Support and adic finiteness for complexes

ABSTRACT

Let X be a chain complex over a commutative noetherian ring R, that is, an object in the derived category 𝒟(R). We investigate the small support and co-support of X, introduced by Foxby and Benson, Iyengar, and Krause. We show that the derived functors $M{\otimes }_R^L-$ and RHom_R(M,−) can detect isomorphisms in 𝒟(R) between complexes with restrictions on their supports or co-supports. In particular, the derived local (co)homology functors RΓ_𝔞(−) and LΛ^𝔞(−) with respect to an ideal 𝔞⊊R have the same ability. Furthermore, we give reprove some results of Benson, Iyengar, and Krause in our setting, with more direct proofs. Also, we include some computations of co-supports, since this construction is still quite mysterious. Lastly, we investigate “𝔞-adically finite” R-complexes, that is, the X∈𝒟_b(R) that are 𝔞-cofinite à la Hartshorne. For instance, we characterize these complexes in terms of a finiteness condition on LΛ^𝔞(X).

KEYWORDS:

• Adically finite complexes
• cofinite complexes; derived categories; Koszul homology
• local cohomology
• local homology

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13B35
• 13D02
• 13D09
• 13D45
• 13J10

Acknowledgments

We are grateful to Srikanth Iyengar for helpful conversations about this work, and to the referee for thoughtful comments.