scholarly journals Ind-abelian categories and quasi-coherent sheaves

2014 ◽  
Vol 157 (3) ◽  
pp. 391-423 ◽  
Author(s):  
DANIEL SCHÄPPI
Keyword(s):  
Tensor Product ◽  
Coherent Sheaves ◽  
Two Factors ◽  
Abelian Categories ◽  
Tannakian Categories ◽  
Resolution Property

AbstractWe study the question of when a category of ind-objects is abelian. Our answer allows a further generalization of the notion of weakly Tannakian categories introduced by the author. As an application we show that, under suitable conditions, the category of coherent sheaves on the product of two schemes with the resolution property is given by the Deligne tensor product of the categories of coherent sheaves of the two factors. To do this we prove that the class of quasi-compact and semi-separated schemes with the resolution property is closed under fiber products.

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 Fourier–Mukai transforms and Bridgeland stability conditions on abelian threefolds II

2016 ◽  
Vol 27 (01) ◽  
pp. 1650007 ◽  
Author(s):  
Antony Maciocia ◽  
Dulip Piyaratne
Keyword(s):  
Type Inequality ◽  
Stability Conditions ◽  
Coherent Sheaves ◽  
Rank One ◽  
Bridgeland Stability Conditions ◽  
Abelian Categories

We show that the conjectural construction proposed by Bayer, Bertram, Macrí and Toda gives rise to Bridgeland stability conditions for a principally polarized abelian threefold with Picard rank one by proving that tilt stable objects satisfy the strong Bogomolov–Gieseker (BG) type inequality. This is done by showing certain Fourier–Mukai transforms (FMTs) give equivalences of abelian categories which are double tilts of coherent sheaves.

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 The “maximal” tensor product of operator spaces

1999 ◽  
Vol 42 (2) ◽  
pp. 267-284 ◽  
Author(s):  
Timur Oikhberg ◽  
Gilles Pisier
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Hilbert Spaces ◽  
Tensor Products ◽  
Operator Space ◽  
Operator Spaces ◽  
Two Factors

In analogy with the maximal tensor product of C*-algebras, we define the “maximal” tensor product E1⊗μE2 of two operator spaces E1 and E2 and we show that it can be identified completely isometrically with the sum of the two Haagerup tensor products: E1⊗hE2 + E2⊗hE1. We also study the extension to more than two factors. Let E be an n-dimensional operator space. As an application, we show that the equality E*⊗μE = E*⊗min E holds isometrically iff E = Rn or E = Cn (the row or column n-dimensional Hilbert spaces). Moreover, we show that if an operator space E is such that, for any operator space F, we have F ⊗min E = F⊗μ E isomorphically, then E is completely isomorphic to either a row or a column Hilbert space.

scholarly journals Which abelian tensor categories are geometric?

2018 ◽  
Vol 2018 (734) ◽  
pp. 145-186 ◽  
Author(s):  
Daniel Schäppi
Keyword(s):  
Characteristic Zero ◽  
Intrinsic Properties ◽  
Tensor Categories ◽  
Projective Varieties ◽  
Coherent Sheaves ◽  
Geometric Objects ◽  
Necessary And Sufficient ◽  
Tannaka Duality ◽  
Tannakian Categories

AbstractFor a large class of geometric objects, the passage to categories of quasi-coherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a characterization of tensor categories in the image of this embedding.However, this notion requires additional structure to be present, namely a fiber functor. For the case of classical Tannakian categories in characteristic zero, Deligne has found intrinsic properties—expressible entirely within the language of tensor categories—which are necessary and sufficient for the existence of a fiber functor. In this paper we generalize Deligne’s result to weakly Tannakian categories in characteristic zero. The class of geometric objects whose tensor categories of quasi-coherent sheaves can be recognized in this manner includes both the gerbes arising in classical Tannaka duality and more classical geometric objects such as projective varieties over a field of characteristic zero.Our proof uses a different perspective on fiber functors, which we formalize through the notion of geometric tensor categories. A second application of this perspective allows us to describe categories of quasi-coherent sheaves on fiber products.

