Question for the audience
By: Mikael Vejdemo JohanssonThe blog is in quite a summer lull, and I’m not doing near my share of keeping it up. I’m blaming my vacation and my marriage ceremony tomorrow.
Anyway, here is a question I was thinking about on my way home. About appropriate terminology for my own research. I’m putting quite a bit of thought into calculation of -structures on Ext-algebras in cases where the usual calculation methods - Homotopy perturbation theory, Merkulov’s method, et.c. - do not really work well since the required data about the endomorphism ring of the appropriate chain complex ends up being much too large. So far I have been calling this local computation, but it struck me that it might end up confusing those more used to local being used for .. say .. localization in various contexts.
On the way I thought about blind computation, to indicate the lack of information compared to the more global methods, but this doesn’t seem to be quite it either.
Thus a question to the readers (and writers?) of this blog: what would be a good word to describe my particular brand of computation of -structures on , for a DG-algebra which gets viewed as a black box, capable of performing calculations, but not of displaying its internals in any good way?
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I think “black-box computation” sounds good.
On a related note, I was wondering if you could point me to some references on the “usual methods” you mention above. I’m interested in the -structure on a particular Ext-algebra (for some coherent sheaves on a projective variety). In particular, I suspect said structure is formal, but am not really sure of how to go about proving it, and can’t seem to find any good references. Any thoughts?
Comment by Ben Webster — August 28, 2007 @ 8:54 pm
[…] Webster asked, in a comment to the post Question for the audience for pointers to literature on the computation of A ∞-structures. Since I’m still out […]
Pingback by The Infinite Seminar » Overview of the computation of A-infinity structures — August 29, 2007 @ 11:20 am
Hi Ben,
If you are lucky, the Ext-algebra in question is ‘intrinsically formal.’ An algebra (A,m) is intrinsically formal if any minimal A-infinity structure (A,m_n) with m_2=m is isomorphic to (A,m) as an A-infinity algebra. A quick way to test this is to compute some Hochschild cohomology (I believe this is also originally due to Kadeishvili). See section 3 in this excellent paper.
Comment by Matt — August 30, 2007 @ 1:32 am