Localization of Rota-Baxter algebras

TL;DR

This paper extends the concept of localization from commutative algebras to commutative Rota-Baxter algebras, establishing their existence, explicit constructions, and related tensor product properties, with applications to algebraic geometry.

Contribution

It introduces the notion of localization for commutative Rota-Baxter algebras, proves its existence, constructs explicit examples, and explores their tensor products and topological structures.

Findings

Existence of localization for commutative Rota-Baxter algebras

Explicit constructions under mild conditions

Rota-Baxter coverings form a Grothendieck topology

Abstract

A commutative Rota-Baxter algebra can be regarded as a commutative algebra that carries an abstraction of the integral operator. With the motivation of generalizing the study of algebraic geometry to Rota-Baxter algebra, we extend the central concept of localization for commutative algebras to commutative Rota-Baxter algebras. The existence of such a localization is proved and, under mild conditions, its explicit constructions are obtained. The existence of tensor products of commutative Rota-Baxter algebras is also proved and the compatibility of localization and tensor product of Rota-Baxter algebras is established. We further study Rota-Baxter coverings and show that they form a Gr\"othendieck topology.