Duality in algebra and topology

We apply ideas from commutative algebra, and Morita theory to algebraic topology using ring spectra. This allows us to prove new duality results in algebra and topology, and to view (1) Poincaré duality for manifolds, (2) Gorenstein duality for commutative rings, (3) Benson–Carlson duality for cohomology rings of finite groups, (4) Poincaré duality for groups and (5) Gross–Hopkins duality in chromatic stable homotopy theory as examples of a single phenomenon.     

Über Die Assoziierten Primideale des Bidualen

Let (R, 𝔪) be a commutative, noetherian, local ring, E the injective hull of the residue field R/𝔪, and M ^○○ = Hom _R (Hom _R (M, E), E) the bidual of an R-module M. We investigate the elements of Ass(M ^○○) as well as those of Coatt(M) = {𝔭 ∈ Spec(R)|𝔭 = Ann _R (Ann _M (𝔭))} and provide criteria for equality in one of the two inclusions Ass(M) ? Ass(M ^○○) ? Coatt(M). If R is a Nagata ring and M a minimax module, i.e., an extension of a finitely generated R-module by an artinian R-module, we show that Ass(M ^○○) = Ass(M) ∪ {𝔭 ∈ Coatt(M)| R/𝔭 is incomplete}.     

Gorenstein Modules and Auslander Categories

In this article, some new characterizations of Gorenstein projective, injective, and flat modules over commutative noetherian local rings are given.     

On the Associated Primes of Matlis Duals of Top Local Cohomology Modules

After motivating the question, we prove various results about the set of associated primes of Matlis duals of top local cohomology modules. In some cases, we can calculate this set. An easy application of this theory is the well-known fact that Krull dimension can be expressed by the vanishing of local cohomology modules.     

The local cohomology modules of Matlis reflexive modules are almost cofinite

We show that if and are Matlis reflexive modules over a complete Gorenstein local domain and is an ideal of such that the dimension of is one, then the modules are Matlis reflexive for all and if . It follows that the Bass numbers of are finite. If is not a domain, then the same results hold for .

     

Generalized Matlis duality

Let be a commutative noetherian ring and let be the minimal injective cogenerator of the category of -modules. A module is said to be reflexive with respect to if the natural evaluation map from to is an isomorphism. We give a classification of modules which are reflexive with respect to . A module is reflexive with respect to if and only if has a finitely generated submodule such that is artinian and is a complete semi-local ring.

     

On the existence of a single test module

We revisit two questions concerning the existence of a single test module by comparing them with similar questions (see Theorem 3.3). As a corollary, we identify domains over which strongly flat modules and torsion-free Whitehead modules coincide (see Corollary 3.6). We obtain several analogous results to the main theorem under stronger hypotheses (see section 4). In particular, we settle a long-standing question concerning a characterization of almost perfect domains (see Corollary 4.4). We also look into the case when the character module of K and the Matlis-dual of K are isomorphic (see Theorem 5.2).     

Matlis duals of top Local cohomology modules

In the first section of this paper we present generalizations of known results on the set of associated primes of Matlis duals of local cohomology modules; we prove these generalizations by using a new technique. In section 2 we compute the set of associated primes of the Matlis dual of , where is a -dimensional local ring and an ideal such that and .

     

Prüfer v-Multiplication Domains and w-Multiplicative Canonical Ideals

Let R be an integral domain. A w-ideal I of R is called a w-multiplicative canonical ideal if (I: (I: J)) = J for each w-ideal J of R. In particular, if R is a w-multiplicative canonical ideal of R, then R is a w-divisorial domain. These are the w-analogues of the concepts of a multiplicative canonical ideal and a divisorial domain, respectively. Motivated by the articles [8 El Baghdadi S., Gabelli , S. ( 2005 ). w-Divisorial domains . J. Algebra 285 : 335 – 355 .[Crossref], [Web of Science ®] , [Google Scholar], 10 Heinzer , W. , Huckaba , J. A. , Papick , I. J. ( 1998 ). m-Canonical ideals in integral domains . Comm. Algebra 26 ( 9 ): 3021 – 3043 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], we study the domains possessing w-multiplicative canonical ideals; in particular, we consider Prüfer v-multiplication domains.