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

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.

On the sheafyness property of spectra of Banach rings

“…‡ Here we discuss only the non-Archimedean version of the theory but with suitable changes one can develop a theory that works uniformly over all Banach rings. See [1][2][3] and [4] where all Banach rings are considered and in particular [6] for a more in-depth analysis.…”

“…We must underline that the Archimedean situation presents further complications that are not present in the non-Archimedean theory. We refer to the (almost finished at the time of writing) pre-print [6] for an in-depth study of this issue. -The theory should work also for other kinds of spaces (and maybe sometimes even become easier).…”