The tangent complex and Hochschild cohomology of En-rings

John Francis*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

44 Scopus citations


In this work, we study the deformation theory of {mathcal En-rings and the {mathcal Enanalogue 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] T-A {mathrm {HH}}-{{mathcal {E}}-{n}}!(A)[n], relating the {mathcal En-tangent complex and {mathcal En-Hochschild cohomology of an {mathcal En-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× {mathrm {Aut}-A {mathrm {Aut}}-{{mathfrak B}n!A}. Here {mathfrak B}n!A is an enriched (infty ,n)-category constructed from A, and {mathcal En-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 En-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.

Original languageEnglish (US)
Pages (from-to)430-480
Number of pages51
JournalCompositio Mathematica
Issue number3
StatePublished - Mar 1 2013


  • Hochschild cohomology
  • Koszul duality
  • algebras
  • categories
  • deformation theory
  • factorization homology
  • operads
  • the tangent complex
  • topological chiral homology

ASJC Scopus subject areas

  • Algebra and Number Theory


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