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A filtered Hochschild-Kostant-Rosenberg theorem for real Hochschild homology

Lucy Yang

Abstract

In this paper, we introduce a notion of derived involutive algebras in $ C_2 $-Mackey functors which simultaneously generalize commutative rings with involution and the (non-equivariant) derived algebras of Bhatt--Mathew and Raksit. We show that the $ \infty $-category of derived involutive algebras admits involutive enhancements of the cotangent complexes, de Rham complex, and de Rham cohomology functors; furthermore, their real Hochschild homology is defined. We identify a filtration on the real Hochschild homology of these derived involutive algebras via a universal property and show that its associated graded may be identified with the involutive de Rham complex. Using $ C_2 $-$ \infty $-categories of Barwick--Dotto--Glasman--Nardin--Shah, we show that our filtered real Hochschild homology specializes to the HKR-filtered Hochschild homology considered by Raksit.

A filtered Hochschild-Kostant-Rosenberg theorem for real Hochschild homology

Abstract

In this paper, we introduce a notion of derived involutive algebras in -Mackey functors which simultaneously generalize commutative rings with involution and the (non-equivariant) derived algebras of Bhatt--Mathew and Raksit. We show that the -category of derived involutive algebras admits involutive enhancements of the cotangent complexes, de Rham complex, and de Rham cohomology functors; furthermore, their real Hochschild homology is defined. We identify a filtration on the real Hochschild homology of these derived involutive algebras via a universal property and show that its associated graded may be identified with the involutive de Rham complex. Using --categories of Barwick--Dotto--Glasman--Nardin--Shah, we show that our filtered real Hochschild homology specializes to the HKR-filtered Hochschild homology considered by Raksit.

Paper Structure

This paper contains 31 sections, 101 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. The Hochschild--Kostant--Rosenberg theorem
  3. The main result
  4. Related work
  5. Outline
  6. Notation & Conventions
  7. Acknowledgements
  8. Genuine equivariance
  9. Parametrized ∞-categories
  10. G-spaces and G-spectra.
  11. Genuine equivariant algebra
  12. Filtered and graded objects
  13. Definitions
  14. Slices and truncations
  15. Bialgebras and their modules
  16. ...and 16 more sections

Key Result

Theorem 1.1.1

Let $B$ be a smooth $A$-algebra. Then there are canonical isomorphisms $\mathrm{HH}_*(B/A) \simeq \Omega^i_{B/A}$. Moreover, the $S^1$-action on $\mathrm{HH}(B/A)$ induces the de Rham differential on $\mathrm{HH}_{*}(B/A) \simeq \Omega^*_{B/A}$.

Theorems & Definitions (305)

  • Theorem 1.1.1: MR142598
  • Theorem 1.2.1
  • Remark 1.2.2: The filtered involutive circle
  • Remark 1.2.3
  • Theorem 1.2.4
  • Remark 1.2.5
  • Theorem 1.2.6: Corollary \ref{['cor:param_gr_fil_day_convolution']}, Proposition \ref{['prop:param_assoc_gr_is_C2_monoidal']}
  • Proposition 1.2.7: Proposition \ref{['prop:regsliceright_complete_sep']}
  • Definition 2.1.2: BDGNS1
  • Remark 2.1.3: LurHTT[Example 2.5]Shah18
  • ...and 295 more