We investigate Buchbaum and Eisenbud's construction of the second symmetric power S-R(2) (X) of a chain complex X of modules over a commutative ring R. We state and prove a number of results from the folklore of the subject for which we know of no good direct references. We also provide several explicit computations and examples. We use this construction to prove the following version of a result of Avramov, Buchweitz, and Sega: let R -> S be a module-finite ring homomorphism such that R is noetherian and local, and such that 2 is a unit in R. Let X be a complex of finite rank free S-modules such that X-n = 0 for each n < 0. If boolean OR(n) Ass(R)(H-n (X circle times(S) X)) subset of Ass(R) and if X-p similar or equal to S-p for each p is an element of Ass(R), then X similar or equal to S.

There is a growing body of research concerned with extending the rich theory of commutative Gorenstein rings to DG (=Differential Graded) algebras. This subject began with the work of Félix, Halperin, and Thomas [7] on a Gorenstein condition for (cochains complexes of) topological spaces, which extends classical Poincaré duality for manifolds. Their study was complemented by that of Avramov and Foxby [3], who adopted a similar definition of the Gorenstein property, but focused on DG algebras arising in commutative ring theory. Further progress along these lines has been made by Dwyer, Greenlees, and Iyengar [6]. In a recent article, Frankild and Jørgensen [13] proposed a new notion of ‘Gorenstein’ DG algebras. The natural problem arises: How does their approach relate to those of Félix-Halperin-Thomas and Avramov-Foxby? The content of one of the main theorems in this paper is that the Avramov and Foxby definition of Gorenstein, for the class of DG algebras considered by them, is equivalent to the one of Frankild-Jørgensen. In order to facilitate the ensuing discussion, we adopt the convention that the term ‘Gorenstein DG algebra’ is used in the sense of Frankild and Jørgensen. Their definition is recalled in Section 4; see also the discussion below.

We study the following question: Given two semidualizing complexes B and C over a commutative noetherian ring R, does the vanishing of Ext(R)(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(R)(n) (B, C) = 0 for all n >> 0. We introduce and investigate an equivalence relation approximate to 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 characterizations of this relation, each of which includes the condition Ext(R)(n) (B, C) = 0 for all n >> 0. Finally, we answer our original question in some special cases.

Consider a local chain Differential Graded algebra, such as the singular chain complex of a pathwise connected topological group. In two previous papers, a number of homological results were proved for such an algebra: An Amplitude Inequality, an Auslander-Buchsbaum Equality, and a Gap Theorem. These were inspired by homological ring theory. By the so-called looking glass principle, one would expect that analogous results exist for simply connected cochain Differential Graded algebras, such as the singular cochain complex of a simply connected topological space. Indeed, this paper establishes such analogous results.

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 a an ideal in the Jacobson radical of R. Let Ra be the a-adic completion of R. If M is a finitely generated R-module such that Ext(R)(i)(M) = 0 for all i not equal 0, then M is a-adically complete.

Let (R;m) and (S;n) be commutative Noetherian local rings, and let ' : R ! S be a ∞at local homomorphism such that mS = n and the induced map on residue flelds R=m ! S=n is an isomorphism. Given a flnitely generated R-module M, we show that M has an S-module structure compatible with the given R-module structure if and only if Ext i (S;M) is flnitely generated as an R-module for each i ‚ 1. We say that an S-module N is extended if there is a flnitely generated R-module M such that N » S ›R M. Given a short exact sequence 0 ! N1 ! N ! N2 ! 0 of flnitely generated S-modules, with two of the three modules N1;N;N2 extended, we obtain conditions forcing the third module to be extended. We show that every flnitely generated module over the Henselization of R is a direct summand of an extended module, but that the analogous result fails for the m-adic completion.

We introduce and study a new construction of the second symmetric power S R (X) of a complex X of modules over a commutative ring R. Our construction has the advantage of being relatively straightforward to define, as it is the cokernel of a certain morphism X ⊗R X → X ⊗R X, defined for any complex of R-modules. We prove that, when 2 is a unit in R, our construction respects homotopy equivalences. For bounded-below complexes of finite-rank free R-modules, we explicitly describe the modules occurring in S R (X), and this description allows us to characterize the complexes X for which S R (X) is trivial or has finite projective dimension. Finally, we provide several explicit computations and examples; for instance, we show that our construction is not isomorphic to versions previously studied.

