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.

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.

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 On the derived category of a graded commutative noetherian ring

2013 ◽  
Vol 373 ◽  
pp. 356-376 ◽  
Author(s):  
Ivo DellʼAmbrogio ◽  
Greg Stevenson
Keyword(s):  
Noetherian Ring ◽  
Derived Category ◽  
Commutative Noetherian Ring

On Grothendieck's local duality theorem

1979 ◽  
Vol 85 (3) ◽  
pp. 431-437 ◽  
Author(s):  
M. H. Bijan-Zadeh ◽  
R. Y. Sharp
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Noetherian Ring ◽  
Duality Theorem ◽  
Great Emphasis ◽  
Triangulated Category ◽  
Elementary Approach ◽  
Commutative Noetherian Ring ◽  
Local Duality ◽  
Dualizing Complexes

In (11) and (12), a comparatively elementary approach to the use of dualizing complexes in commutative algebra has been developed. Dualizing complexes were introduced by Grothendieck and Hartshorne in (2) for use in algebraic geometry; the approach to dualizing complexes in (11) and (12) differs from that of Grothendieck and Hartshorne in that it avoids use of the concepts of triangulated category, derived category, and localization of categories, and instead places great emphasis on the concept of quasi-isomorphism of complexes of modules over a commutative Noetherian ring.

Primary decomposition of homogeneous ideal in idealization of a module

2018 ◽  
Vol 55 (3) ◽  
pp. 345-352
Author(s):  
Tran Nguyen An
Keyword(s):  
Noetherian Ring ◽  
Primary Decomposition ◽  
Associated Primes ◽  
Homogeneous Ideal ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Irreducible Decomposition

Let R be a commutative Noetherian ring, M a finitely generated R-module, I an ideal of R and N a submodule of M such that IM ⫅ N. In this paper, the primary decomposition and irreducible decomposition of ideal I × N in the idealization of module R ⋉ M are given. From theses we get the formula for associated primes of R ⋉ M and the index of irreducibility of 0R ⋉ M.

scholarly journals On a class of dual Rickart modules

2020 ◽  
Vol 72 (7) ◽  
pp. 960-970
Author(s):  
R. Tribak
Keyword(s):  
Direct Summand ◽  
Noetherian Ring ◽  
Commutative Noetherian Ring ◽  
Direct Sum ◽  
Finite Set ◽  
Rickart Modules

UDC 512.5 Let R be a ring and let Ω R be the set of maximal right ideals of R . An R -module M is called an sd-Rickart module if for every nonzero endomorphism f of M , ℑ f is a fully invariant direct summand of M . We obtain a characterization for an arbitrary direct sum of sd-Rickart modules to be sd-Rickart. We also obtain a decomposition of an sd-Rickart R -module M , provided R is a commutative noetherian ring and A s s ( M ) ∩ Ω R is a finite set. In addition, we introduce and study ageneralization of sd-Rickart modules.

Model structures, recollements and duality pairs

2021 ◽  
Author(s):  
Wenjing Chen ◽  
Zhongkui Liu
Keyword(s):  
Projective Modules ◽  
Cotorsion Pair ◽  
Gorenstein Rings ◽  
Injective Modules ◽  
Flat Modules ◽  
Duality Pairs ◽  
Cotorsion Pairs ◽  
Gorenstein Flat ◽  
Model Structures

In this paper, we construct some model structures corresponding Gorenstein [Formula: see text]-modules and relative Gorenstein flat modules associated to duality pairs, Frobenius pairs and cotorsion pairs. By investigating homological properties of Gorenstein [Formula: see text]-modules and some known complete hereditary cotorsion pairs, we describe several types of complexes and obtain some characterizations of Iwanaga–Gorenstein rings. Based on some facts given in this paper, we find new duality pairs and show that [Formula: see text] is covering as well as enveloping and [Formula: see text] is preenveloping under certain conditions, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-injective modules and [Formula: see text] denotes the class of Gorenstein [Formula: see text]-flat modules. We give some recollements via projective cotorsion pair [Formula: see text] cogenerated by a set, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-projective modules. Also, many recollements are immediately displayed through setting specific complete duality pairs.

Some Results on the Domination Number of a Zero-divisor Graph

2014 ◽  
Vol 57 (3) ◽  
pp. 573-578 ◽  
Author(s):  
Sima Kiani ◽  
Hamid Reza Maimani ◽  
Reza Nikandish
Keyword(s):  
Noetherian Ring ◽  
Zero Divisor ◽  
Domination Number ◽  
Total Domination ◽  
Ore Extension ◽  
Commutative Noetherian Ring ◽  
Zero Divisor Graph ◽  
Domination Numbers

AbstractIn this paper, we investigate the domination, total domination, and semi-total domination numbers of a zero-divisor graph of a commutative Noetherian ring. Also, some relations between the domination numbers of Γ(R/I) and Γ1(R), and the domination numbers of Γ(R) and Γ(R[x, α, δ]), where R[x, α, δ] is the Ore extension of R, are studied.

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