The homotopy theory of complete modules

Abstract

Given a commutative ring R and finitely generated ideal I, one can consider the classes of I-adically complete, L I 0 -complete and derived I-complete complexes. Under a mild assumption on the ideal I called weak pro-regularity, these three notions of completions interact well. We consider the classes of I-adically complete, L I 0 -complete and derived I-complete complexes and prove that they present the same homotopy theory. Given a ring homomorphism R → S, we then give necessary and sufficient conditions for the categories of complete R-complexes and the categories of complete S-complexes to have equivalent homotopy theories. This recovers and generalizes a result of Sather-Wagstaff and Wicklein on extended local (co)homology.