We show that the set S(R) of shift-isomorphism classes of semidualizing complexes over a local ring R admits a nontrivial metric. We investigate the interplay between the metric and several algebraic operations. Motivated by the dagger duality isometry, we prove the following: If K,L are homologically bounded below and degreewise finite R-complexes such that K⊗RLK⊗RLL is semidualizing, then K is shift-isomorphic to R. In investigating the existence of nontrivial open balls in S(R), we prove that S(R) contains elements that are not comparable in the reflexivity ordering if and only if it contains at least three distinct elements.

The main result asserts that a local commutative Noetherian ring is Gorenstein, if it possesses a non-zero cyclic module of finite Gorenstein injective dimension. from this follows a classical result by Peskine and Szpiro stating that the ring is Gorenstein, if it admits a non-zero cyclic module of finite (classical) injective dimension. The main result applies to local homomorphisms of local rings and yields the next: if the source is a homomorphic image of a Gorenstein local ring and the target has finite Gorenstein injective dimension over the source, then the source is a Gorenstein ring. This, in turn, applies to the Frobenius endomorphism when the local ring is of prime equicharacteristic and is a homomorphic image of a Gorenstein local ring.

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.

Let R be a commutative, noetherian, local ring. Topological Q-vector spaces modelled on full subcategories of the derived category of R are constructed in order to study intersection multiplicities.

Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical counterparts, these dimensions do not immediately come with practical and robust criteria for finiteness, not even over commutative noetherian local rings. In this paper we enlarge the class of rings known to admit good criteria for finiteness of Gorenstein dimensions: It now includes, for instance, the rings encountered in commutative algebraic geometry and, in the non-commutative realm, $k$--algebras with a dualizing complex.

There is a growing body of research concerned with extending the rich theory of commutative Gorenstein rings to DG (=Differential Graded) algebras. This subject began with the work of Félix, Halperin, and Thomas [7] on a Gorenstein condition for (cochains complexes of) topological spaces, which extends classical Poincaré duality for manifolds. Their study was complemented by that of Avramov and Foxby [3], who adopted a similar definition of the Gorenstein property, but focused on DG algebras arising in commutative ring theory. Further progress along these lines has been made by Dwyer, Greenlees, and Iyengar [6]. In a recent article, Frankild and Jørgensen [13] proposed a new notion of ‘Gorenstein’ DG algebras. The natural problem arises: How does their approach relate to those of Félix-Halperin-Thomas and Avramov-Foxby? The content of one of the main theorems in this paper is that the Avramov and Foxby definition of Gorenstein, for the class of DG algebras considered by them, is equivalent to the one of Frankild-Jørgensen. In order to facilitate the ensuing discussion, we adopt the convention that the term ‘Gorenstein DG algebra’ is used in the sense of Frankild and Jørgensen. Their definition is recalled in Section 4; see also the discussion below.

Recently, Dwyer and Greenless established a Morita-like equivalence between categories consisting of complete modules and torsion modules. It turns out that these categories contain certain full subcategories which may be viewed as "perturbed" Auslander and Bass classes; Auslander and Bass classes are used in the study of so-called Gorenstein dimensions. This observation allows us to prove that any ideal in a commutative, local, Noetherian ring can detect whether or not the underlying ring is Gorenstein.

There has been much speculation about the structure of the set of shift-isomorphism classes of semidualizing complexes over a local ring. In this paper we show that this set can be given the structure of a nontrivial metric space. We investigate the interplay between the metric and several standard algebraic operations, and we provide a new characterization of Gorenstein rings that is motivated by this interplay. In the process, we obtain new results describing the behavior of reflexivity over homomorphisms of finite flat dimension.

Consider a local chain Differential Graded algebra, such as the singular chain complex of a pathwise connected topological group.In two previous papers, a number of homological results were proved for such an algebra: An Amplitude Inequality, an Auslander–Buchsbaum Equality, and a Gap Theorem. These were inspired by homological ring theory.By the so-called looking glass principle, one would expect that analogous results exist for simply connected cochain Differential Graded algebras, such as the singular cochain complex of a simply connected topological space.Indeed, this paper establishes such analogous results.

. We study the vanishing properties of local homology of complexes of modules without assuming that its homology is artinian. Using vanishing results for local homology and cohomology we prove new vanishing results for Ext- and Tor-modules.

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 Jørgensen. 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.