scholarly journals Lifting of recollements and gluing of partial silting sets

2021 ◽  
pp. 1-49
Author(s):  
Manuel Saorín ◽  
Alexandra Zvonareva
Keyword(s):  
Finite Length ◽  
Explicit Construction ◽  
Derived Categories ◽  
Triangulated Categories ◽  
Artin Algebras ◽  
Stable Categories ◽  
Silting Objects ◽  
Torsion Free

This paper focuses on recollements and silting theory in triangulated categories. It consists of two main parts. In the first part a criterion for a recollement of triangulated subcategories to lift to a torsion torsion-free triple (TTF triple) of ambient triangulated categories with coproducts is proved. As a consequence, lifting of TTF triples is possible for recollements of stable categories of repetitive algebras or self-injective finite length algebras and recollements of bounded derived categories of separated Noetherian schemes. When, in addition, the outer subcategories in the recollement are derived categories of small linear categories the conditions from the criterion are sufficient to lift the recollement to a recollement of ambient triangulated categories up to equivalence. In the second part we use these results to study the problem of constructing silting sets in the central category of a recollement generating the t-structure glued from the silting t-structures in the outer categories. In the case of a recollement of bounded derived categories of Artin algebras we provide an explicit construction for gluing classical silting objects.

scholarly journals ON HOMOLOGICAL FROBENIUS COMPLEXES AND BIMODULES

2014 ◽  
Vol 56 (3) ◽  
pp. 629-642
Author(s):  
J. R. GARCÍA ROZAS ◽  
LUIS OYONARTE ◽  
BLAS TORRECILLAS
Keyword(s):  
Associative Ring ◽  
Natural Generalization ◽  
Derived Categories ◽  
Triangulated Categories ◽  
Ring Homomorphisms ◽  
Frobenius Ring ◽  
Gorenstein Categories

AbstractWe introduce the concept of homological Frobenius functors as the natural generalization of Frobenius functors in the setting of triangulated categories, and study their structure in the particular case of the derived categories of those of complexes and modules over a unital associative ring. Tilting complexes (modules) are examples of homological Frobenius complexes (modules). Homological Frobenius functors retain some of the nice properties of Frobenius ones as the ascent theorem for Gorenstein categories. It is shown that homological Frobenius ring homomorphisms are always Frobenius.

scholarly journals Completing perfect complexes

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1387-1427 ◽  
Author(s):  
Henning Krause
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Direct Construction ◽  
Coherent Sheaves ◽  
Perfect Complexes ◽  
Coherent Ring ◽  
Abelian Categories ◽  
Finitely Presented

Abstract This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra. The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. The final appendix by Bernhard Keller introduces the concept of a morphic enhancement for triangulated categories and provides a foundation for completing a triangulated category.

scholarly journals On the abelianization of derived categories and a negative solution to Rosický’s problem

2012 ◽  
Vol 149 (1) ◽  
pp. 125-147 ◽  
Author(s):  
Silvana Bazzoni ◽  
Jan Šťovíček
Keyword(s):  
Regular Cardinal ◽  
Large Family ◽  
Global Dimension ◽  
Derived Categories ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Negative Solution ◽  
Compactly Generated

AbstractWe prove for a large family of rings R that their λ-pure global dimension is greater than one for each infinite regular cardinal λ. This answers in the negative a problem posed by Rosický. The derived categories of such rings then do not satisfy, for any λ, the Adams λ-representability for morphisms. Equivalently, they are examples of well-generated triangulated categories whose λ-abelianization in the sense of Neeman is not a full functor for any λ. In particular, we show that given a compactly generated triangulated category, one may not be able to find a Rosický functor among the λ-abelianization functors.

scholarly journals Reduction of triangulated categories and maximal modification algebras for cAn singularities

2018 ◽  
Vol 2018 (738) ◽  
pp. 149-202 ◽  
Author(s):  
Osamu Iyama ◽  
Michael Wemyss
Keyword(s):  
Complete Classification ◽  
Global Dimension ◽  
Triangulated Categories ◽  
Endomorphism Rings ◽  
Ground Field ◽  
Finite Global Dimension ◽  
Base Ring ◽  
Crepant Resolutions ◽  
Cluster Tilting ◽  
Stable Categories

