Expanding {C}ech cohomology for quantales

We expand {C}ech cohomology of a topological space X with values in a presheaf on X to {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 (θ, τ) 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 qth {C}ech cohomology groups of X with values on the constant sheaf F on X is isomorphic to the qth {C}ech cohomology groups of the ring $C(X)$ with values on a sheaf F ∘ τ on $C(X)$.