scholarly journals A reconstruction theorem for abelian categories of twisted sheaves

2016 ◽  
Vol 2016 (712) ◽  
Author(s):  
Benjamin Antieau
Keyword(s):  
Coherent Sheaves ◽  
Abelian Categories ◽  
Reconstruction Theorem ◽  
Twisted Sheaves

AbstractWe use an idea of Rosenberg to prove a reconstruction theorem for abelian categories of α-twisted quasi-coherent sheaves on quasi-compact and quasi-separated schemes

Derived Categories of Coherent Sheaves

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Tensor Product ◽  
Final Section ◽  
Abelian Category ◽  
Inverse Image ◽  
Derived Categories ◽  
Derived Category ◽  
Coherent Sheaves ◽  
Serre Functor

The discussion of the previous chapter is applied to the derived category of the abelian category of coherent sheaves. The Serre functor is introduced, and particular spanning classes are constructed. The usual geometric functors, direct and inverse image, tensor product, and global sections, are derived and extended to functors between derived categories. The compatibilities between them are reviewed. The final section focuses on the Grothendieck-Verdier duality.

Slow Down You’re Moving Too Fast

2015 ◽  
Vol 27 (2) ◽  
pp. 53-63 ◽  
Author(s):  
Annie Lang ◽  
Nancy Schwartz ◽  
Sharon Mayell
Keyword(s):  
Older Adults ◽  
Cognitive Aging ◽  
Attention And Memory ◽  
Younger Adults ◽  
Media Messages ◽  
Two Factors ◽  
The Difference

The study reported here compared how younger and older adults processed the same set of media messages which were selected to vary on two factors, arousing content and valence. Results showed that older and younger adults had similar arousal responses but different patterns of attention and memory. Older adults paid more attention to all messages than did younger adults. However, this attention did not translate into greater memory. Older and younger adults had similar levels of memory for slow-paced messages, but younger adults outperformed older adults significantly as pacing increased, and the difference was larger for arousing compared with calm messages. The differences found are in line with predictions made based on the cognitive-aging literature.

Effects of an Anxiety-Specific Psychometric Factor on Fear Conditioning and Fear Generalization

2017 ◽  
Vol 225 (3) ◽  
pp. 200-213 ◽  
Author(s):  
Christian Baumann ◽  
Miriam A. Schiele ◽  
Martin J. Herrmann ◽  
Tina B. Lonsdorf ◽  
Peter Zwanzger ◽  
...  
Keyword(s):  
Risk Factor ◽  
Anxiety Disorders ◽  
Fear Conditioning ◽  
Longitudinal Design ◽  
Principal Component ◽  
Anxiety And Depression ◽  
Specific Factor ◽  
High Anxious ◽  
Two Factors ◽  
Fear Generalization

Abstract. Conditioning and generalization of fear are assumed to play central roles in the pathogenesis of anxiety disorders. Here we investigate the influence of a psychometric anxiety-specific factor on these two processes, thus try to identify a potential risk factor for the development of anxiety disorders. To this end, 126 healthy participants were examined with questionnaires assessing symptoms of anxiety and depression and with a fear conditioning and generalization paradigm. A principal component analysis of the questionnaire data identified two factors representing the constructs anxiety and depression. Variations in fear conditioning and fear generalization were solely associated with the anxiety factor characterized by anxiety sensitivity and agoraphobic cognitions; high-anxious individuals exhibited stronger fear responses (arousal) during conditioning and stronger generalization effects for valence and UCS-expectancy ratings. Thus, the revealed psychometric factor “anxiety” was associated with enhanced fear generalization, an assumed risk factor for anxiety disorders. These results ask for replication with a longitudinal design allowing to examine their predictive validity.

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