Approximations of complete modules by complete big Cohen-Macaulay modules over a Cohen-Macaulay local ring
type: Item
instance of: scholarly article
title: Approximations of complete modules by complete big Cohen-Macaulay modules over a Cohen-Macaulay local ring
zbMATH Open document ID: 1176.13002
author: Anne-Marie Simon
published in: Algebras and Representation Theory
publication date: 2009-09-01T00:00:00Z
review text: Let \(\mathcal T\) be a class of objects in an additive category \(\mathcal A\). A morphism \(f: M \rightarrow T\) with \(M \in {\mathcal A}\) and \(T \in {\mathcal T}\) is called a \(\mathcal T\)-\textit{preenvelope} if \(\text{Hom}_{\mathcal A}(T,T^{\prime}) \rightarrow \text{Hom}_{\mathcal A}(M,T^{\prime})\) is surjective, \(\forall T^{\prime} \in {\mathcal T}\). A \(\mathcal T\)-preenvelope is called a \(\mathcal T\)-\textit{envelope} when any endomorphism \(h\) of \(T\) with \(h \circ f = f\) is an automorphism. Dually, one has the notion of \(\mathcal T\)-\textit{cover}. Now, let \((A,\mathfrak{m},k)\) be a Cohen-Macaulay noetherian local ring, \(\mathcal X\) the class of complete big Cohen-Macaulay \(A\)-modules plus the null module and \(\mathcal Y\) the class of complete \(A\)-modules of finite injective dimension. In the paper under review, the author shows that any complete \(A\)-module, not necessarily finitely generated, admits a surjective \(\mathcal X\)-cover with kernel in \(\mathcal Y\) and an injective \(\mathcal Y\)-envelope with cokernel in \(\mathcal X\). This result extends to complete modules the theory developed by \textit{M. Auslander} and \textit{R. O. Buchweitz} [Mém. Soc. Math. Fr. 38, 5--37 (1989; Zbl 0697.13005)] for finitely generated modules. The above result allows the author to extend to complete modules some other known facts concerning finitely generated ones. For example, she shows that every element of \({\mathcal X} \cap {\mathcal Y}\) is of the form \(W\otimes_{\widehat A}F\), where \(W\) is the dualizing module of \(\widehat A\) and \(F\) is a complete flat \(A\)-module, thus extending a result of \textit{R. Y. Sharp} [Proc. Lond. Math. Soc., III Ser. 25, 303--328 (1972; Zbl 0244.13015)]. The author also shows that if \(M\) is a complete \(A\)-module and \((0) \neq Y \in {\mathcal Y}\) then: \[ \text{dim}\, A = \text{inf}\, \{ i\, | \, \text{Ext}^i_A(k,M) \neq 0\} + \text{sup}\, \{ i\, | \, \text{Ext}^i_A(M,Y) \neq 0 \}. \] T…
reviewed by: Iustin Coandă
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 13D30
Mathematics Subject Classification ID: 13J10
Mathematics Subject Classification ID: 13C11
zbMATH DE Number: 5600720
zbMATH Keywords: Cohen-Macaulay module
zbMATH Keywords: local ring
zbMATH Keywords: complete module
zbMATH Keywords: envelope
zbMATH Keywords: cover
MaRDI profile type: Publication
full work available at URL: s10468-009-9151-8
OpenAlex ID: W2149172269
cites work: Q5563439
cites work: The homological theory of maximal Cohen-Macaulay approximations
cites work: Q3037578
cites work: Q4274969
cites work: Power series rings and projectivity
cites work: Some remarks on index and generalized Loewy length of a Gorenstein local ring
cites work: Der kanonische Modul eines Cohen-Macaulay-Rings. (The canonical moduls of a Cohen-Macaulay-ring)
cites work: Dimension projective finie et cohomologie locale. Applications à la demonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck
cites work: Finitely Generated Modules of Finite Injective Dimension Over Certain Cohen-Macaulay Rings†
cites work: Some homological properties of complete modules
cites work: adic-completion and some dual homological results
cites work: Reduced Bass numbers, Auslanders δ-invariant and certain homological conjectures
cites work: Q3995214
DOI: 10.1007/S10468-009-9151-8
Recommended article: A generalization of Auslander's last theorem
Recommended article: INJECTIVE COVERS OVER COMMUTATIVE NOETHERIAN RINGS OF GLOBAL DIMENSION AT MOST TWO II
Recommended article: The structure of preenvelopes with respect to maximal Cohen-Macaulay modules
Recommended article: Envelopes and covers by modules of finite injective and projective dimensions
Recommended article: Envelopes, covers and semidualizing modules
Recommended article: Flat covers of modules
Recommended article: Finitely generated cotorsion modules
Recommended article: Q4466594
Recommended article: Tilting preenvelopes and cotilting precovers
Recommended article: Rings characterized by (pre)envelopes and (pre)covers of their modules∗
description: scientific article; zbMATH DE number 5600720