Abstract In this paper we define and study triangulated categories in which the Hom-spaces have Krull dimension at most one over some base ring (hence they have a natural 2-step filtration), and each factor of the filtration satisfies some Calabi–Yau type property. If \mathcal{C} is such a category, we say that \mathcal{C} is Calabi–Yau with \dim\mathcal{C}\leq 1 . We extend the notion of Calabi–Yau reduction to this setting, and prove general results which are an analogue of known results in cluster theory. Such categories appear naturally in the setting of Gorenstein singularities in dimension three as the stable categories \mathop{\underline{\textup{CM}}}R of Cohen–Macaulay modules. We explain the connection between Calabi–Yau reduction of \mathop{\underline{\textup{CM}}}R and both partial crepant resolutions and \mathbb{Q} -factorial terminalizations of \operatorname{Spec}R , and we show under quite general assumptions that Calabi–Yau reductions exist. In the remainder of the paper we focus on complete local cA_{n} singularities R. By using a purely algebraic argument based on Calabi–Yau reduction of \mathop{\underline{\textup{CM}}}R , we give a complete classification of maximal modifying modules in terms of the symmetric group, generalizing and strengthening results in [I. Burban, O. Iyama, B. Keller and I. Reiten, Cluster tilting for one-dimensional hypersurface singularities, Adv. Math. 217 2008, 6, 2443–2484], [H. Dao and C. Huneke, Vanishing of Ext, cluster tilting and finite global dimension of endomorphism rings, Amer. J. Math. 135 2013, 2, 561–578], where we do not need any restriction on the ground field. We also describe the mutation of modifying modules at an arbitrary (not necessarily indecomposable) direct summand. As a corollary when k=\mathbb{C} we obtain many autoequivalences of the derived category of the \mathbb{Q} -factorial terminalizations of \operatorname{Spec}R .

scholarly journals Discreteness of silting objects and t-structures in triangulated categories

2018 ◽  
Vol 118 (1) ◽  
pp. 1-42 ◽  
Author(s):  
Takahide Adachi ◽  
Yuya Mizuno ◽  
Dong Yang
Keyword(s):  
Triangulated Categories ◽  
Silting Objects

Perfect complexes on Deligne-Mumford stacks and applications

2009 ◽  
Vol 4 (3) ◽  
pp. 559-603 ◽  
Author(s):  
Amalendu Krishna
Keyword(s):  
Full Subcategory ◽  
Derived Categories ◽  
Derived Category ◽  
Triangulated Categories ◽  
Localization Theorem ◽  
Coherent Sheaves ◽  
Perfect Complexes ◽  
Tensor Triangulated Categories ◽  
Resolution Property

AbstractFor a tame Deligne-Mumford stack X with the resolution property, we show that the Cartan-Eilenberg resolutions of unbounded complexes of quasicoherent sheaves are K-injective resolutions. This allows us to realize the derived category of quasi-coherent sheaves on X as a reflexive full subcategory of the derived category of X-modules.We then use the results of Neeman and recent results of Kresch to establish the localization theorem of Thomason-Trobaugh for the K-theory of perfect complexes on stacks of above type which have coarse moduli schemes. As a byproduct, we get a generalization of Krause's result about the stable derived categories of schemes to such stacks.We prove Thomason's classification of thick triangulated tensor subcategories of D(perf / X). As the final application of our localization theorem, we show that the spectrum of D(perf / X) as defined by Balmer, is naturally isomorphic to the coarse moduli scheme of X, answering a question of Balmer for the tensor triangulated categories arising from Deligne-Mumford stacks.

scholarly journals Birational geometry of moduli spaces of perverse coherent sheaves on blow-ups

2021 ◽  
Author(s):  
Naoki Koseki
Keyword(s):  
Moduli Spaces ◽  
Blow Up ◽  
Derived Categories ◽  
Model Program ◽  
Minimal Model Program ◽  
Blow Down ◽  
Coherent Sheaves ◽  
Framed Sheaves ◽  
Wall Crossing Formula ◽  
Torsion Free

AbstractIn order to study the wall-crossing formula of Donaldson type invariants on the blown-up plane, Nakajima–Yoshioka constructed a sequence of blow-up/blow-down diagrams connecting the moduli space of torsion free framed sheaves on projective plane, and that on its blow-up. In this paper, we prove that Nakajima–Yoshioka’s diagram realizes the minimal model program. Furthermore, we obtain a fully-faithful embedding between the derived categories of these moduli spaces.

scholarly journals Rank functions on triangulated categories

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Joseph Chuang ◽  
Andrey Lazarev
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Rank Function ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Graded Algebras ◽  
Artinian Rings ◽  
Differential Graded Algebras ◽  
Skew Fields ◽  
Rank Functions

Abstract We introduce the notion of a rank function on a triangulated category 𝒞 {\mathcal{C}} which generalizes the Sylvester rank function in the case when 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ ( A ) {\mathcal{C}=\mathsf{Perf}(A)} is the perfect derived category of a ring A. We show that rank functions are closely related to functors into simple triangulated categories and classify Verdier quotients into simple triangulated categories in terms of particular rank functions called localizing. If 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ ( A ) {\mathcal{C}=\mathsf{Perf}(A)} as above, localizing rank functions also classify finite homological epimorphisms from A into differential graded skew-fields or, more generally, differential graded Artinian rings. To establish these results, we develop the theory of derived localization of differential graded algebras at thick subcategories of their perfect derived categories. This is a far-reaching generalization of Cohn’s matrix localization of rings and has independent interest.

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