Extended Local Cohomology and Local Homology

Abstract

Abstract. We present an in-depth exploration of the module structures of local (co)homology modules (moreover, for complexes) over the completion R a of a commutative noetherian ring R with respect to a proper ideal a. In particular, we extend Greenlees-May Duality and MGM Equivalence to track behavior over R a , not just over R. We apply this to the study of two recent versions of homological finiteness for complexes, and to certain isomorphisms, with a view toward further applications. We also discuss subtleties and simplifications in the computations of these functors.

Adic finiteness: Bounding homology and applications

“…Sections 3-5 contains numerous results akin to Theorem 1.1, with various derived functors and finiteness conditions. It should be reiterated that these results are all applied in our subsequent work, especially in [25]. We also note that many of the results of Section 3 are new even when the adically finite condition is replaced with the more restrictive assumption of being in D f b (R).…”

“…We end with an indication of some of the applications of our boundedness results, following Foxby and Iyengar [10]. See also [25] for other applications. In particular, the quantities fd Q (R) and fd Q (F ) are simultaneously finite.…”

“…This work is part 3 of a series of papers exploring notions of support and finiteness of R-complexes. It builds on our previous papers [27,29], and it is used in the papers [25,26,28]. It is heavily influenced by Foxby and Iyengar's paper [10] and the non-local extension in [11].…”

“…Here RΓ b (R) is the derived local cohomology complex with respect to b. Further applications are contained in the subsequent [25].…”

Injective DG-modules over non-positive DG-rings

“…of functors D(A) → D(A). See [21] and [22,Section 3] for a study of these functors over commutative rings.…”