On the derived category of perverse sheaves

1987 ◽  
pp. 27-41 ◽  
Author(s):  
A. A. Beilinson
Keyword(s):  
Derived Category ◽  
Perverse Sheaves

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 Purity and decomposition theorems for staggered sheaves

2012 ◽  
Vol 11 (4) ◽  
pp. 695-745
Author(s):  
Pramod N. Achar ◽  
David Treumann
Keyword(s):  
Decomposition Theorem ◽  
Derived Category ◽  
Perverse Sheaves ◽  
Perverse Sheaf ◽  
Coherent Sheaves ◽  
Direct Sum ◽  
Constructible Sheaves ◽  
Decomposition Theorems ◽  
Pure Complex

AbstractTwo major results in the theory of ℓ-adic mixed constructible sheaves are the purity theorem (every simple perverse sheaf is pure) and the decomposition theorem (every pure object in the derived category is a direct sum of shifts of simple perverse sheaves). In this paper, we prove analogues of these results for coherent sheaves. Specifically, we work with staggered sheaves, which form the heart of a certain t-structure on the derived category of equivariant coherent sheaves. We prove, under some reasonable hypotheses, that every simple staggered sheaf is pure, and that every pure complex of coherent sheaves is a direct sum of shifts of simple staggered sheaves.

scholarly journals Under the Umbrella: Goal-Derived Category Construction and Product Category Nesting

2021 ◽  
pp. 000183922110123
Author(s):  
Johnny Boghossian ◽  
Robert J. David
Keyword(s):  
Process Model ◽  
Product Category ◽  
Derived Category ◽  
The State ◽  
Product Attributes ◽  
Product Categories ◽  
Role Of The State ◽  
Market Intermediaries ◽  
Category Construction

Categories are organized vertically, with product categories nested under larger umbrella categories. Meaning flows from umbrella categories to the categories beneath them, such that the construction of a new umbrella category can significantly reshape the categorical landscape. This paper explores the construction of a new umbrella category and the nesting beneath it of a product category. Specifically, we study the construction of the Quebec terroir products umbrella category and the nesting of the Quebec artisanal cheese product category under this umbrella. Our analysis shows that the construction of umbrella categories can unfold entirely separately from that of product categories and can follow a distinct categorization process. Whereas the construction of product categories may be led by entrepreneurs who make salient distinctive product attributes, the construction of umbrella categories may be led by “macro actors” removed from the market. We found that these macro actors followed a goal-derived categorization process: they first defined abstract goals and ideals for the umbrella category and only subsequently sought to populate it with product categories. Among the macro actors involved, the state played a central role in defining the meaning of the Quebec terroir category and mobilizing other macro actors into the collective project, a finding that suggests an expanded role of the state in category construction. We also found that market intermediaries are important in the nesting of product categories beneath new umbrella categories, notably by projecting identities onto producers consistent with the goals of the umbrella category. We draw on these findings to develop a process model of umbrella category construction and product category nesting.

scholarly journals Equivalence of the Derived Category of a Variety with a Singularity Category

2012 ◽  
Vol 2013 (12) ◽  
pp. 2787-2808 ◽  
Author(s):  
Mehmet Umut Isik
Keyword(s):  
Derived Category ◽  
Singularity Category

Explicit Models for Perverse Sheaves, II

2007 ◽  
Vol 11 (2) ◽  
pp. 149-178 ◽  
Author(s):  
F. Gudiel-Rodríguez ◽  
L. Narváez-Macarro
Keyword(s):  
Perverse Sheaves

Perverse sheaves on rank stratifications

1999 ◽  
Vol 96 (2) ◽  
pp. 317-362 ◽  
Author(s):  
Tom Braden ◽  
Mikhail Grinberg
Keyword(s):  
Perverse Sheaves

scholarly journals Simple Helices on Fano Threefolds

2011 ◽  
Vol 54 (3) ◽  
pp. 520-526
Author(s):  
A. Polishchuk
Keyword(s):  
Braid Group ◽  
Vector Bundles ◽  
Coherent Sheaf ◽  
Derived Category ◽  
Fano Threefolds ◽  
Fano Threefold

AbstractBuilding on the work of Nogin, we prove that the braid groupB4acts transitively on full exceptional collections of vector bundles on Fano threefolds withb2= 1 andb3= 0. Equivalently, this group acts transitively on the set of simple helices (considered up to a shift in the derived category) on such a Fano threefold. We also prove that on threefolds withb2= 1 and very ample anticanonical class, every exceptional coherent sheaf is locally free.

scholarly journals Vanishing theorems for perverse sheaves on abelian varieties, revisited

2017 ◽  
Vol 24 (1) ◽  
pp. 63-84 ◽  
Author(s):  
Bhargav Bhatt ◽  
Christian Schnell ◽  
Peter Scholze
Keyword(s):  
Abelian Varieties ◽  
Vanishing Theorems ◽  
Perverse Sheaves

scholarly journals Cohomology of groups and splendid equivalences of derived categories

2001 ◽  
Vol 131 (3) ◽  
pp. 459-472 ◽  
Author(s):  
ALEXANDER ZIMMERMANN
Keyword(s):  
Group Ring ◽  
Local Structure ◽  
Cohomology Ring ◽  
Derived Categories ◽  
Derived Category ◽  
Restriction Map ◽  
Group Rings ◽  
Cohomology Rings ◽  
The Impact ◽  
Cohomology Of Groups

In an earlier paper we studied the impact of equivalences between derived categories of group rings on their cohomology rings. Especially the group of auto-equivalences TrPic(RG) of the derived category of a group ring RG as introduced by Raphaël Rouquier and the author defines an action on the cohomology ring of this group. We study this action with respect to the restriction map, transfer, conjugation and the local structure of the group G.

scholarly journals FACTORIZATION THEOREMS FOR MORPHISMS OF ORDERED GROUPOIDS AND INVERSE SEMIGROUPS

2001 ◽  
Vol 44 (3) ◽  
pp. 549-569 ◽  
Author(s):  
Benjamin Steinberg
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Classical Theory ◽  
Derived Category ◽  
Inverse Semigroups ◽  
Subject Classification

AbstractAdapting the theory of the derived category to ordered groupoids, we prove that every ordered functor (and thus every inverse and regular semigroup homomorphism) factors as an enlargement followed by an ordered fibration. As an application, we obtain Lawson’s version of Ehresmann’s Maximum Enlargement Theorem, from which can be deduced the classical theory of idempotent-pure inverse semigroup homomorphisms and $E$-unitary inverse semigroups.AMS 2000 Mathematics subject classification: Primary 20M18; 20L05; 20M17

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