Homotopy epimorphisms and derived tate’s acyclicity for commutative C*-algebras

Abstract

Abstract We study homotopy epimorphisms and covers formulated in terms of derived Tate’s acyclicity for commutative $C^*$-algebras and algebras of continuous functions valued in non-Archimedean valued fields. We prove that a homotopy epimorphism between commutative $C^*$-algebras precisely corresponds to a closed immersion between the compact Hausdorff topological spaces associated with them and a cover of a commutative $C^*$-algebra precisely corresponds to a topological cover of the compact Hausdorff topological space associated with 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.