scholarly journals LOCAL HOMOLOGY AND GORENSTEIN FLAT MODULES

2012 ◽  
Vol 11 (02) ◽  
pp. 1250022
Author(s):  
FATEMEH MOHAMMADI AGHJEH MASHHAD ◽  
KAMRAN DIVAANI-AAZAR
Keyword(s):  
Noetherian Ring ◽  
Derived Category ◽  
Natural Isomorphism ◽  
Local Rings ◽  
Commutative Noetherian Ring ◽  
Dualizing Complex ◽  
Local Homology ◽  
Flat Modules ◽  
The Right ◽  
Gorenstein Flat

Let R be a commutative Noetherian ring, 𝔞 be an ideal of R and [Formula: see text] denote the derived category of R-modules. We investigate the theory of local homology in conjunction with Gorenstein flat modules. Let X be a homologically bounded to the right complex and Q be a bounded to the right complex of Gorenstein flat R-modules such that Q and X are isomorphic in [Formula: see text]. We establish a natural isomorphism LΛ𝔞(X) ≃ Λ𝔞(Q) in [Formula: see text] which immediately asserts that sup LΛ𝔞(X) ≤ Gfd RX. This isomorphism yields several conseQuences. For instance, in the case R possesses a dualizing complex, we show that Gfd RLΛ𝔞(X) ≤ Gfd RX. Also, we establish a criterion for regularity of Gorenstein local rings.

Gorenstein Flat Complexes with Respect to a Semidualizing Module

2015 ◽  
Vol 22 (02) ◽  
pp. 259-270
Author(s):  
Li Liang ◽  
Chunhua Yang
Keyword(s):  
Noetherian Ring ◽  
Commutative Noetherian Ring ◽  
Semidualizing Module ◽  
Flat Modules ◽  
Gorenstein Flat

In this paper, we introduce and study G C-flat complexes over a commutative Noetherian ring, where C is a semidualizing module. We prove that G C-flat complexes are actually the complexes of G C-flat modules. This complements a result of Yang and Liang. As an application, we get that every complex has a [Formula: see text]-cover, where [Formula: see text] is the class of G C-flat complexes. We also give a characterization of complexes of modules in [Formula: see text] that are defined by Sather-Wagstaff, Sharif and White.

scholarly journals Minimal complexes of cotorsion flat modules

2019 ◽  
Vol 124 (1) ◽  
pp. 15-33 ◽  
Author(s):  
Peder Thompson
Keyword(s):  
Noetherian Ring ◽  
Derived Category ◽  
Homotopy Equivalence ◽  
Commutative Noetherian Ring ◽  
Partial Converse ◽  
Flat Modules ◽  
Cotorsion Pairs

Let $R$ be a commutative noetherian ring. We give criteria for a complex of cotorsion flat $R$-modules to be minimal, in the sense that every self homotopy equivalence is an isomorphism. To do this, we exploit Enochs' description of the structure of cotorsion flat $R$-modules. More generally, we show that any complex built from covers in every degree (or envelopes in every degree) is minimal, as well as give a partial converse to this in the context of cotorsion pairs. As an application, we show that every $R$-module is isomorphic in the derived category over $R$ to a minimal semi-flat complex of cotorsion flat $R$-modules.

Buchweitz’s equivalences for Gorenstein flat modules with respect to semidualizing modules

2019 ◽  
pp. 2150006
Author(s):  
Jiangsheng Hu ◽  
Yuxian Geng ◽  
Jinyong Wu ◽  
Huanhuan Li
Keyword(s):  
Noetherian Ring ◽  
Full Subcategory ◽  
Stable Category ◽  
Commutative Noetherian Ring ◽  
Singularity Category ◽  
Flat Modules ◽  
Frobenius Category ◽  
Semidualizing Modules ◽  
Structure Formula ◽  
Gorenstein Flat

Let [Formula: see text] be a commutative Noetherian ring and [Formula: see text] a semidualizing [Formula: see text]-module. We obtain an exact structure [Formula: see text] and prove that the full subcategory [Formula: see text] of [Formula: see text] is a Frobenius category with [Formula: see text] the subcategory of projective and injective objects, where [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]) is the subcategory of [Formula: see text]-Gorenstein flat (respectively, [Formula: see text]-flat [Formula: see text]-cotorsion) [Formula: see text]-modules. Then the stable category [Formula: see text] of [Formula: see text] and the singularity category [Formula: see text] of [Formula: see text] are also considered. As a consequence, we get that there is a Buchweitz’s equivalence [Formula: see text] if and only if [Formula: see text] is a part of some AB-context.

