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

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.