Parametrizing recollement data for triangulated categories

Abstract. Recollements of derived module categories are investigated, using a new technique, ladders of recollements, which are maximal mutation sequences. The position in the ladder is shown to control whether a recollement restricts from unbounded to another level of derived category. Ladders also turn out to control derived simplicity on all levels. An algebra is derived simple if its derived category cannot be deconstructed, that is, if it is not the middle term of a non-trivial recollement whose outer terms are again derived categories of algebras. Derived simplicity on each level is characterised in terms of heights of ladders.These results are complemented by providing new classes of examples of derived simple rings, in particular indecomposable commutative rings, as well as by a finite dimensionsal counterexample to the Jordan-Hölder problem for derived module categories. Moreover, recollements are used to compute homological and K-theoretic invariants.

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“…By [44, 5.2.9], [45], or [5, Theorem 2.2], the recollement is generated by the compact exceptional object T = j ! (C) ∈ D(ModA).…”

Section: Recollements Of Derived Categories Suppose There Is a Recolmentioning

confidence: 99%

Ladders and simplicity of derived module categories

et al. 2017

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“…Firstly, it turns out that it actually does not matter for Definition 2.1 which of these functors we derive. Secondly, we need that C ⊗ L B C → C is a quasi-isomorphism when we view − ⊗ B − as a functor C(B) × C(B op ⊗ k C) → C(C) because this is the interpretation in [NS09] and yields crucial Proposition 2.5. Now we need to prove that our notion of a homological epimorphism is well defined in the following sense:…”

Section: Homological Epimorphisms For Dg Algebrasmentioning

confidence: 99%

Smashing localizations of rings of weak global dimension at most one

Bazzoni

Šťovíček

2017

Advances in Mathematics

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Abstract. We show for a ring R of weak global dimension at most one that there is a bijection between the smashing subcategories of its derived category and the equivalence classes of homological epimorphisms starting in R. If, moreover, R is commutative, we prove that the compactly generated localizing subcategories correspond precisely to flat epimorphisms. We also classify smashing localizations of the derived category of any valuation domain, and provide an easy criterion for the Telescope Conjecture (TC) for any commutative ring of weak global dimension at most one. As a consequence, we show that the TC holds for any commutative von Neumann regular ring R, and it holds precisely for those Prüfer domains which are strongly discrete.

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“…Afterwards, his result was extended to differential graded rings [30] and unbounded derived categories [46]. All these criterions determine recollements in terms of two exceptional objects.…”

Section: Introductionmentioning

confidence: 99%

On the Recollements of Functor Categories

Asadollahi

Hafezi

Vahed

2015

Appl Categor Struct

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This paper is devoted to the study of recollements of functor categories in different levels. In the first part of the paper, we start with a small category S and a maximal object s of S and construct a recollement of Mod-S in terms of Mod-End S (s) and Mod-(S \ {s}) in four different levels. In case S is a finite directed category, by iterating this argument, we get chains of recollements having some interesting applications. In the second part, we start with a recollement of rings and construct a recollement of their path rings, with respect to a finite quiver. Third part of the paper presents some applications, including recollements of triangular matrix rings, an example of a recollement in Gorenstein derived level and recollements of derived categories of N -complexes.

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Parametrizing recollement data for triangulated categories

Cited by 63 publications

References 32 publications

Ladders and simplicity of derived module categories

Ladders and simplicity of derived module categories

Smashing localizations of rings of weak global dimension at most one

On the Recollements of Functor Categories

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