• Tags
  • #homotopy #arithmetic-geometry/prisms
• Refs:
  • Piper, Algebraic Deformation Theory #resources/papers
• Links:
  • Algebraic K Theory
  • Connes exact sequence
  • Segal conjecture
  • syntomic
  • THH
  • HKR

Hochschild Homology

Idea: for \(\mathsf{C}\in \mathsf{Cat}\), define \({\operatorname{HH}}(\mathsf{C}) = {\mathbf{R}}\mathop{\mathrm{Aut}}_{\mathsf{C}}(\operatorname{id}_{\mathsf{C}}) = \operatorname{Ext} (\operatorname{id}, \operatorname{id})\).

Motivations

• Idea: a cohomology theory for bimodules over a DGA.

• Study deformations of algebras and rings

  • When proving stuff about algebras – try with polynomial algebras first, essentially the simplest case.
  • The Hochschild homology of an \(R{\hbox{-}}R{\hbox{-}}\)bimodule reflects some ring-theoretic stuff. about \(R\)?

• From physics: we would like to describe all supersymmetric deformations of Yang-Mills that come from a Lagrangian.

• Occurence in homological mirror symmetry:

• Computing various homological invariants of associative algebras (such as Tor and Ext of various modules, Hochschild (co)homology, cyclic homology etc.) has been an active research topic in ring theory for many years. More recently (about 15 years ago), ring theorists became interested in associative algebras up to homotopy, or Yoneda Ext-algebras.

  • This offers two different perspectives on associative algebras: homological invariants are “abelian” (i. e. arise when one works with an additive category, e.g. chain complexes of modules over a ring), while homotopical invariants are “non-abelian” (i. e. arise from non-additive categories, like the category of all DGAs).
  • However, these two perspectives are closely related, and it is often possible to recover homological information from the homotopical one, and the other way round. For experts in homotopical algebra on a larger scale (beyond the associative ring theory), this philosophy is already present in works of Stasheff and Hinich on homotopy algebras.

Definitions

• If \(k\in \mathsf{Ring}\) and \(X\in {\mathsf{sSet}}\) a geometric realization \({\left\lvert {X} \right\rvert}\) with coefficients in \(k\).

• Let $A\in {}_{R} \mathsf{Alg} $ and write \(A^e \coloneqq A\otimes_R { {A}^{\operatorname{op}}}\).

• Let \(A \in \mathsf{Alg} _{/R}\) such that \(A\) is projective as an \(R{\hbox{-}}\)module and \(M\in ({A}, {A}){\hbox{-}}\mathsf{biMod}\). Then define the Hochschild complex as the bar construction \begin{align*} C_n(A, M) := M \otimes A^{\otimes n} \end{align*}

• For \(A\in{\mathsf{dg \mathsf{Alg} } }\) and \(M\in ({A}, {A}){\hbox{-}}\mathsf{biMod}\), identify $M\in {\mathsf{Mod}}_{A^e} $ and \(A\in {}_{A^e}{\mathsf{Mod}}\) and define Hochshild homology as \begin{align*}{\operatorname{HH}}_*(A; M) \coloneqq\operatorname{Tor}_{\scriptscriptstyle\bullet}^{A^e}(A, M) = A \overset{\mathbb{L}}{ \otimes} _{A^e}M .\end{align*}

• Identify \(M\in {}_{A^e}{\mathsf{Mod}}\) and define the Hochschild cohomology as \begin{align*}{\operatorname{HH}}^*(A; M) \coloneqq{ \operatorname{Ext} _{A^e}^{\scriptscriptstyle\bullet}}(A, M) = \mathop{\mathrm{\mathbb{R}Hom}}_{A^e}(A, M)\end{align*}

• Notational convention: \({\operatorname{HH}}^*(A) \coloneqq{\operatorname{HH}}^*(A; A)\) identifying \(A \in ({A}, {A}){\hbox{-}}\mathsf{biMod}\).

• In homotopy:

  • The higher Hochschild cohomology of an E_n algebra \(A\) is defined as the derived mapping object of \(({E_n}, {A}){\hbox{-}}\mathsf{biMod}\): \begin{align*} \mathop{\mathrm{\mathbb{R}Hom}}_{ {}_{E_n-A}{\mathsf{Mod}}}(A, A) .\end{align*}

Properties

• Connes operator \(B\):
  • \({\operatorname{HH}}_{*}(A)\) admits a degree-1 operator \(B\) with \(B^{2}=0\) due to Connes, arising from the cyclic permutation of the \(n+1\) \(A\) factors in the \(n\)th level of \(C H_{*}(A, A)\).
• Cup product:
  • The cup product \(f \smile g\) of cochains \(f: A^{\otimes p} \rightarrow A\) and \(g: A^{\otimes q} \rightarrow A\) is the map \(\mu(f \otimes g): A^{\otimes p+q} \rightarrow A\) given by applying \(f\) to the first \(p\) tensor factors and \(g\) to the remaining \(q\) tensor factors, and then multiplying the two \(A\) output tensor factors.
  • The cup product can be described on the derived level via the Yoneda or composition product on \({\mathbf{R}}\operatorname{Hom}_{A^{e}}(A, A)\).
• The Gerstenhaber bracket making it into a Lie algebra:
  • 
• Poincare duality:
  • 

Remarks on the bar construction

HH of a category

#todo

Derived approach

#todo

Etale/Galois descent

Examples

• For $R\in {}_{k} \mathsf{Alg} $, \(H_1(R,R) = \Omega_{R/k}\), the module of relative differentials . If \(Q\subseteq R\) then there is an algebraic decomposition of this homology analogous to the Hodge decomposition for complex manifolds.

• Simplest Hopf algebra

Relation to Bott Periodicity and computation for \({ \mathbf{F} }_p\):