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, $L_0^I$-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.

arXiv comment20pp, v2: version accepted in Journal of Algebra

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• Dedualizing complexes and MGM duality
• The Left Localization Principle, completions, and cofree $G$-spectra
• Remarks on derived complete modules and complexes
• Derived completion for comodules
• Adic reduction to the diagonal and a relation between cofiniteness and derived completion
• Approximation of subcategories by abelian subcategories