Expanding \v{C}ech cohomology for quantales

TL;DR

This paper generalizes cech cohomology from topological spaces to commutative rings with unity by leveraging the structure of quantales, establishing an isomorphism between their respective cohomology groups.

Contribution

It introduces a novel approach to cech cohomology for rings using quantale theory and functor adjunctions, extending classical topological methods.

Findings

Establishes an isomorphism between cech cohomology of spaces and rings.

Develops a functorial framework connecting open sets and ideals via quantales.

Provides a new perspective on cohomology using algebraic and order-theoretic structures.

Abstract

We expand \v{C}ech cohomology of a topological space $X$ with values in a presheaf on $X$ to \v{C}ech cohomology of a commutative ring with unity $R$ with values in a presheaf on $R$. The strategy is to observe that both the set of open subsets of $X$ and the set of ideals of $R$ provide examples of a (semicartesian) quantale. We study a particular pair of (adjoint) functors $(\theta, \tau)$ between the quantale of open subsets of $X$ and the quantale of ideals of $C(X)$, the ring of real-valued continuous functions on $X$. This leads to the main result of this paper: the $q$th \v{C}ech cohomology groups of $X$ with values on the constant sheaf $F$ on $X$ is isomorphic to the $q$th \v{C}ech cohomology groups of the ring $C(X)$ with values on a sheaf $F \circ \tau$ on $C(X)$.