Cartier isomorphism for unital associative algebras

TL;DR

This paper generalizes the Cartier isomorphism to unital associative algebras over fields of odd positive characteristic, replacing differential forms with Hochschild homology and the de Rham differential with the Connes-Tsygan differential.

Contribution

It introduces a non-commutative version of the Cartier isomorphism for associative algebras, extending classical concepts to a broader algebraic context.

Findings

Constructed a non-commutative Cartier isomorphism for unital associative algebras.

Replaced differential forms with Hochschild homology classes.

Substituted de Rham differential with Connes-Tsygan differential.

Abstract

Given an associative unital algebra $A$ over a perfect field $k$ of odd positive characteristic, we construct a non-commutative generalization of the Cartier isomorphism for $A$. The role of differential forms is played by Hochschild homology classes, and de Rham diferential is replaced with the Connes-Tsygan differential.