Corings, their dual rings and relative (co)Hochschild cohomology

TL;DR

This paper establishes an isomorphism between Cartier and relative Hochschild cohomology for finitely generated projective corings, extending to $B_{}$-algebras, with applications to entwining structures and equivariant cohomology.

Contribution

It demonstrates a novel isomorphism between Cartier and relative Hochschild cohomology for certain corings and extends this to the algebraic structure of $B_{}$-algebras, with applications to entwining structures.

Findings

Cartier cohomology is isomorphic to relative Hochschild cohomology for finitely generated projective corings.

The isomorphism extends to the level of $B_{}$-algebras of chain complexes.

Application to entwining structures yields a description of equivariant cohomology as relative Hochschild cohomology.

Abstract

We show for a coring which is finitely generated projective as a left module that the Cartier cohomology is isomorphic to the relative Hochschild cohomology of the right algebra. Furthermore, we show that this isomorphism lifts to the level of $B_{\infty}$-algebras of the chain complexes, by showing that the opposite $B_{\infty}$-algebra of the relative Hochschild cochains of the right algebra is isomorphic to the $B_{\infty}$-algebra of Cartier cochains. Lastly, we apply this to entwining structures where the coalgebra is finite-dimensional, to get a description of the equivariant cohomology of the entwining structure as the relative Hochschild cohomology of the twisted convolution algebra.