An endomorphism of an Azumaya algebra of constant rank over a formally smooth center admits a multiplicative first-order flat lift if and only if it preserves the Poisson structure on the center induced by the lift of the algebra.
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Hochschild cohomology and lifts of endomorphisms
An endomorphism of an Azumaya algebra of constant rank over a formally smooth center admits a multiplicative first-order flat lift if and only if it preserves the Poisson structure on the center induced by the lift of the algebra.