Anders Frankild

REFLEXIVITY AND RING HOMOMORPHISMS OF FINITE FLAT DIMENSION

In this article we present a systematic study of the reflexivity properties of homologically finite complexes with respect to semidualizing complexes in the setting of nonlocal rings. One primary focus is the descent of these properties over ring homomorphisms of finite flat dimension, presented in terms of inequalities between generalized G-dimensions. Most of these results are new even when the ring homomorphism is local. The main tool for these analyses is a nonlocal version of the amplitude inequality of Iversen, Foxby, and Iyengar. We provide numerous examples demonstrating the need for certain hypotheses and the strictness of many inequalities.

Dualizing Differential Graded Modules and Gorenstein Differential Graded Algebras

The paper explores dualizing differential graded (DG) modules over DG algebras. The focus is on DG algebras that are commutative local, and finite. One of the main results established is that, for this class of DG algebras, a finite DG module is dualizing precisely when its Bass number is 1. As a corollary, one obtains that the Avramov–Foxby notion of Gorenstein DG algebras coincides with that due to Frankild and Jorgensen. One other key result is that, under suitable hypotheses, any two dualizing DG modules are quasiisomorphic up to a suspension. In addition, it is established that a number of naturally occurring DG algebras possess dualizing DG modules.

Relations between semidualizing complexes

We study the following question: Given two semidualizing com- plexes B and C over a commutative noetherian ring R, does the vanishing of Ext n(B, C) for n ≫ 0 imply that B is C-reflexive? This question is a natural generalization of one studied by Avramov, Buchweitz, and Sega. We begin by providing conditions equivalent to B being C-reflexive, each of which is slightly stronger than the condition Ext n(B, C) = 0 for all n ≫ 0. We introduce and investigate an equivalence relation ≈ on the set of isomorphism classes of semidualizing complexes. This relation is defined in terms of a natural action of the derived Picard group and is well-suited for the study of semidualizing complexes over nonlocal rings. We identify numerous alternate characteriza- tions of this relation, each of which includes the condition Ext n(B, C) = 0 for all n ≫ 0. Finally, we answer our original question in some special cases.

Homological identities for differential graded algebras

Our original motivation for this paper was to answer [4, Question (3.10)] on ga the sequence of Bass numbers of a Differential Graded Algebra (DGA). We do Section 3.1. [4, Question (3.10)] asks for a sort of No Holes theorem for Bass numbers of D More precisely, it asks for a certain bound on the length of gaps in the sequence o numbers; namely, that if one has μ = 0,μ +1= · · · = μ +g = 0, andμ +g+1 = 0, theng is at most equal to the degree of the highest nonvanishing homology of the DGA. T the best possible bound one can hope for, as shown in [4, Example (3.9)]. We provide this bound in Section 3.1 and thereby answer the question. Our m works for several important classes of DGAs, among them DG fibres of ring homo phisms, Koszul complexes, and singular chain DGAs of the form C ∗(G; k) wherek is a field andG a path connected topological monoid with dim kH∗(G; k) <∞ (see Remark 2.2). Section 3.1 arises as corollary to a more general Gap theorem, Theore which is the natural generalization to the world of DGAs of the classical No Holes rem from homological ring theory (see [10,12,18,23,25]).

Gorenstein differential graded algebras

AbstractWe propose a definition of Gorenstein Differential Graded Algebra. In order to give examples, we introduce the technical notion of Gorenstein morphism. This enables us to prove the following: Theorem:Let A be a noetherian local commutative ring, let L be a bounded complex of finitely generated projective A-modules which is not homotopy equivalent to zero, and let ɛ=HomA(L, L)be the endomorphism Differential Graded Algebra of L. Then ɛ is a Gorenstein Differential Graded Algebra if and only if A is a Gorenstein ring. Theorem:Let A be a noetherian local commutative ring with a sequence of elementsa=(a1,…,an)in the maximal ideal, and let K(a)be the Koszul complex ofa.Then K(a)is a Gorenstein Differential Graded Algebra if and only if A is a Gorenstein ring. Theorem:Let A be a noetherian local commutative ring containing a field k, and let X be a simply connected topological space with dimkH*(X;k)<∞,which has poincaré duality over k. Let C*(X;A)be the singular cochain Differential Graded Algebra of X with coefficients in A. Then C*(X; A)is a Gorenstein Differential Graded Algebra if and only if A is a Gorenstein ring.The second of these theorems is a generalization of a result by Avramov and Golod from [4].

Detecting completeness from Ext-vanishing

Motivated by work of C. U. Jensen, R.-O. Buchweitz, and H. Flenner, we prove the following result. Let R be a commutative noetherian ring and α an ideal in the Jacobson radical of R. Let Ra be the α-adic completion of R. If M is a finitely generated R-module such that Ext i R (M) = 0 for all i ≠ 0, then M is α-adically complete.

Foxby equivalence, complete modules, and torsion modules

Abstract Taking the idea from classical Foxby equivalence, we develop an equivalence theory for derived categories over differential graded algebras. Both classical Foxby equivalence and the Morita equivalence for complete modules and torsion modules developed by Dwyer and Greenlees arise as special cases.