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The tangent complex and Hochschild cohomology of $\mathcal {E}_n$-rings

Part of: Homological algebra Local theory Homology and cohomology theories

Published online by Cambridge University Press:  10 December 2012

John Francis
John Francis*
Affiliation:
Department of Mathematics, Northwestern University, Evanston, IL 60208-2370, USA (email: [email protected])
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Abstract

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In this work, we study the deformation theory of ${\mathcal {E}}_n$-rings and the ${\mathcal {E}}_n$ analogue of the tangent complex, or topological André–Quillen cohomology. We prove a generalization of a conjecture of Kontsevich, that there is a fiber sequence $A[n-1] \rightarrow T_A\rightarrow {\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)[n]$, relating the ${\mathcal {E}}_n$-tangent complex and ${\mathcal {E}}_n$-Hochschild cohomology of an ${\mathcal {E}}_n$-ring $A$. We give two proofs: the first is direct, reducing the problem to certain stable splittings of configuration spaces of punctured Euclidean spaces; the second is more conceptual, where we identify the sequence as the Lie algebras of a fiber sequence of derived algebraic groups, $B^{n-1}A^\times \rightarrow {\mathrm {Aut}}_A\rightarrow {\mathrm {Aut}}_{{\mathfrak B}^n\!A}$. Here ${\mathfrak B}^n\!A$ is an enriched $(\infty ,n)$-category constructed from $A$, and ${\mathcal {E}}_n$-Hochschild cohomology is realized as the infinitesimal automorphisms of ${\mathfrak B}^n\!A$. These groups are associated to moduli problems in ${\mathcal {E}}_{n+1}$-geometry, a less commutative form of derived algebraic geometry, in the sense of the work of Toën and Vezzosi and the work of Lurie. Applying techniques of Koszul duality, this sequence consequently attains a nonunital ${\mathcal {E}}_{n+1}$-algebra structure; in particular, the shifted tangent complex $T_A[-n]$ is a nonunital ${\mathcal {E}}_{n+1}$-algebra. The ${\mathcal {E}}_{n+1}$-algebra structure of this sequence extends the previously known ${\mathcal {E}}_{n+1}$-algebra structure on ${\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)$, given in the higher Deligne conjecture. In order to establish this moduli-theoretic interpretation, we make extensive use of factorization homology, a homology theory for framed $n$-manifolds with coefficients given by ${\mathcal {E}}_n$-algebras, constructed as a topological analogue of Beilinson and Drinfeld’s chiral homology. We give a separate exposition of this theory, developing the necessary results used in our proofs.

Keywords

-algebrasdeformation theorythe tangent complexHochschild cohomologyfactorization homologytopological chiral homologyKoszul dualityoperads-categories

MSC classification

Secondary: 14B12: Local deformation theory, Artin approximation, etc. 55N35: Other homology theories 18G55: Homotopical algebra
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 3 , March 2013 , pp. 430 - 480
Copyright
Copyright © 2012 The Author(s)

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