Homotopy Epimorphisms and Derived Tate's Acyclicity for Commutative C*-algebras

TL;DR

This paper explores the relationship between homotopy epimorphisms, covers, and derived Tate's acyclicity in commutative C*-algebras, establishing topological correspondences and descent properties.

Contribution

It characterizes homotopy epimorphisms as closed immersions and covers as topological covers, linking algebraic and topological structures in C*-algebras.

Findings

Homotopy epimorphisms correspond to closed immersions.

Covers correspond to topological covers with finite subcovers.

Derived descent for Banach modules is established.

Abstract

We study homotopy epimorphisms and covers formulated in terms of derived Tate's acyclicity for commutative C*-algebras and their non-Archimedean counterparts. We prove that a homotopy epimorphism between commutative C*-algebras precisely corresponds to a closed immersion between the compact Hausdorff topological spaces associated to them, and a cover of a commutative C*-algebra precisely corresponds to a topological cover of the compact Hausdorff topological space associated to it by closed immersions admitting a finite subcover. This permits us to prove derived and non-derived descent for Banach modules over commutative C*-algebras.