Hochschild homology

date: 2021-04-26 21:49 modification date: Friday 28th January 2022 15:51:31 title: Hochschild homology aliases: Hochschild homology aliases: [“HH”, “THH”, “Topological Hochschild homology”, “Topological Hochschild cohomology”, “topological Hochschild cohomology”, “topological periodic cyclic homology”, “cyclic homology”, “negative cyclic homology”]

Tags: #subjects/homotopy #subjects/arithmetic-geometry/prisms Refs: - [ ] File:attachments/1214427886.pdf

Links: Algebraic K Theory Connes exact sequence

Hochschild Homology

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 \begin{align*} C_n(A, M) := M \otimes A^{\otimes n} \end{align*}

The Hochschild homology of an \(R{\hbox{-}}R{\hbox{-}}\)bimodule reflects some ring-theoretic stuff.

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

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

attachments/Pasted%20image%2020220207222715.png attachments/Pasted%20image%2020220207222726.png attachments/Pasted%20image%2020220207222747.png attachments/Pasted%20image%2020220207222805.png

  • When \(M=A\), considered as a bimodule over itself, we write \({\operatorname{HH}}_{*}(A)\) for \({\operatorname{HH}}_{*}(A, A)\) and \({\operatorname{HH}}^{*}(A)\) for \({\operatorname{HH}}^{*}(A, A)\).

  • There is a 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 \({\mathbb{R}}\operatorname{Hom}_{A^{e}}(A, A)\).

  • Bracket operations: attachments/Pasted%20image%2020220207223257.png

  • Poincare duality attachments/Pasted%20image%2020220207225024.png # Defining THH

  • Take \(A \in \mathsf{C}\), where \(\mathcal C\) is a “nice” Monoidal category, and \(A\) is an algebra object in it. We’ll call the monoidal operation \(\otimes\).

  • We’ll make a simplicial object \({THH}_{\bullet}(A)\):

    • \(THH_n(A) = A^{\otimes n+1}\). If it’s to be simplicial, need to specify the face/degeneracy maps:
    • Face maps: collapse by cyclic multiplication
    • Degeneracy maps: use the unit of \(A\), can replace any tensor symbol with it. Have a unit map that goes from the unit to \(A\), so somehow this gets you “up” one level (?)

  • Now take its geometric realization \({\left\lvert {THH_{-}(A)} \right\rvert}\)

    • In general, take \(\mathrm{hocolim}_\Delta THH_{-}(A)\)

Unsorted

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 differential graded associative algebras). 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.

  • See the HKR theorem

  • When proving stuff about algebras – try with polynomial algebras first, essentially the simplest case.

  • Simplest Hopf algebra

  • Hochschild cohomology of \(A\) is a derived mapping object \begin{align*} \mathop{\mathrm{\mathbb{R}Hom}}_{({A}, {A}){\hbox{-}}\mathsf{biMod}} (A, A) .\end{align*}

The higher Hochschild cohomology of an E_n algebra \(A\) is the derived mapping object of \(E_n{\hbox{-}}A{\hbox{-}}\)modules \begin{align*} \mathop{\mathrm{\mathbb{R}Hom}}_{\mathsf{E_n-A}{\hbox{-}}\mathsf{Mod}}(A, A) .\end{align*} attachments/Pasted%20image%2020220213204029.png

Use in physics: we would like to describe all supersymmetric deformations of Yang-Mills that come from a Lagrangian

🗓️ Timeline
  • Prismatic cohomology
    HH corresponds to prismatic cohomology
    THH corresponds to Hodge-Tate cohomology.

    The Relationship Between THH and K-Theory

    Can say \begin{align*} H^i(\Delta_{{\mathbb{Z}}_p}) = \begin{cases} {\mathbb{Z}}_p & i=0 \\ \prod_{{\mathbb{N}}} {\mathbb{Z}}_p & i=1. \end{cases} \end{align*} Can compute using THH? \(\pi_{-1}( {\operatorname{TP}}({\mathbb{Z}}_p) )\) is where the \(i=1\) part comes from.

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#subjects/homotopy #subjects/arithmetic-geometry/prisms