Expanding Čech cohomology for quantales

Abstract

We expand Čech cohomology of a topological space $X$ with values in a presheaf on $X$ to Čech 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 $(\hat{I}¸,\ddot{I}„)$ 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 Čech cohomology groups of $X$ with values on the constant sheaf $F$ on $X$ is isomorphic to the $q$th Čech cohomology groups of the ring $C(X)$ with values on a sheaf $F\circ \ddot{I}„$ on $C(X)$.

20 pages. We have removed one of the examples and the notion of a "basis" for a quantale; we will discuss them elsewhere with more details