scholarly journals Building Modules From the Singular Locus

2015 ◽  
Vol 116 (1) ◽  
pp. 23
Author(s):  
Jesse Burke ◽  
Lars Winther Christensen ◽  
Ryo Takahashi
Keyword(s):  
Noetherian Ring ◽  
Singular Locus ◽  
Prime Ideals ◽  
Local Rings ◽  
Isolated Singularity ◽  
Finitely Generated Module ◽  
Commutative Noetherian Ring ◽  
Exact Number ◽  
Building Process ◽  
Number Of Iterations

A finitely generated module over a commutative noetherian ring of finite Krull dimension can be built from the prime ideals in the singular locus by iteration of three procedures: taking extensions, direct summands, and cosyzygies. In 2003 Schoutens gave a bound on the number of iterations required to build any module, and in this note we determine the exact number. This building process yields a stratification of the module category, which we study in detail for local rings that have an isolated singularity.

scholarly journals Totally acyclic complexes and locally Gorenstein rings

2018 ◽  
Vol 17 (03) ◽  
pp. 1850039
Author(s):  
Lars Winther Christensen ◽  
Kiriko Kato
Keyword(s):  
Noetherian Ring ◽  
Noetherian Rings ◽  
Commutative Noetherian Ring ◽  
Gorenstein Rings ◽  
Dualizing Complex ◽  
Injective Modules ◽  
Acyclic Complex ◽  
Acyclic Complexes

A commutative noetherian ring with a dualizing complex is Gorenstein if and only if every acyclic complex of injective modules is totally acyclic. We extend this characterization, which is due to Iyengar and Krause, to arbitrary commutative noetherian rings, i.e. we remove the assumption about a dualizing complex. In this context Gorenstein, of course, means locally Gorenstein at every prime.

A dualizing complex for Stanley–Reisner rings

1984 ◽  
Vol 96 (2) ◽  
pp. 203-212 ◽  
Author(s):  
Hans-Gert Gräbe
Keyword(s):  
Noetherian Ring ◽  
Commutative Noetherian Ring ◽  
Dualizing Complex

AbstractIn this paper we construct a (multihomogeneous) dualizing complex for A [Δ], the Stanley–Reisner ring of Δ over an arbitrary commutative noetherian ring A with identity, admitting a dualizing complex. In addition, some consequences are discussed.

Triangulated Equivalences Involving Gorenstein Projective Modules

2017 ◽  
Vol 60 (4) ◽  
pp. 879-890 ◽  
Author(s):  
Yuefei Zheng ◽  
Zhaoyong Huang
Keyword(s):  
Noetherian Ring ◽  
Derived Category ◽  
Projective Modules ◽  
Injective Dimension ◽  
Stable Category ◽  
Dualizing Complex ◽  
Injective Modules ◽  
Gorenstein Projective Modules ◽  
Left And Right

AbstractFor any ring R, we show that, in the bounded derived category Db(Mod R) of left R-modules, the subcategory of complexes with finite Gorenstein projective (resp. injective) dimension modulo the subcategory of complexes with finite projective (resp. injective) dimension is equivalent to the stable category (resp. ) of Gorenstein projective (resp. injective) modules. As a consequence, we get that if R is a left and right noetherian ring admitting a dualizing complex, then and are equivalent.

scholarly journals On the vanishing and the positivity of intersection multiplicities over local rings with small non complete intersection loci

1994 ◽  
Vol 136 ◽  
pp. 133-155 ◽  
Author(s):  
Kazuhiko Kurano
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
Noetherian Ring ◽  
Minimal Prime Ideal ◽  
Local Rings ◽  
Regular Local Ring ◽  
Commutative Noetherian Ring ◽  
Intersection Multiplicities ◽  
Minimal Prime

Throughout this paperAis a commutative Noetherian ring of dimensiondwith the maximal ideal m and we assume that there exists a regular local ringSsuch thatAis a homomorphic image ofS, i.e.,A=S/Ifor some idealIofS. Furthermore we assume thatAis equi-dimensional, i.e., dimA= dimA/for any minimal prime idealofA. We put.

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