Complete modules and torsion modules

Abstract

Abstract. Suppose that R is a ring and that A is a chain complex over R. Inside the derived category of differential graded R-modules there are naturally defined subcategories of A-torsion objects and of A-complete objects. Under a finiteness condition on A, we develop a Morita theory for these subcategories, find conceptual interpretations for some associated algebraic functors, and, in appropriate commutative situations, identify the associated functors as local homology or local cohomology. Some of the results are suprising even in the case R = Z and A = Z/p.

The Cellularization Principle for Quillen adjunctions

“…We establish (2) in the next paragraph. One can make very similar arguments for (3) or one can deduce (3) from (2). To deduce (3) We now return to the Quillen equivalence in (2).…”

Differential graded categories

“…In the independently obtained [40], W. Dwyer and J. Greenlees give elegant descriptions via dg endomorphism rings of categories of complete, respectively torsion, modules. Their results are applied in a unifying study of duality phenomena in algebra and topology in [41].…”

Affine equivalence and Gorensteinness

“…Hartshorne [13] -R Foxby/Yassemi [17] Trivial R ᑾ (R) -Dwyer/Greenlees [7] The first three X's in the diagram are:…”

“…Here A E(k) and B E(k) are the Auslander and Bass classes of E(k) which are defined in a way analogous to A D and B D above, see [11, (3.3)], and A ᒊ comp and A tors ᒊ are the categories of so-called derived complete and derived torsion complexes with respect to ᒊ, see [7] or remark 1.2 below.…”

“…Remark 1.2. In [7] is considered the following situation: Given a ring, S, and a bounded complex of finitely generated projective S-left-modules, A, one can construct the endomorphism Differential Graded Algebra, E = Hom S (A, A), and A becomes a Differential Graded E -left-module whose E -structure is compatible with its S-structure. Likewise, the complex A = Hom S (A, S) is a bounded complex of finitely generated projective S-rightmodules, and becomes a Differential Graded E -right-module whose E -structure is compatible with its S-structure.…”