The homotopy theory of complete modules

TL;DR

This paper explores the relationships between different notions of module completion over a ring, establishing their homotopy equivalence under certain conditions and analyzing how these theories behave under ring homomorphisms.

Contribution

It proves the equivalence of various completion theories for modules and complexes under weak pro-regularity, and characterizes when their homotopy categories are equivalent across ring extensions.

Findings

Different notions of module completion share the same homotopy theory under weak pro-regularity.

Necessary and sufficient conditions for homotopy equivalence of complete complexes across ring homomorphisms.

Generalizes previous results on extended local (co)homology.

Abstract

Given a commutative ring $R$ and finitely generated ideal $I$, one can consider the classes of $I$-adically complete, $L_0^I$-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, $L_0^I$-complete and derived $I$-complete complexes and prove that they present the same homotopy theory. Given a ring homomorphism $R \to S$, 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.