Silting Modules over Commutative Rings

Hügel, Lidia Angeleri; Hrbek, Michal

doi:10.1093/imrn/rnw147

Cited by 16 publications

(30 citation statements)

References 19 publications

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“…One of the main results of this paper is based on some recent results of Hrbek and Angeleri Hügel-Hrbek [11,1]. We show that whenever u : R −→ U is a homological ring epimorphism and U is an R-module of projective dimension 1, it follows that U is a flat R-module.…”

Section: Introductionmentioning

confidence: 81%

“…The R-module U ⊕ U/R is 1-silting by [13,Example 6.5], and a 2-term projective resolution of the complex U ⊕ K • is the related silting complex. Hence C = Hom Z (U ⊕ U/R, Q/Z) is a cosilting R-module of cofinite type [1,Corollary 3.6]. The cosilting class associated with C consists of all the u-torsion-free R-modules, and the torsion class in the cosilting torsion pair is the class of all u-torsion R-modules.…”

Section: When Is the Class Of Torsion Modules Hereditary?mentioning

confidence: 99%

“…The cosilting class associated with C consists of all the u-torsion-free R-modules, and the torsion class in the cosilting torsion pair is the class of all u-torsion R-modules. By [1,Lemma 4.2], any cosilting torsion pair of cofinite type in the category of modules over a commutative ring is hereditary. Once again, by Lemma 5.1 (4) =⇒ (6) we can conclude that U is a flat R-module.…”

Section: When Is the Class Of Torsion Modules Hereditary?mentioning

confidence: 99%

“…Proof. The argument is based on some results from the papers[11,1]. Assume first that u is injective.…”

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confidence: 99%

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Matlis category equivalences for a ring epimorphism

Bazzoni

Positselski

2020

Journal of Pure and Applied Algebra

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Under mild assumptions, we construct the two Matlis additive category equivalences for an associative ring epimorphism u : R −→ U . Assuming that the ring epimorphism is homological of flat/projective dimension 1, we discuss the abelian categories of u-comodules and u-contramodules and construct the recollement of unbounded derived categories of R-modules, U -modules, and complexes of R-modules with u-co/contramodule cohomology. Further assumptions allow to describe the third category in the recollement as the unbounded derived category of the abelian categories of u-comodules and u-contramodules. For commutative rings, we also prove that any homological epimorphism of projective dimension 1 is flat. Injectivity of the map u is not required.The following result can be also found in [5, Corollary 4.4].Corollary 7.3. Let u : R −→ U be a homological ring epimorphism. Assume that fd U R ≤ 1 and pd R U ≤ 1. Suppose further that (U/R) ⊗ R J = 0 for all injective left R-modules J and Hom R (U/R, F ) = 0 for all projective left R-modules F . Then for every conventional derived category symbol ⋆ = b, +, −, or ∅, there is a triangulated equivalence between the derived categories of the abelian categories R-mod u-co and R-mod u-ctra of left u-comodules and left u-contramodules,Proof. According to Corollary 6.2 and Theorem 7.1(a-b), we have a chain of triangulated equivalences D ⋆ (R-mod u-co ) ∼ = D ⋆ u-co (R-mod) ∼ = D ⋆ u-ctra (R-mod) ∼ = D ⋆ (R-mod u-ctra ).

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Section: Introductionmentioning

confidence: 81%

Section: When Is the Class Of Torsion Modules Hereditary?mentioning

confidence: 99%

Section: When Is the Class Of Torsion Modules Hereditary?mentioning

confidence: 99%

“…Proof. The argument is based on some results from the papers[11,1]. Assume first that u is injective.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Matlis category equivalences for a ring epimorphism

Bazzoni

Positselski

2020

Journal of Pure and Applied Algebra

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show abstract

“…In [15], it is shown that, for finitedimensional algebras of finite representation type, silting modules are in bijection F. Pop with universal localizations. Recently, in [4], the authors give a classification of silting modules over commutative rings, establishing a bijective correspondence of them with Gabriel filters of finite type. The dual notion, of cosilting module, was independently introduced in [7,17], as a generalization of the concept of cotilting module.…”

Section: Introductionmentioning

confidence: 99%