ABSTRACT
Let X be a chain complex over a commutative noetherian ring R, that is, an object in the derived category π(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 and RHomR(M,β) can detect isomorphisms in π(R) between complexes with restrictions on their supports or co-supports. In particular, the derived local (co)homology functors RΞπ(β) and LΞπ(β) with respect to an ideal πβR 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 βπ-adically finiteβ R-complexes, that is, the Xβπb(R) that are π-cofinite Γ la Hartshorne. For instance, we characterize these complexes in terms of a finiteness condition on LΞπ(X).
Acknowledgments
We are grateful to Srikanth Iyengar for helpful conversations about this work, and to the referee for thoughtful comments.