scholarly journals Perverse, Hodge and motivic realizations of étale motives

2016 ◽  
Vol 152 (6) ◽  
pp. 1237-1285 ◽  
Author(s):  
Florian Ivorra
Keyword(s):  
Abelian Category ◽  
Derived Category ◽  
Original Version ◽  
Complex Numbers ◽  
Triangulated Categories ◽  
Basic Lemma

Let $k=\mathbb{C}$ be the field of complex numbers. In this article we construct Hodge realization functors defined on the triangulated categories of étale motives with rational coefficients. Our construction extends to every smooth quasi-projective $k$-scheme, the construction done by Nori over a field, and relies on the original version of the basic lemma proved by Beĭlinson. As in the case considered by Nori, the realization functor factors through the bounded derived category of a perverse version of the Abelian category of Nori motives.

Dimension and level of triangulated categories

2020 ◽  
pp. 2150122
Author(s):  
Xiaoyan Yang ◽  
Jingwen Shen
Keyword(s):  
Abelian Category ◽  
Derived Categories ◽  
Derived Category ◽  
Triangulated Categories ◽  
Cotorsion Pair

For the bounded derived category of an abelian category, bounds of the dimension with respect to a complete hereditary cotorsion pair are given. We also characterize levels of DG-modules and study how levels involved in a recollement of derived categories over DG-rings are related.

scholarly journals Irregular perverse sheaves

2021 ◽  
Vol 157 (3) ◽  
pp. 573-624
Author(s):  
Tatsuki Kuwagaki
Keyword(s):  
Abelian Category ◽  
Derived Category ◽  
Perverse Sheaves ◽  
Algebraic Version ◽  
Constructible Sheaves ◽  
Straightforward Generalization ◽  
Novikov Ring

We introduce irregular constructible sheaves, which are ${\mathbb {C}}$-constructible with coefficients in a finite version of the Novikov ring $\Lambda$ and special gradings. We show that the bounded derived category of cohomologically irregular constructible complexes is equivalent to the bounded derived category of holonomic ${\mathcal {D}}$-modules by a modification of D’Agnolo and Kashiwara's irregular Riemann–Hilbert correspondence. The bounded derived category of cohomologically irregular constructible complexes is equipped with the irregular perverse $t$-structure, which is a straightforward generalization of usual perverse $t$-structure, and we prove that its heart is equivalent to the abelian category of holonomic ${\mathcal {D}}$-modules. We also develop the algebraic version of the theory.

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.

Cut cotorsion pairs

2021 ◽  
pp. 1-38
Author(s):  
Mindy Huerta ◽  
Octavio Mendoza ◽  
Marco A. Pérez
Keyword(s):  
General Framework ◽  
Abelian Category ◽  
Triangulated Categories ◽  
Finitistic Dimension ◽  
Coherent Sheaves ◽  
Chain Complexes ◽  
Cotorsion Pairs ◽  
Gorenstein Homological Algebra ◽  
Serre Subcategories ◽  
Dimension Conjecture

Abstract We present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations restricted to certain subcategories. We also exhibit some connections between cut cotorsion pairs and Auslander–Buchweitz approximation theory, by considering relative analogs for Frobenius pairs and Auslander–Buchweitz contexts. Several applications are given in the settings of relative Gorenstein homological algebra, chain complexes, and quasi-coherent sheaves, as well as to characterize some important results on the Finitistic Dimension Conjecture, the existence of right adjoints of quotient functors by Serre subcategories, and the description of cotorsion pairs in triangulated categories as co-t-structures.

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.

scholarly journals Quillen model structures for relative homological algebra

2002 ◽  
Vol 133 (2) ◽  
pp. 261-293 ◽  
Author(s):  
J. DANIEL CHRISTENSEN ◽  
MARK HOVEY
Keyword(s):  
Homotopy Theory ◽  
Homological Algebra ◽  
Abelian Category ◽  
Derived Categories ◽  
Derived Category ◽  
Category Structure ◽  
Model Category ◽  
Projective Class ◽  
Chain Complexes ◽  
Model Structures

An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category [Ascr ] is exactly the information needed to do homological algebra in [Ascr ]. The main result is that, under weak hypotheses, the category of chain complexes of objects of [Ascr ] has a model category structure that reflects the homological algebra of the projective class in the sense that it encodes the Ext groups and more general derived functors. Examples include the ‘pure derived category’ of a ring R, and derived categories capturing relative situations, including the projective class for Hochschild homology and co-homology. We characterize the model structures that are cofibrantly generated, and show that this fails for many interesting examples. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class, and give examples that include equivariant homotopy theory and bounded below derived categories.

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 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.

Derived Categories: A Quick Tour

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Abelian Category ◽  
Derived Categories ◽  
Derived Category ◽  
Homotopy Category ◽  
Intermediate Step ◽  
Spectral Sequences ◽  
Separate Section ◽  
Derived Functors ◽  
Left And Right

This chapter briefly outlines the main steps in the construction of the derived category of an arbitrary abelian category. The homotopy category of complexes is considered as an intermediate step, which is then localized with respect to quasi-isomorphisms. Left and right derived functors are explained in general, and particular examples are studied in more detail. Spectral sequences are treated in a separate section.

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.

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