Smashing localizations of rings of weak global dimension at most one

Bazzoni, Silvana; Šťovíček, Jan

doi:10.1016/j.aim.2016.09.028

Cited by 18 publications

(29 citation statements)

References 56 publications

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“…The t-structures among these which are stable (that is, the t-structures which consist of a pair of triangulated subcategories) are precisely the ones associated to a smashing localization of the derived category. In this way, our present results generalize those of [8] to the non-stable case. As in the stable case [8], we confine for the most part to the commutative setting, and give a full classification of definable coaisles in the local case, that is, over valuation domains.…”

supporting

confidence: 85%

“…Although the failure of the Telescope Conjecture is usually viewed as a pathological behavior, there are rings for which the Telescope Conjecture does not hold in general, but still a full classification of smashing localizations is possible, and a simple ring theoretic criterion is available characterizing when the Telescope Conjecture is true. This is a result due to the first author andŠťovíček [8]:…”

Section: Introductionmentioning

confidence: 87%

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Definable coaisles over rings of weak global dimension at most one

Bazzoni¹,

Hrbek²

2021

Publicacions Matemàtiques

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In the setting of the unbounded derived category D(R) of a ring R of weak global dimension at most one we consider t-structures with a definable coaisle. The t-structures among these which are stable (that is, the t-structures which consist of a pair of triangulated subcategories) are precisely the ones associated to a smashing localization of the derived category. In this way, our present results generalize those of [8] to the non-stable case. As in the stable case [8], we confine for the most part to the commutative setting, and give a full classification of definable coaisles in the local case, that is, over valuation domains. It turns out that, unlike in the stable case of smashing subcategories, the definable coaisles do not always arise from homological ring epimorphisms. We also consider a non-stable version of the Telescope Conjecture for t-structures and give a ring-theoretic characterization of the commutative rings of weak global dimension at most one for which it is satisfied.

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supporting

confidence: 85%

Section: Introductionmentioning

confidence: 87%

Definable coaisles over rings of weak global dimension at most one

Bazzoni¹,

Hrbek²

2021

Publicacions Matemàtiques

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“…Note that the fact that A is noetherian is needed only in the very last step of the proof to infer that A ⊗ A − reflects isomorphisms. This may fail for non-noetherian commutative rings as may the conclusion of Proposition 4.5 (see [2,Theorem 7.2]). Proof.…”

Section: Universal Localisationsmentioning

confidence: 99%

Flat ring epimorphisms and universal localizations of commutative rings

Hügel

Marks

Šťovíček

et al. 2020

The Quarterly Journal of Mathematics

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We study different types of localizations of a commutative noetherian ring. More precisely, we provide criteria to decide: (a) if a given flat ring epimorphism is a universal localization in the sense of Cohn and Schofield; and (b) when such universal localizations are classical rings of fractions. In order to find such criteria, we use the theory of support and we analyse the specialization closed subset associated to a flat ring epimorphism. In case the underlying ring is locally factorial or of Krull dimension one, we show that all flat ring epimorphisms are universal localizations. Moreover, it turns out that an answer to the question of when universal localizations are classical depends on the structure of the Picard group. We furthermore discuss the case of normal rings, for which the divisor class group plays an essential role to decide if a given flat ring epimorphism is a universal localization. Finally, we explore several (counter)examples which highlight the necessity of our assumptions.

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“…We should mention that, in the same assumptions as ours, the equivalence of derived categories (2) was obtained in [5,Corollary 4.4] as a particular case of a general result about derived decomposition of abelian categories. The general approach in [5] is based on the technique of complete Ext-orthogonal pair in abelian categories, which was introduced by Krause andŠt'ovíček in [12] (see also [4]). The same argument as in the present paper, going back to [18] and [20], is used in [5] in order to prove that the triangulated functors induced by the embeddings of abelian subcategories are fully faithful.…”

Section: Introductionmentioning

confidence: 99%

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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Smashing localizations of rings of weak global dimension at most one

Cited by 18 publications

References 56 publications

Definable coaisles over rings of weak global dimension at most one

Definable coaisles over rings of weak global dimension at most one

Flat ring epimorphisms and universal localizations of commutative rings

Matlis category equivalences for a ring epimorphism

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