Lifting and Restricting Recollement Data

Nicolás, Pedro Azara; Saorı́n, Manuel

doi:10.1007/s10485-009-9198-z

Cited by 18 publications

(14 citation statements)

References 29 publications

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“…The following result was first proved by the second named author for bounded derived categories [23], and it was then further developed by several authors [19,30] (note that in [23] a condition has been misstated, see [32] for a discussion). The versions of this result in [23] and in [19] are assuming that all triangulated categories are derived categories of (differential graded) rings; therefore, the exceptional objects that appear there are images of two of the rings.…”

Section: Recollements Induced By Single Objectsmentioning

confidence: 95%

Recollements and tilting objects

Hügel

Koenig

Liu

2011

Journal of Pure and Applied Algebra

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a b s t r a c tWe study connections between recollements of the derived category D(Mod R) of a ring R and tilting theory. We first provide constructions of tilting objects from given recollements, recovering several different results from the literature. Secondly, we show how to construct a recollement from a tilting module of projective dimension one. By Nicolás and Saorín (2009) [31], every recollement of D(Mod R) is associated to a differential graded homological epimorphism λ : R → S. We will focus on the case where λ is a homological ring epimorphism or even a universal localization. Our results will be employed in a forthcoming paper in order to investigate stratifications of D(Mod R).

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Section: Recollements Induced By Single Objectsmentioning

confidence: 95%

Recollements and tilting objects

Hügel

Koenig

Liu

2011

Journal of Pure and Applied Algebra

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“…In [41] we prove that all of them restrict to give the recollements of right bounded derived categories of [30].…”

Section: Homological Epimorphisms Of Dg Categoriesmentioning

confidence: 97%

“…This parametrization yields, together with B. Keller's Morita theory for derived categories [23], a generalization of some results of [11,21] and offers an unbounded and abstract version of S. König's theorem [30] on recollements of right bounded derived categories of algebras. The paper [41] could be viewed as a continuation of Section 3.4. There we study the problem of descending the parametrization from unbounded to right bounded derived categories, and the corresponding lifting.…”

mentioning

confidence: 99%

Parametrizing recollement data for triangulated categories

Nicolás

Saorı́n

2009

Journal of Algebra

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We give a general parametrization of all the recollement data for a triangulated category with a set of generators. From this we deduce a characterization of when a ℵ 0 -perfectly generated (or aisled) triangulated category is a recollement of triangulated categories generated by a single compact object. Also, we use homological epimorphisms to give a complete and explicit description of all the recollement data for (or smashing subcategories of) the derived category of a k-flat dg category. In the final part we give a bijection between smashing subcategories of compactly generated triangulated categories and certain ideals of the subcategory of compact objects, in the spirit of H. Krause's work [Henning Krause, Cohomological quotients and smashing localizations, Amer. J. Math. 127 (2005Math. 127 ( ) 1191Math. 127 ( -1246. This bijection implies the following weak version of the generalized smashing conjecture: in a compactly generated triangulated category every smashing subcategory is generated by a set of Milnor colimits of compact objects.

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“…The existence of a recollement at

D^{b}

‐level occurs in the following special case (see ): Let

R

be a ring and

J = R e R

be an ideal generated by an idempotent element

e

R

such that

_{R} J

is projective and finitely generated and that

J_{R}

has finite projective dimension. Then there exists a recollement among

D^{b} false(e R e false), D^{b} false(R false)

and

D^{b} false(R / J false)

.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

Recollements of derived categories III: finitistic dimensions

Chen

2017

J. London Math. Soc.

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In this paper, we study homological dimensions of algebras linked by recollements of derived module categories, and establish a series of new upper bounds and relationships among their finitistic or global dimensions. This is closely related to a longstanding conjecture, the finitistic dimension conjecture, in representation theory and homological algebra. Further, we apply our results to a series of situations of particular interest: exact contexts, ring extensions, trivial extensions, pullbacks of rings, and algebras induced from Auslander-Reiten sequences. In particular, we not only extend and amplify Happel's reduction techniques for finitistic dimenson conjecture to more general contexts, but also generalise some recent results in the literature.

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Lifting and Restricting Recollement Data

Cited by 18 publications

References 29 publications

Recollements and tilting objects

Recollements and tilting objects

Parametrizing recollement data for triangulated categories

Recollements of derived categories III: finitistic dimensions

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