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Computing the Gerstenhaber Bracket
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Aaron Bergman
science forum addict


Joined: 24 Mar 2005
Posts: 94

PostPosted: Fri Jul 07, 2006 2:50 am    Post subject: Computing the Gerstenhaber Bracket Reply with quote

I've been working on doing some computations in Hochschild cohomology,
and all the definitions I've seen of the Gerstenhaber bracket define it
either hopelessly abstractly or in terms of the bar resolution. I have a
different resolution I've been using to compute HH^*, and I was hoping
to be able to compute the Gerstenhaber bracket (or HH^2 to a specific
formal deformation of the algebra) without having to compute an explicit
quasi-isomorphism to the bar resolution. Any ideas or references would
be appreciated.

Thanx,
Aaron
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Agustí Roig
science forum beginner


Joined: 18 Jul 2005
Posts: 11

PostPosted: Fri Jul 07, 2006 7:27 am    Post subject: Re: Computing the Gerstenhaber Bracket Reply with quote

Aaron Bergman ha escrit:

Quote:
I've been working on doing some computations in Hochschild cohomology,
and all the definitions I've seen of the Gerstenhaber bracket define it
either hopelessly abstractly or in terms of the bar resolution. I have a
different resolution I've been using to compute HH^*, and I was hoping
to be able to compute the Gerstenhaber bracket (or HH^2 to a specific
formal deformation of the algebra) without having to compute an explicit
quasi-isomorphism to the bar resolution. Any ideas or references would
be appreciated.



I don't know if I understand the problem: do you mean you have a
resolution that allows you the computation of the Gerstenhaber bracket
and you want an explicit quasi-isomorphism with the bar resolution
because your are not sure that your resolution actually computes
Hochschild cohomology?

If this is the case, you don't need this explicit quasi-isomorphism:
since Hochschild cohomology is a derived functor (an Ext; Weibel, "An
introduction to homological algebra", lemma 9.1.3), all you have to
check is if your resolution is projective (in an apropriate sense).
Then, the usual theorem of comparison of projective resolutions does
the work for you.

Agustí Roig
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mariano.suarezalvarez@gma
science forum addict


Joined: 28 Apr 2006
Posts: 58

PostPosted: Fri Jul 07, 2006 5:59 pm    Post subject: Re: Computing the Gerstenhaber Bracket Reply with quote

Aaron Bergman wrote:
Quote:
I've been working on doing some computations in Hochschild cohomology,
and all the definitions I've seen of the Gerstenhaber bracket define it
either hopelessly abstractly or in terms of the bar resolution. I have a
different resolution I've been using to compute HH^*, and I was hoping
to be able to compute the Gerstenhaber bracket (or HH^2 to a specific
formal deformation of the algebra) without having to compute an explicit
quasi-isomorphism to the bar resolution. Any ideas or references would
be appreciated.

I do not think anyone has figured out how to compute
the bracket from an arbitrary projective resolution.
Indeed, most computations that I am aware of explicitly
construct those dreaded quasi-isomorphisms.

Cheers,

-- m
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