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.

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.

DEGENERATION-LIKE ORDERS IN TRIANGULATED CATEGORIES

2005 ◽  
Vol 04 (05) ◽  
pp. 587-597 ◽  
Author(s):  
BERNT TORE JENSEN ◽  
XIUPING SU ◽  
ALEXANDER ZIMMERMANN
Keyword(s):  
Partial Order ◽  
Natural Generalization ◽  
Derived Categories ◽  
Derived Category ◽  
Closed Field ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional

In an earlier paper we defined a relation ≤Δ between objects of the derived category of bounded complexes of modules over a finite dimensional algebra over an algebraically closed field. This relation was shown to be equivalent to the topologically defined degeneration order in a certain space [Formula: see text] for derived categories. This space was defined as a natural generalization of varieties for modules. We remark that this relation ≤Δ is defined for any triangulated category and show that under some finiteness assumptions on the triangulated category ≤Δ is always a partial order.

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 Rouquier's Cocovering Theorem and Well-generated Triangulated Categories

2010 ◽  
Vol 8 (1) ◽  
pp. 31-57 ◽  
Author(s):  
Daniel Murfet
Keyword(s):  
Regular Cardinal ◽  
Derived Category ◽  
Compact Objects ◽  
New Technique ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
A New Technique

AbstractWe study cocoverings of triangulated categories, in the sense of Rouquier, and prove that for any regular cardinal α the condition of α-compactness, in the sense of Neeman, is local with respect to such cocoverings. This was established for ordinary compactness by Rouquier. Our result yields a new technique for proving that a given triangulated category is well-generated. As an application we describe the α-compact objects in the unbounded derived category of a quasi-compact and semi-separated scheme.

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 Bar Category of Modules and Homotopy Adjunction for Tensor Functors

2020 ◽  
Author(s):  
Rina Anno ◽  
Timothy Logvinenko
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Triangulated Categories ◽  
Intended Application ◽  
Adjunction Theory ◽  
Left And Right ◽  
Category Of Modules

Abstract Given a differentially graded (DG)-category ${{\mathcal{A}}}$, we introduce the bar category of modules ${\overline{\textbf{{Mod}}}-{\mathcal{A}}}$. It is a DG enhancement of the derived category $D({{\mathcal{A}}})$ of ${{\mathcal{A}}}$, which is isomorphic to the category of DG ${{\mathcal{A}}}$-modules with ${A_{\infty }}$-morphisms between them. However, it is defined intrinsically in the language of DG categories and requires no complex machinery or sign conventions of ${A_{\infty }}$-categories. We define for these bar categories Tensor and Hom bifunctors, dualisation functors, and a convolution of twisted complexes. The intended application is to working with DG-bimodules as enhancements of exact functors between triangulated categories. As a demonstration, we develop a homotopy adjunction theory for tensor functors between derived categories of DG categories. It allows us to show in an enhanced setting that given a functor $F$ with left and right adjoints $L$ and $R$, the functorial complex $FR \xrightarrow{F{\operatorname{act}}{R}} FRFR \xrightarrow{FR{\operatorname{tr}} - {\operatorname{tr}}{FR}} FR \xrightarrow{{\operatorname{tr}}} {\operatorname{Id}}$ lifts to a canonical twisted complex whose convolution is the square of the spherical twist of $F$. We then write down four induced functorial Postnikov systems computing this convolution.

Derived Equivalences and Recollements of Differential Graded Algebras and Schemes

2016 ◽  
Vol 23 (03) ◽  
pp. 385-408
Author(s):  
Xinhong Chen ◽  
Ming Lu
Keyword(s):  
Derived Categories ◽  
Graded Algebras ◽  
Coherent Sheaves ◽  
Derived Equivalences ◽  
Differential Graded Algebras ◽  
Triangular Algebras

In this paper, we first prove for two differential graded algebras (DGAs) A, B which are derived equivalent to k-algebras Λ, Γ, respectively, that [Formula: see text]. In particular, [Formula: see text]. Secondly, for two quasi-compact and separated schemes X, Y and two algebras A, B over k which satisfy [Formula: see text] and [Formula: see text], we show that [Formula: see text] and [Formula: see text]. Finally, we prove that if X is a quasi-compact and separated scheme over k, then [Formula: see text] admits a recollement relative to [Formula: see text], and we describe the functors in the recollement explicitly. This recollement induces a recollement of bounded derived categories of coherent sheaves and a recollement of singularity categories. When the scheme X is derived equivalent to a DGA or algebra, then the recollement which we get corresponds to the recollement of DGAs in [14] or the recollement of upper triangular algebras in [7].

scholarly journals Dimensions of triangulated categories

2008 ◽  
Vol 1 (2) ◽  
pp. 193-256 ◽  
Author(s):  
Raphaël Rouquier
Keyword(s):  
Finite Dimension ◽  
Derived Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Coherent Sheaves ◽  
Finite Dimensional

AbstractWe define a dimension for a triangulated category. We prove a representability Theorem for a class of functors on finite dimensional triangulated categories. We study the dimension of the bounded derived category of an algebra or a scheme and we show in particular that the bounded derived category of coherent sheaves over a variety has a finite dimension.

scholarly journals Dimensions of triangulated categories

2007 ◽  
Vol 1 (2) ◽  
pp. 193-256 ◽  
Author(s):  
Raphaël Rouquier
Keyword(s):  
Finite Dimension ◽  
Derived Category ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Coherent Sheaves ◽  
Finite Dimensional

AbstractWe define a dimension for a triangulated category. We prove a representability Theorem for a class of functors on finite dimensional triangulated categories. We study the dimension of the bounded derived category of an algebra or a scheme and we show in particular that the bounded derived category of coherent sheaves over a variety has a finite dimension.

scholarly journals THE DERIVED CATEGORY OF AN ÉTALE EXTENSION AND THE SEPARABLE NEEMAN–THOMASON THEOREM

2014 ◽  
Vol 15 (3) ◽  
pp. 613-623 ◽  
Author(s):  
Paul Balmer
Keyword(s):  
Derived Categories ◽  
Derived Category ◽  
Triangulated Categories ◽  
Localization Theorem ◽  
Separable Extensions

We prove that étale morphisms of schemes yield separable extensions of derived categories. We then generalize the Neeman–Thomason localization theorem to separable extensions of triangulated categories.

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