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When we are dealing with ordinary things or dg things (where thing = algebra or category), I think I understand how HH^2 corresponds to 1st order deformations and HH^3 corresponds to obstructions.

One often hears (or at least I often hear) that HH^* corresponds to A-infinity deformations. I am wondering whether there is any reference which works this out precisely. EDIT: This seems to be incorrect (depending on what we mean by "deformation"). See Damien's answer. And see David Ben-Zvi's comment.

Kevin H. Lin
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    Let A be an algebra. If you write down what exactly it means to have an `$A_\infty$ structure on $A\oplus A$ (with the first $A$ in degree zero and the second $A$ in some degree) which extends that of $A$, you get the Hochschild cocycle condition. – Mariano Suárez-Álvarez Jul 14 '10 at 20:59
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    In what sense is this incorrect? the claim is that the moduli stack attached to the shifted Hochschild complex, with its $L_\infty$ structure, is the deformation space of your $A_\infty$-algebra. For the latest words on this general deformation theory see Kontsevich-Soibelman's book and Lurie's ICM. If you interpret this statement correctly it will give the assertion Damien says - points over graded rings correspond to graded points of the Hochschild complex.. – David Ben-Zvi Jul 15 '10 at 03:26
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    If you take "deformations" to mean "'derived' deformations", then it is correct. If you take "deformations" to mean "non-'derived' deformations", then it is incorrect. ---- Is this correct? ;-) – Kevin H. Lin Jul 15 '10 at 04:11

3 Answers3

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Well. Even in the case of a DG (or $A_\infty$) algebra $A$, infinitesimal (i.e. 1st order) deformations are classified by $HH^2(A,A)$. Namely, the structure maps (a-k-a Taylor components) of an $A_\infty$-algebra, viewed as elements of the Hochschild cochain complex, do have total degree $2$.

I think that one recovers the full Hochschild cohomolgy $HH^*(A,A)$ by considering "derived" infinitesimal deformations (namely, deformations for which the deformation parameter is allowed to have non zero degree).

In other words, and making use of funny words, $HH^*(A,A)$ is the tangent to the derived stack of associative (better, $A_\infty$) algebras at the point $A$. While $HH^2(A,A)$ can be viewed as the tangent to the coarse moduli space. As an indermediate statement between those two, in his PhD thesis Mathieu Anel computed the tangent complex to the 2-stack of associative algebras (not in the derived context): he found that it is precisely a 2 step complex, obtain as a truncation of the Hochschild complex. See http://arxiv.org/abs/math/0607385 (in french, sorry).

DamienC
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  • How can I view the "Taylor coefficients" or structure maps $m_n$ of an $A_\infty$ algebra as elements of the Hochschild cochain complex? – Kevin H. Lin Jul 14 '10 at 22:41
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    Oh, I see. We have $m_n : V^{\otimes n} \to V[2-n]$, which has "Hochschild degree" $n$ but "homological degree" $2-n$, hence total degree $n+2-n = 2$. – Kevin H. Lin Jul 14 '10 at 23:10
  • Yes, exactly! :-) – DamienC Jul 16 '10 at 20:57
  • Do you know any references that talk about "derived" deformations? Is any of this written up anywhere yet? – Kevin H. Lin Jul 22 '10 at 07:28
  • I would advise you to take a look at Marco Manetti's paper "Extended deformation functors" (IMRN) or "Deformation theory via differential graded Lie algebras" (math.AG/0507284).

    Another excellent reference (but difficult to read, at least for me) is of course DAG-IV by Jacob Lurie : http://www.math.harvard.edu/~lurie/papers/DAG-IV.pdf

     – DamienC Jul 24 '10 at 19:06
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    Thanks, Damien. By the way, Lurie has a new paper that is probably relevant to this discussion: http://www.math.harvard.edu/~lurie/papers/moduli.pdf – Kevin H. Lin Jul 25 '10 at 21:29
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You might want to look at 0705.3719.

Vít Tuček
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I'll have to read it more carefully, but this paper of Penkava and Schwarz seems to do it: http://arxiv.org/abs/hep-th/9408064

Kevin H. Lin
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