Commutative bidifferential algebra

TL;DR

This paper introduces a systematic study of commutative rings with biderivations, exploring their geometric properties, extending foundational results, and addressing a bidifferential version of the Dixmier-Moeglin equivalence problem.

Contribution

It develops foundational theory for commutative rings with biderivations, including extension results, a base extension theory, and a bidifferential Dixmier-Moeglin equivalence.

Findings

Extended biderivations to localizations and extensions.

Established a base extension theory for B-varieties.

Formulated a bidifferential Dixmier-Moeglin equivalence.

Abstract

Motivated by the Poisson Dixmier-Moeglin equivalence problem, a systematic study of commutative unitary rings equipped with a {\em biderivation}, namely a binary operation that is a derivation in each argument, is here begun, with an eye toward the geometry of the corresponding {\em $B$-varieties}. Foundational results about extending biderivations to localisations, algebraic extensions and transcendental extensions are established. 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 bidifferential version of the Dixmier-Moeglin equivalence problem is articulated.