2024/04/19 by Ana Luiza Tenório, Tenório, Ana Luiza, Peter Arndt +3
Mathematics · #18F75 55N30 13D03 #Algebraic structures and combinatorial models #Category Theory (math.CT) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2404.12884
openalex publication_date 2024/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (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 Čech cohomology groups of X with values on the constant sheaf F on X is isomorphic to the qth Čech cohomology groups of the ring C(X) with values on a sheaf F ∘ τ on C(X).