Cohomologically Cofinite Complexes

Abstract

Let A be a commutative noetherian ring, and 𝔞 an ideal in it. In this paper we continue the study, begun in [11], of the derived 𝔞-adic completion and the derived 𝔞-torsion functors. Here are our results: (1) a structural characterization of bounded above cohomologically complete complexes; (2) the Cohomologically Complete Nakayama Theorem; and (3) a characterization of cohomologically cofinite complexes.

Key Words:

• Adic completion
• Derived functors
• Torsion

2010 Mathematics Subject Classification:

• Primary: 13D07
• Secondary: 13B35; 13C12; 13D09; 18E30

ACKNOWLEDGMENTS

We wish to thank Joseph Lipman, Ana Jeremias, and Leo Alonso for helpful discussions. We are also grateful to the anonymous referee who read the paper carefully and made some useful suggestions.

Notes

¹If we were to replace “A is noetherian” with “𝔞 is finitely generated” in Setup 1.1, then everything from there to this corollary would remain valid, except for flatness (in Theorem 1.5(2) and Corollary 1.8(1)), about which we do not know.

Communicated by S. Kleiman.