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Global Homotopies for Differential Hochschild Cohomologies

Marvin Dippell, Chiara Esposito, Jonas Schnitzer, Stefan Waldmann

Abstract

We construct global homotopies to compute differential Hochschild cohomologies in differential geometry. This relies on two different techniques: a symbol calculus from differential geometry and a coalgebraic version of the van Est theorem. Not only do we obtain an improved version of the Hochschild-Kostant-Rosenberg theorem but we also compute Hochschild cohomologies for related scenarios, e.g. for principal bundles, and invariant versions.

Global Homotopies for Differential Hochschild Cohomologies

Abstract

We construct global homotopies to compute differential Hochschild cohomologies in differential geometry. This relies on two different techniques: a symbol calculus from differential geometry and a coalgebraic version of the van Est theorem. Not only do we obtain an improved version of the Hochschild-Kostant-Rosenberg theorem but we also compute Hochschild cohomologies for related scenarios, e.g. for principal bundles, and invariant versions.

Paper Structure

This paper contains 37 sections, 71 theorems, 283 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Coalgebra Cohomology of the Symmetric Algebra
  3. (Lie) Coalgebras, their Comodules and Cohomology
  4. The Bialgebra Structure of $\Sym V$
  5. Coalgebra Cohomology
  6. Chevalley-Eilenberg Cohomology
  7. The Van Est Double Complex
  8. Relation to Chevalley-Eilenberg Complex
  9. Relation to the Coalgebra Complex
  10. The Van Est Maps
  11. Functoriality
  12. Infinitesimal Functoriality
  13. The Van Est Complex with Coefficients
  14. Perturbing the Van Est Double Complex
  15. Coalgebra Cohomology for the Reduced Symmetric Coalgebra
  16. ...and 22 more sections

Key Result

lemma 1

Let $V$ be an $\ring{R}$-module.

Figures (4)

  • Figure 1: The augmentation of the columns of the van Est double complex
  • Figure 2: The augmentation of the rows of the van Est double complex.
  • Figure 3: The double complex with the augmentation maps and the homotopies.
  • Figure 4: Homotopy retract $(C^\bullet,\D_C)$ and $(D^\bullet,\D_D)$.

Theorems & Definitions (139)

  • lemma 1
  • proof
  • definition 1: Coalgebra complex
  • proposition 1
  • proof
  • definition 2: $\Sym V$-comodule
  • proposition 2
  • proof
  • definition 3: Coalgebra complex with coefficients
  • definition 4: $V$-Lie coaction
  • ...and 129 more