Approximations and Endomorphism Algebras of Modules

Göbel, Rüdiger; Trlifaj, Jan

doi:10.1515/9783110218114

Cited by 132 publications

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“…As C is finitely presented, the class C ⊥ is definable by [10,Theorem 13.26]. We finish the proof by showing that D σ ′ is closed under direct sums and submodules.…”

Section: Definability and Finite Typementioning

confidence: 85%

“…The class D σ is definable by the assumption, and the definability of the class D is proved by implication (i) → (ii) of this Theorem. By [10,Lemma 6.9], it is enough to show that M ∈ D implies M ∈ D σ for M pure-injective.…”

Section: Definability and Finite Typementioning

confidence: 99%

“…By Lemma 2.1, we have that Ext 1 R (C i , M ) = 0 and M ∈ D σ ′ i , where σ ′ i is given by restricting the codomain of σ i to G i . Since M is pure-injective, we have by [10,Lemma 6.28] the following isomorphism:…”

Section: Definability and Finite Typementioning

confidence: 99%

“…(ii) → (iv): Let T ′ be a finitely presented silting module such that Gen (T ′ ) = D. Then T ′ is a finitely presented tilting R/A-module, which is projective by [10,Proposition 13.2]). Hence, D = Mod-R/A, and A is an idempotent ideal.…”

Section: Further G Is Of Finite Type If It Has a Filter Basis Consismentioning

confidence: 99%

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Silting Modules over Commutative Rings

Hügel

Hrbek

2016

Int Math Res Notices

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“…As C is finitely presented, the class C ⊥ is definable by [10,Theorem 13.26]. We finish the proof by showing that D σ ′ is closed under direct sums and submodules.…”

Section: Definability and Finite Typementioning

confidence: 85%

Section: Definability and Finite Typementioning

confidence: 99%

Section: Definability and Finite Typementioning

confidence: 99%

Section: Further G Is Of Finite Type If It Has a Filter Basis Consismentioning

confidence: 99%

See 2 more Smart Citations

Silting Modules over Commutative Rings

Hügel

Hrbek

2016

Int Math Res Notices

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“…A pair of classes of objects (A, B) is called a cotorsion pair or cotorsion theory (see [17,22]) if A ⊥ = B and ⊥ B = A where [3] Covers and envelopes 387 and…”

Section: Introductionmentioning

confidence: 99%

SOME COVERS AND ENVELOPES IN THE CHAIN COMPLEX CATEGORY OF R-MODULES

Wang¹,

Liu²

2011

J. Aust. Math. Soc.

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We study the existence of some covers and envelopes in the chain complex category of R-modules. Let (A, B) be a cotorsion pair in R-Mod and let E A stand for the class of all exact complexes with each term in A. We prove that (E A, E A ⊥ ) is a perfect cotorsion pair whenever A is closed under pure submodules, cokernels of pure monomorphisms and direct limits and so every complex has an E A-cover.As an application we show that every complex of R-modules over a right coherent ring R has an exact Gorenstein flat cover. In addition, the existence of A-covers and B-envelopes of special complexes is considered where A and B denote the classes of all complexes with each term in A and B, respectively. 2010 Mathematics subject classification: primary 18G35; secondary 55U15, 03C35.

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A model structure approach to the finitistic dimension conjectures

Cortés-Izurdiaga

Estrada

Asensio

2012

Mathematische Nachrichten

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We explore the interlacing between model category structures attained to classes of modules of finite Xdimension, for certain classes of modules X . As an application we give a model structure approach to the finitistic dimension conjectures and present a new conceptual framework in which these conjectures can be studied.Let Λ be a finite dimensional algebra over a field k (or more generally, let Λ be an Artin ring). The big finitistic dimension of Λ, Findim(Λ), is defined as the supremum of the projective dimensions of all modules having finite projective dimension. And the little finitistic dimension of Λ, findim(Λ), is defined in a similar way by restricting to the subclass of all finitely generated modules of finite projective dimension. In 1960, Bass stated the so-called Finitistic Dimension Conjectures: (I) Findim(Λ) = findim(Λ), and (II) findim(Λ) is finite. The first conjecture was proved to be false by Huisgen-Zimmermann in [19], but the second one still remains open. It has been proved to be true, for instance, for finite-dimensional monomial algebras [16], for Artin algebras with vanishing cube radical [20], or Artin algebras with representation dimension bounded by 3 [22].In [21] and [5], Huisgen-Zimmermann, Smalø, Auslander and Reiten proved that the finitistic dimension conjectures hold for Artin algebras in which the class P < ∞ of all finitely generated modules of finite projective dimension is contravariantly finite (equivalently, it is a precovering class in the sense of [9], [15]). In general, P < ∞ does not need to be contravariantly finite, even for Artin algebras satisfying the finitistic dimension conjectures. But, as Angeleri-Hügel and Trlifaj have noticed in [3], it cogenerates a cotorsion pair (F, C) in which the class F is precovering in R -Mod. By means of this idea, the authors are able to extend Auslander-Reiten's approach to arbitrary artinian rings and obtain a general criterium for an artinian ring to satisfy the finitistic dimension conjectures in terms of Tilting Theory (see [3]). This type of arguments has also been recently extended to more general homologies induced by arbitrary hereditary cotorsion pairs (see [1]).On the other hand, Hovey has recently shown in [18] that there exists a quite strong relation between the construction of hereditary cotorsion pairs in module categories and the existence of model structures in the sense of Quillen in the associated categories of unbounded chain complexes. Moreover, under very general hypotheses, the cohomology functors defined from these model structures coincide with the absolute cohomology functors defined from the injective model structure (in the sense of [18, Example 3.2])]. Recall that a model category is a category with three distinguished classes of morphisms (fibrations, cofibrations and weak equivalences) satisfying a certain number of axioms. We refer to [17] for a complete definition and main properties of model categories. One of the main advantages of these model categories is that they allow the construction of the ...

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Approximations and Endomorphism Algebras of Modules

Cited by 132 publications

References 0 publications

Silting Modules over Commutative Rings

Silting Modules over Commutative Rings

SOME COVERS AND ENVELOPES IN THE CHAIN COMPLEX CATEGORY OF R-MODULES

A model structure approach to the finitistic dimension conjectures

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