Remarks on derived complete modules and complexes

Leonid Positselski

Mathematische Nachrichten, 2023, vol. 296, issue 2, 811-839

Abstract: Let R be a commutative ring and I⊂R$I\subset R$ 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 ones. 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.

Date: 2023
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