Commutative bidifferential algebra

Motivated by the Poisson Dixmier-Moeglin equivalence prob-lem, a systematic study of commutative unitary rings equipped with a biderivation, namely a binary operation that is a derivation in each argument, is here begun, with an eye toward the geometry of the corresponding B-varieties. Foun-dational results about extending biderivations to localisations, algebraic extensions and transcendental extensions are estab-lished. Resolving a deficiency in Poisson algebraic geometry, a theory of base extension is achieved, and it is shown that dominant B-morphisms admit generic B-fibres. A bidifferen-tial version of the Dixmier-Moeglin equivalence problem is articulated.Crown Copyright (c) 2022 Published by Elsevier Inc. All rights reserved.

Automated summary

Motivated by the Poisson Dixmier-Moeglin equivalence problem, this paper initiates a systematic study of commutative unitary rings equipped with a biderivation, exploring the geometry of the corresponding B-varieties. Foundational results regarding the extension of biderivations to localisations, algebraic extensions, and transcendental extensions are established. A base extension theory is achieved to address a deficiency in Poisson algebraic geometry, demonstrating that dominant B-morphisms have generic B-fibres. A bidifferential version of the Dixmier-Moeglin equivalence problem is articulated.