Commutative bidifferential algebra (Q2168839)

The article gives an introduction to the theory of biderivations on commutative rings. A \textit{biderivation} on a ring \(R\) is defined to be binary operator \(\{\cdot ,\cdot \}: R\times R \to R\) such that for each \(r\in R\), the maps \(\{r,\cdot\}\) and \(\{\cdot, r\}\) are derivations on \(R\). The main family of examples are Poisson brackets. However, contrary to Poisson brackets, the family of biderivations behaves better under localisations and extensions of rings, as the extension properties from derivations usually carry over (see Theorem 2.8, Theorem 2.10, and Theorem 2.13). For example, a biderivation on a field extends uniquely to its separable closure (Corollary 2.12). After giving the basic definitions and proving the main properties on extending biderivations, the authors handle tensor products of rings with biderivations. These definitions and results are then translated to affine algebraic geometry. The authors' motivation for their study of bidifferential algebras comes from their approach to probably attack the Dixmier-Moeglin equivalence problem for Poisson varieties. For this, they also transfer this equivalence problem to the bidifferential setting, and pin it down to answering the following question which is the direct generalisation of the open question in the Poisson Dixmier-Moeglin equivalence problem. For stating that problem, fix a bidifferential field \((k, \{\cdot,\cdot\})\) of characteristic zero, and consider a finitely generated bidifferential extension \((R,\{\cdot,\cdot\})\) of \(k\) without zero-divisors (called \emph{affine bidifferential \(k\)-algebra} in the article). Denote by \(\operatorname{Spec}_B(R)\) the set of prime ideals of \(R\) that are stable under the biderivation (so called \emph{prime bidifferential ideals}). Further call a prime differential ideal \(P\in \operatorname{Spec}_B(R)\) \begin{itemize} \item \textit{\(B\)-locally closed} if it is strictly contained in the intersection of all prime differential ideals strictly containing \(P\), i.e., if \(P\ne \bigcap\limits_{P\subsetneq Q\in \operatorname{Spec}_B(R)} Q\). \item \textit{\(B\)-rational} if the constants of \(\operatorname{Frac}(R/P)\) are contained in the algebraic closure \(k^{\mathrm{alg}}\) of \(k\). \end{itemize} Here, an element \(s\) of a bidifferential ring \((S,\{\cdot,\cdot\})\) is called \textit{constant}, if both derivations \(\{s,\cdot\}\) and \(\{\cdot, s\}\) are identically zero. \textbf{Problem} (see Problem 4.4 in the article) For which affine bidifferential \(k\)-algebras \(R\) is every \(B\)-rational prime bidifferential ideal \(P\) also \(B\)-locally closed? The converse implication (\(B\)-locally closed \(\Rightarrow\) \(B\)-rational) was already known to hold for all affine Poisson algebras and is shown for affine bidifferential algebras by the authors. That's why the stated question is called an equivalence problem. Even for affine Poisson algebras, there are known cases where the questioned implication does not hold, and known cases where it holds (cf. the paragraph after Problem 4.4). It is further conjectured that the equivalence holds for affine Poisson-Hopf algebras. In the final part, the authors elaborate on the properties of \(B\)-rationality and \(B\)-locally closedness, and explain a possible approach to attack the equivalence problem.