On localizing topological algebras

TL;DR

This paper explores a class of topological algebras that generalize structure sheaf algebras using sheaf-theoretic localization, with potential applications in differential and algebraic geometry.

Contribution

It introduces geometric topological algebras based on Gel'fand transforms, highlighting their cohomological properties and stability under inductive limits.

Findings

Sheaf-theoretic localization underpins the structure of these algebras.

Geometric topological algebras possess special cohomological features.

The class of these algebras is closed under inductive limits.

Abstract

Through the subsequent discussion we consider a certain particular sort of (topological) algebras, which may substitute the `` structure sheaf algebras'' in many--in point of fact, in all--the situations of a geometrical character that occur, thus far, in several mathematical disciplines, as for instance, differential and/or algebraic geometry, complex (geometric) analysis etc. It is proved that at the basis of this type of algebras lies the sheaf-theoretic notion of (functional) localization, which, in the particular case of a given topological algebra, refers to the respective ``Gel'fand transform algebra'' over the spectrum of the initial algebra. As a result, one further considers the so-called ``geometric topological algebras'', having special cohomological properties, in terms of their ``Gel'fand sheaves'', being also of a particular significance for (abstract) differential-geometric applications; yet, the same class of algebras is still ``closed'', with respect to appropriate inductive limits, a fact which thus considerably broadens the sort of the topological algebras involved, hence, as we shall see, their potential applications as well.