Remarks on derived complete modules and complexes

Abstract

Let $R$ be a commutative ring and $I\subset R$ a finitely generated ideal. We discuss two definitions of derived $I$-adically complete (also derived $I$-torsion) complexes of $R$-modules which appear in the literature: the idealistic and the sequential one. The two definitions are known to be equivalent for a weakly proregular ideal $I$; we show that they are different otherwise. We argue that the sequential approach works well, but the idealistic one needs to be reinterpreted or properly understood. We also consider $I$-adically flat $R$-modules.

arXiv commentLaTeX 2e with xy-pic and tikz-cd; 40 pages, 3+2 commutative diagrams; v.3: abstract rewritten, last section "Conclusion" expanded with two commutative diagrams inserted; v.4: Proposition 1.3(d), Lemma 1.4, Examples 1.8, Remarks 2.4, 5.6, 6.1, 7.3 inserted, the definition of a contraherent cosheaf spelled out in Section 2; v.5: references updated

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Cited by in corpus (7)

• Contramodules
• Derived, coderived, and contraderived categories of locally presentable abelian categories
• Differential graded Koszul duality: an introductory survey
• Prismatic $G$-displays and descent theory
• The homotopy theory of complete modules
• Flat commutative ring epimorphisms of almost Krull dimension zero
• Homological full-and-faithfulness of comodule inclusion and contramodule forgetful functors