Simplicial Hochschild cochains as an Amitsur complex

TL;DR

This paper demonstrates an isomorphism between the Hochschild cochain complex of certain algebra extensions and the Amitsur complex of a related coring, unifying different algebraic structures in a differential graded algebra framework.

Contribution

It establishes a novel isomorphism linking Hochschild cochains with the Amitsur complex for depth two algebra extensions, extending to various classes like Hopf-Galois extensions.

Findings

Hochschild cochains form a differential graded algebra isomorphic to the Amitsur complex.

The result applies to finite dimensional, H-separable, and Hopf-Galois extensions.

Unifies different algebraic complexes under a common framework.

Abstract

It is shown that the cochain complex of relative Hochschild A-valued cochains of a depth two extension A | B under cup product is isomorphic as a differential graded algebra with the Amitsur complex of the coring S = End {}_BA_B over the centralizer R = A^B with grouplike element 1_S, which itself is isomorphic to the Cartier complex of S with coefficients in the (S,S)-bicomodule R^e. This specializes to finite dimensional algebras, H-separable extensions and Hopf-Galois extensions.