Computing in complete local equicharacteristic Noetherian rings via topological rewriting on commutative formal power series

TL;DR

This paper extends the analogy between Gr"obner basis theory and rewriting systems to formal power series in complete local rings, using topological rewriting to characterize standard bases and confluence properties.

Contribution

It introduces topological rewriting methods to analyze standard bases in formal power series rings, providing new proofs and extending existing algebraic characterizations.

Findings

Characterization of standard bases via topological confluence.

Equivalence of generalized confluence properties in formal power series.

Extension of algebraic rewriting theory to topological setting.

Abstract

In commutative algebra, the theory of Gr\"obner bases enables one to compute in any finitely generated algebra over a given computable field. For non-finitely generated algebras however, other methods have to be pursued. For instance, it follows from the Cohen structure theorem that standard bases of formal power series ideals offer a similar prospect but for complete local equicharacteristic rings whose residue field is computable. Using the language of rewriting theory, one can characterise Gr\"obner bases in terms of confluence of the induced rewriting system. It has been shown, so far via purely algebraic tools, that an analogous characterisation holds for standard bases with a generalised notion of confluence. Subsequently, that result is utilised to prove that two generalised confluence properties, where one is actually in general strictly stronger than the other, are actually equivalent in the context of formal power series. In the present paper, we propose alternative proofs making use of tools purely from the new theory of topological rewriting to recover both the characterisation of standard bases and the equivalence between generalised confluence properties. The objective is to extend the analogy between Gr\"obner basis theory together with classical algebraic rewriting theory and standard basis theory with topological rewriting theory.