scholarly journals ON A DUALITY THEOREM OF WAKAMATSU

2008 ◽  
Vol 78 (2) ◽  
pp. 343-350
Author(s):  
ZHAOYONG HUANG
Keyword(s):  
Nonnegative Integer ◽  
Duality Theorem ◽  
Tilting Module ◽  
Coherent Ring ◽  
Finitely Presented ◽  
Generalized Tilting Module

AbstractLet R be a left coherent ring, S a right coherent ring and RU a generalized tilting module, with S=End(RU) satisfying the condition that each finitely presented left R-module X with ExtRi(X,U)=0 for any i≥1 is U-torsionless. If M is a finitely presented left R-module such that ExtRi(M,U)=0 for any i≥0 with $i \neq n$ (where n is a nonnegative integer), then $\mathrm {Ext}_S^n( \mathrm {Ext}_R^n(M, U), U)\cong M$ and ExtSi(ExtRn(M,U),U)=0 for any i≥0 with $i\neq n$. A duality is thus induced between the category of finitely presented holonomic left R-modules and the category of finitely presented holonomic right S-modules.

scholarly journals PROPERLY STRATIFIED ALGEBRAS AND TILTING

2005 ◽  
Vol 92 (1) ◽  
pp. 29-61 ◽  
Author(s):  
ANDERS FRISK ◽  
VOLODYMYR MAZORCHUK
Keyword(s):  
Projective Dimension ◽  
Finitistic Dimension ◽  
Tilting Module ◽  
Tilting Modules ◽  
Category Of Modules ◽  
Cotilting Modules ◽  
Contravariantly Finite ◽  
Properly Stratified Algebras ◽  
Stratified Algebra ◽  
Generalized Tilting Module

We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question of when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of properly stratified algebras for which the characteristic tilting and cotilting modules coincide is closed under taking the Ringel dual. Studying stratified algebras whose Ringel dual is properly stratified, we discover a new Ringel-type duality for such algebras, which we call the two-step duality. This duality arises from the existence of a new (generalized) tilting module for stratified algebras with properly stratified Ringel dual. We show that this new tilting module has a lot of interesting properties; for instance, its projective dimension equals the projectively defined finitistic dimension of the original algebra, it guarantees that the category of modules of finite projective dimension is contravariantly finite, and, finally, it allows one to compute the finitistic dimension of the original algebra in terms of the projective dimension of the characteristic tilting module.

Commutative rings and modules that are Nil*-coherent or special Nil*-coherent

2017 ◽  
Vol 16 (10) ◽  
pp. 1750187 ◽  
Author(s):  
Karima Alaoui Ismaili ◽  
David E. Dobbs ◽  
Najib Mahdou
Keyword(s):  
Commutative Rings ◽  
Finitely Generated ◽  
Basic Properties ◽  
Unital Ring ◽  
Reduced Ring ◽  
Coherent Ring ◽  
Ring Extensions ◽  
Coherent Rings ◽  
Finitely Presented

Recently, Xiang and Ouyang defined a (commutative unital) ring [Formula: see text] to be Nil[Formula: see text]-coherent if each finitely generated ideal of [Formula: see text] that is contained in Nil[Formula: see text] is a finitely presented [Formula: see text]-module. We define and study Nil[Formula: see text]-coherent modules and special Nil[Formula: see text]-coherent modules over any ring. These properties are characterized and their basic properties are established. Any coherent ring is a special Nil[Formula: see text]-coherent ring and any special Nil[Formula: see text]-coherent ring is a Nil[Formula: see text]-coherent ring, but neither of these statements has a valid converse. Any reduced ring is a special Nil[Formula: see text]-coherent ring (regardless of whether it is coherent). Several examples of Nil[Formula: see text]-coherent rings that are not special Nil[Formula: see text]-coherent rings are obtained as byproducts of our study of the transfer of the Nil[Formula: see text]-coherent and the special Nil[Formula: see text]-coherent properties to trivial ring extensions and amalgamated algebras.

scholarly journals Completing perfect complexes

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1387-1427 ◽  
Author(s):  
Henning Krause
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Direct Construction ◽  
Coherent Sheaves ◽  
Perfect Complexes ◽  
Coherent Ring ◽  
Abelian Categories ◽  
Finitely Presented

Abstract This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra. The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. The final appendix by Bernhard Keller introduces the concept of a morphic enhancement for triangulated categories and provides a foundation for completing a triangulated category.

(m,n)-Injective and (m,n)-coherent endomorphism rings of modules

2019 ◽  
Vol 19 (03) ◽  
pp. 2050048
Author(s):  
Lixin Mao
Keyword(s):  
Endomorphism Ring ◽  
Endomorphism Rings ◽  
Text Coherence ◽  
Coherent Ring ◽  
The Right ◽  
Positive Integers ◽  
Finitely Presented

Let [Formula: see text] and [Formula: see text] be fixed positive integers. [Formula: see text] is called a right [Formula: see text]-injective ring if every right [Formula: see text]-homomorphism from an [Formula: see text]-generated submodule of the right [Formula: see text]-module [Formula: see text] to [Formula: see text] extends to one from [Formula: see text] to [Formula: see text]; [Formula: see text] is called a right [Formula: see text]-coherent ring if each [Formula: see text]-generated submodule of the right [Formula: see text]-module [Formula: see text] is a finitely presented right [Formula: see text]-module. Let [Formula: see text] be a right [Formula: see text]-module. We study the [Formula: see text]-injectivity and [Formula: see text]-coherence of the endomorphism ring [Formula: see text] of [Formula: see text]. Some applications are also given.

