Hochschild cohomology and lifts of endomorphisms

TL;DR

This paper investigates conditions under which algebra endomorphisms can be lifted to first-order flat lifts, linking the problem to Hochschild cohomology classes and properties of Azumaya algebras.

Contribution

It introduces a canonical Hochschild cohomology class associated with lifts and characterizes liftability via the vanishing of this class and Poisson structure preservation.

Findings

Endomorphisms lift iff the associated Hochschild cohomology class vanishes.

For Azumaya algebras, liftability is equivalent to preserving the Poisson structure.

The paper establishes a cohomological criterion for lifting endomorphisms.

Abstract

We study when algebra endomorphisms can be lifted to first-order flat lifts. To a first-order flat lift of an algebra and an endomorphism, we associate a canonical class in Hochschild cohomology with coefficients in a naturally twisted bimodule. The cohomology class vanishes exactly when the endomorphism admits a multiplicative lift. For an Azumaya algebra of constant rank over a formally smooth center, we prove that an endomorphism lifts if and only if the induced endomorphism of the center preserves the Poisson structure given by the lift of the algebra.