Derived decompositions of abelian categories I

Chen, Haishan; Xi, Changchang

doi:10.48550/arxiv.1804.10759

Cited by 4 publications

(9 citation statements)

References 16 publications

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“…The condition (3) (for uncountable direct limits) is equivalent to (7) The conditions ( 12) and ( 13) are equivalent by Lemma 13.4(a). The equivalence of all the eleven conditions (1-6) and (8)(9)(10)(11)(12) is provable in the same way as the equivalence of the eleven conditions in Theorem 13.3 (2)(3)(4)(6)(7)(8)(9)(10)(11)(12)(13).…”

Section: If the Topological Ring R Satisfies One Of The Conditions (A...mentioning

confidence: 71%

“…(5) the class B proj is covering in B; (6) any countable direct limit of copies of the projective generator R ∈ B has a projective cover in B; By the Artin-Wedderburn classification of simple Artinian rings, ( 11) implies (10). In fact, the only difference between (11) and (ii) is that the sets Υ γ can be infinite in (ii); the class of topological rings S in (11) is obtained by such class in (ii) by imposing the condition that all the sets Υ γ are finite.…”

Section: Perfect Decompositionsmentioning

confidence: 99%

“…The argument is largely based in the following result, which is a particular case of [10,Corollary 4.4] or [8,Corollary 7.3].…”

Section: When Is the Left Tilting Class Covering?mentioning

confidence: 99%

“…Proof. The additional assumptions of [10,Corollary 4.4] or [8,Corollary 7.3] hold for all injective ring epimorphisms by [8,Example 7.4].…”

Section: When Is the Left Tilting Class Covering?mentioning

confidence: 99%

See 3 more Smart Citations

Covers and direct limits: a contramodule-based approach

Bazzoni,

Positselski

2019

Preprint

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We present applications of contramodule techniques to the Enochs conjecture about covers and direct limits, both in the categorical tilting context and beyond. In the n-tilting-cotilting correspondence situation, if A is a Grothendieck abelian category and the related abelian category B is equivalent to the category of contramodules over a topological ring R belonging to one of certain four classes of topological rings (e. g., R is commutative), then the left tilting class is covering in A if and only if it is closed under direct limits in A, and if and only if the topological ring R is pro-perfect. More generally, if M is a module satisfying a certain telescope Hom exactness condition (e. g., M is Σ-pure-Ext 1 -self-orthogonal or self-pure-projective) and the topological ring R of endomorphisms of M belongs to one of the four classes, then the class Add(M ) is closed under direct limits if and only if its is covering, and if and only if M has perfect decomposition. The 1-tilting modules and objects arising from injective ring epimorphisms of projective dimension 1 form a class of examples which we discuss.

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Section: If the Topological Ring R Satisfies One Of The Conditions (A...mentioning

confidence: 71%

Section: Perfect Decompositionsmentioning

confidence: 99%

“…The argument is largely based in the following result, which is a particular case of [10,Corollary 4.4] or [8,Corollary 7.3].…”

Section: When Is the Left Tilting Class Covering?mentioning

confidence: 99%

“…Proof. The additional assumptions of [10,Corollary 4.4] or [8,Corollary 7.3] hold for all injective ring epimorphisms by [8,Example 7.4].…”

Section: When Is the Left Tilting Class Covering?mentioning

confidence: 99%

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Covers and direct limits: a contramodule-based approach

Bazzoni,

Positselski

2019

Preprint

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“…Notably, the method in this article sheds some new light on when the kernel of A T ⊗ L B − is triangle equivalent to the derived module category of a ring, that is, when a good titling module T is homological (see [11]). Also, the method establishes a connection of homological tilting modules with derived decompositions of module categories (see [13] or Section 4.3 for Definition). We may state these observations as a corollary.…”

Section: Introductionmentioning

confidence: 99%

Symmetric subcategories, tilting modules and derived recollements

Chen¹,

Xi²

2021

Preprint

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For any good tilting module T over a ring A, there exists an n-symmetric subcategory E of a module category such that the derived category of the endomorphism ring of T is a recollement of the derived categories of E and A in the sense of Beilinson-Bernstein-Deligne. Thus the kernel of the total left-derived tensor functor induced by the tilting module is triangle equivalent to the derived category of E .

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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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Derived decompositions of abelian categories I

Cited by 4 publications

References 16 publications

Covers and direct limits: a contramodule-based approach

Covers and direct limits: a contramodule-based approach

Symmetric subcategories, tilting modules and derived recollements

Matlis category equivalences for a ring epimorphism

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