
From: Andrey Lazarev
To: Don Davis
Both constructions are instances of a third one which I will now
describe.
Let C be a filtered chain complex with an approximate differential d
of level n. By this I understand an endomorphism of C of degree -1
such that its square is not zero but rather raises filtration by n.
Let us write d as
d=d_0+...+d_n. Here d_n raises filtration by n, d_{n-1} raises
filtration by n-1... d_0 preserves filtartion.
Let's assume that d_0 is an honest differential, i.e. with square
zero. Now the problem is to find a d_{n+1} which raises filtration by
n+1 and such that d_0+...+d_{n+1} is an approximate differential of
level n+1. Let's call this a perturbation of d. Then we can introduce
an equivalence relation on perturbations and the equivalence classes
will correspond to homology of End(C) with respect to d_0.
This construction needs a slight modification if we want our
differentials to be compatible with, say, an algebra or coalgebra
structure on C. In that case End(C) needs to be replaced with Der(C)
or Coder(C) (derivations or coderivations).
Now for deformation theory take C to be the bar-construction BA[[t]]
of an algebra A[[t]] where A is an algebra or an A_\infty algebra.
The filtration is by powers of t.
The (A_\infty)-structure determines a differential d_0 on BA[[t]].
Its perturbations will correspond to Hochschild cohomology of A which
is is Coder(BA[[t]])
For Robinson's theory let A be an A_n-algebra, its tensor coalgebra
T\Sigma A is filtered by tensor powers. (we need the differential on
A itself to be zero in order for Robinson's theory to work). The A_n
structure determines an approximate differential on T\Sigma A of an
appropriate level. In this case the analogue of our d_0 is the
Hochschild differential corresponding to m_2:A tensor A-->A. Since A
has vanishing differential m_2 is actually associative and determines
an honest differential on T\Sigma A. Finally take Coder(T\Sigma A);
the obstructions to perturbations will lie in cohomology of this
complex wrt d_0 which is Hochschild cohomology of A.
Hope this helps,
Andrey
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Can someone help me out here?
Hochschild cohomology seems to come up in two ways in the
theory of A_infty structures:
First way: As the target recepticle for
the obstructions to going from A_n-structures to
A_{n+1}-structures (relative to a fixed A_{n-1}-structure).
I'm thinking here about the work of Alan Robinson.
Second way: In Gerstenhaber deformation theory of rings,
the Hochschild cohomology codifies the infinitesimal deformations of
associative structures on rings. Also, my understanding is that the
"newer" deformation theory for A_infty algebra structures
also has obstructions living in Hochschild cohomology groups.
My Question: What relationship is there (if any) between
the first way and the second way?
That is, what's the connection between
Robinson's work and deformation theory?
John Klein
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Andrey Lazarev
A.Lazarev@bristol.ac.uk
Phone +44 117 928 7997
School of Mathematics
University of Bristol
Bristol BS8 1TW UK
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