Braided Hochschild cohomology and Hopf actions

TL;DR

This paper explores the structure of braided Hochschild cohomology in braided monoidal categories, linking it to classical Hochschild cohomology of smash products and applications in deformation theory and algebraic geometry.

Contribution

It introduces a graded ring structure on braided Hochschild cohomology, establishes a canonical isomorphism with Hochschild cohomology of smash products, and applies these results to deformation theory and algebraic structures.

Findings

Braided Hochschild cohomology admits a braided commutative graded ring structure.

A canonical isomorphism relates Hochschild cohomology of smash products to braided Hochschild cohomology invariants.

Results have implications for deformation theory and finite group actions on schemes.

Abstract

We show that the braided Hochschild cohomology, of an algebra in a suitably algebraic braided monoidal category, admits a graded ring structure under which it is braided commutative. We then give a canonical identification between the usual Hochschild cohomology ring of a smash product and the (derived) invariants of its braided Hochschild cohomology ring. We apply our results to identify the associative formal deformation theory of a smash product with its formal deformation theory as a module algebra over the given Hopf algebra (when the Hopf algebra is sufficiently semisimple). As a second application we deduce some structural results for the usual Hochschild cohomology of a smash product, and discuss specific implications for finite group actions on smooth affine schemes.