scholarly journals Hodge realizations of 1-motives and the derived Albanese

2012 ◽  
Vol 10 (2) ◽  
pp. 371-412 ◽  
Author(s):  
Vadim Vologodsky
Keyword(s):  
Derived Category ◽  
Left Adjoint ◽  
Triangulated Category ◽  
Rational Form ◽  
Adjoint Functor ◽  
Deligne's Conjecture

AbstractWe prove that the embedding of the derived category of 1-motives up to isogeny into the triangulated category of effective Voevodsky motives, as well as its left adjoint functor LAlbℚ, commute with the Hodge realization. This result yields a new proof of the rational form of Deligne's conjecture on 1-motives.

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.

Regular and Strongly Finitary Structures Over Strongly Algebroidal Categories

1978 ◽  
Vol 30 (02) ◽  
pp. 250-261 ◽  
Author(s):  
Günter Matthiessen
Keyword(s):  
Left Adjoint ◽  
Adjoint Functor ◽  
Reduced Products

Most properties an algebraist needs in categories are reflected by regular functors, introduced in [6]. If is a regular and strongly finitary functor and has some nice properties, it can be shown that the left adjoint functor of G helps to characterize finitary and strongly finitary objects of . The property of being algebroidal can be lifted from to if a certain condition holds in . As an application, the implicational hull of subcategories can be constructed with the help of reduced products.

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 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 The Path Algebra as a Left Adjoint Functor

2018 ◽  
Vol 23 (1) ◽  
pp. 33-52 ◽  
Author(s):  
Kostiantyn Iusenko ◽  
John William MacQuarrie
Keyword(s):  
Path Algebra ◽  
Left Adjoint ◽  
Adjoint Functor

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 CO-STABILITY MANIFOLD OF A TRIANGULATED CATEGORY

2012 ◽  
Vol 55 (1) ◽  
pp. 161-175 ◽  
Author(s):  
PETER JØRGENSEN ◽  
DAVID PAUKSZTELLO
Keyword(s):  
Derived Category ◽  
Stability Conditions ◽  
Closed Field ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Dual Numbers ◽  
Locally Finite ◽  
Algebraically Closed Field ◽  
Stability Manifold

AbstractStability conditions on triangulated categories were introduced by Bridgeland as a ‘continuous’ generalisation of t-structures. The set of locally-finite stability conditions on a triangulated category is a manifold that has been studied intensively. However, there are mainstream triangulated categories whose stability manifold is the empty set. One example is Dc(k[X]/(X2)), the compact derived category of the dual numbers over an algebraically closed field k. This is one of the motivations in this paper for introducing co-stability conditions as a ‘continuous’ generalisation of co-t-structures. Our main result is that the set of nice co-stability conditions on a triangulated category is a manifold. In particular, we show that the co-stability manifold of Dc(k[X]/(X2)) is ℂ.

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.

Last Jewish Writings, 1915–1918

2018 ◽  
Author(s):  
Frederick C. Beiser
Keyword(s):  
Western World ◽  
The West ◽  
Rational Form ◽  
Jewish Communities ◽  
Special Obligation ◽  
Moral Content ◽  
The Many ◽  
At Home

The Jewish writings of these final years develop themes of the earlier years. Cohen continues to explore one of his favorite topics: the affinity of German and Jewish character. Despite his cosmopolitan conception of Judaism, Cohen still thought that the Jews were most at home in Germany. Yet, despite his belief in the special affinity between Germans and Jews, Cohen still shows his cosmopolitanism by his sympathy for the Ostjuden; he maintains that they should be freed from the many immigration controls imposed on them. Cohen continues to worry about the growing weakening of Jewish communities in Germany, and argues, as Socrates did in the Crito, that people have a special obligation to stay within the communities which nurtured them. In a remarkable 1916 lecture on Plato and the prophets Cohen argues that they are the two major ethical voices in the Western world: Plato gave the West a rational form while the prophets gave it moral content. Cohen now reduces his earlier striving for a unity of religions down to the demand for a unity of conscience.

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