scholarly journals Some results on coherent rings II

1967 ◽  
Vol 8 (2) ◽  
pp. 123-126 ◽  
Author(s):  
Morton E. Harris
Keyword(s):  
Finitely Generated ◽  
Basic Theorem ◽  
Coherent Ring ◽  
Coherent Rings ◽  
Finitely Presented

According to Bourbaki [1, pp. 62–63, Exercise 11], a left (resp. right) A-module M is said to be pseudo-coherent if every finitely generated submodule of M is finitely presented, and is said to be coherent if it is both pseudo-coherent and finitely generated. This Bourbaki reference contains various results on pseudo-coherent and coherent modules. Then, in [1, p. 63, Exercise 12], a ring which as a left (resp. right) module over itself is coherent is said to be a left (resp. right) coherent ring, and various results on and examples of coherent rings are presented. The result stated in [1, p. 63, Exercise 12a] is a basic theorem of [2] and first appeared there. A variety of results on and examples of coherent rings and modules are presented in [3].

scholarly journals A note on existence of envelopes and covers

1996 ◽  
Vol 54 (3) ◽  
pp. 383-390 ◽  
Author(s):  
Jianlong Chen ◽  
Nanqing Ding
Keyword(s):  
Projective Cover ◽  
Direct Products ◽  
Coherent Ring ◽  
Finitely Presented

We prove the following results for a ring R. (a) If C is a class of right R-modules closed under direct summands and isomorphisms, then every right R-module has an epic C-envelope if and only if C is closed under direct products and submodules. (b) If R is left T-coherent and pure injective as a right R-module, then every T-finitely presented right R-module has a T-flat envelope, (c) Let R be a left T-coherent ring and injective right R-modules be T-flat. If every finitely presented left R-module has a flat envelope, then every T-finitely presented right R-module has a projective cover.

scholarly journals Closing the category of finitely presented functors under images made constructive

2020 ◽  
Vol 2 ◽  
pp. 4
Author(s):  
Sebastian Posur
Keyword(s):  
Data Structure ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Basic Problem ◽  
Explicit Construction ◽  
Graded Modules ◽  
Additive Category ◽  
Abelian Case ◽  
Coherent Ring ◽  
Finitely Presented

For an additive category P we provide an explicit construction of a category Q(P) whose objects can be thought of as formally representing im(γ)im(ρ)∩im(γ) for given morphisms γ:A→B and ρ:C→B in P, even though P does not need to admit quotients or images. We show how it is possible to calculate effectively within Q(P), provided that a basic problem related to syzygies can be handled algorithmically. We prove an equivalence of Q(P) with the smallest subcategory of the category of contravariant functors from P to the category of abelian groups Ab which contains all finitely presented functors and is closed under the operation of taking images. Moreover, we characterize the abelian case: Q(P) is abelian if and only if it is equivalent to fp(Pop,Ab), the category of all finitely presented functors, which in turn, by a theorem of Freyd, is abelian if and only if P has weak kernels.The category Q(P) is a categorical abstraction of the data structure for finitely presented R-modules employed by the computer algebra system Macaulay2, where R is a ring. By our generalization to arbitrary additive categories, we show how this data structure can also be used for modeling finitely presented graded modules, finitely presented functors, and some not necessarily finitely presented modules over a non-coherent ring.

J-COHERENT RINGS

2009 ◽  
Vol 08 (02) ◽  
pp. 139-155 ◽  
Author(s):  
NANQING DING ◽  
YUANLIN LI ◽  
LIXIN MAO
Keyword(s):  
Jacobson Radical ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Flat Modules ◽  
Coherent Ring ◽  
Necessary And Sufficient ◽  
Coherent Rings ◽  
Finitely Presented

Let R be a ring. Recall that a left R-module M is coherent if every finitely generated submodule of M is finitely presented. R is a left coherent ring if the left R-module RR is coherent. In this paper, we say that R is left J-coherent if its Jacobson radical J(R) is a coherent left R-module. J-injective and J-flat modules are introduced to investigate J-coherent rings. Necessary and sufficient conditions for R to be left J-coherent are given. It is shown that there are many similarities between coherent and J-coherent rings. J-injective and J-flat dimensions are also studied.

RELATIVE COHERENCE OF RINGS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250047
Author(s):  
LIXIN MAO ◽  
NANQING DING
Keyword(s):  
Left Ideal ◽  
Torsion Theory ◽  
Hereditary Torsion Theory ◽  
Injective Modules ◽  
Coherent Ring ◽  
Coherent Rings ◽  
Finitely Presented

Let R be a ring and τ a hereditary torsion theory for the category of all left R-modules. A right R-module M is called τ-flat if Tor 1(M, R/I) = 0 for any τ-finitely presented left ideal I. A left R-module N is said to be τ-f-injective in case Ext 1(R/I, N) = 0 for any τ-finitely presented left ideal I. R is called a left τ-coherent ring in case every τ-finitely presented left ideal is finitely presented. τ-coherent rings are characterized in terms of, among others, τ-flat and τ-f-injective modules. Some known results are extended.

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