scholarly journals STEENROD OPERATIONS ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC STACKS

2021 ◽  
pp. 1-48
Author(s):  
Federico Scavia
Keyword(s):  
Finite Groups ◽  
Reductive Groups ◽  
Orthogonal Groups ◽  
De Rham Cohomology ◽  
Algebraic Stacks ◽  
Steenrod Operations ◽  
De Rham

Abstract Building upon work of Epstein, May and Drury, we define and investigate the mod p Steenrod operations on the de Rham cohomology of smooth algebraic stacks over a field of characteristic $p>0$ . We then compute the action of the operations on the de Rham cohomology of classifying stacks for finite groups, connected reductive groups for which p is not a torsion prime and (special) orthogonal groups when $p=2$ .

Algebraic stacks

2001 ◽  
Vol 111 (1) ◽  
pp. 1-31 ◽  
Author(s):  
Tomás L Gómez
Keyword(s):  
Algebraic Stacks

The pro-étale topology for algebraic stacks

2020 ◽  
Vol 48 (9) ◽  
pp. 3761-3770
Author(s):  
Chang-Yeon Chough
Keyword(s):  
Algebraic Stacks

scholarly journals Geometricity for derived categories of algebraic stacks

2016 ◽  
Vol 22 (4) ◽  
pp. 2535-2568 ◽  
Author(s):  
Daniel Bergh ◽  
Valery A. Lunts ◽  
Olaf M. Schnürer
Keyword(s):  
Derived Categories ◽  
Algebraic Stacks

scholarly journals Moduli of Bridgeland semistable objects on 3-folds and Donaldson–Thomas invariants

2019 ◽  
Vol 2019 (747) ◽  
pp. 175-219 ◽  
Author(s):  
Dulip Piyaratne ◽  
Yukinobu Toda
Keyword(s):  
Finite Type ◽  
Equivalent Form ◽  
Stability Conditions ◽  
Weak Stability ◽  
Algebraic Stacks ◽  
Moduli Stacks ◽  
Étale Quotients

Abstract In this paper we show that the moduli stacks of Bridgeland semistable objects on smooth projective 3-folds are quasi-proper algebraic stacks of finite type if they satisfy the Bogomolov–Gieseker (BG for short) inequality conjecture proposed by Bayer, Macrì and the second author. The key ingredients are the equivalent form of the BG inequality conjecture and its generalization to arbitrary very weak stability conditions. This result is applied to define Donaldson–Thomas invariants counting Bridgeland semistable objects on smooth projective Calabi–Yau 3-folds satisfying the BG inequality conjecture, for example on étale quotients of abelian 3-folds.

scholarly journals On the de Rham cohomology of differential and algebraic stacks

2005 ◽  
Vol 198 (2) ◽  
pp. 583-622 ◽  
Author(s):  
Kai A. Behrend
Keyword(s):  
De Rham Cohomology ◽  
Algebraic Stacks ◽  
De Rham

scholarly journals Sheaves of categories with local actions of Hochschild cochains

2019 ◽  
Vol 155 (08) ◽  
pp. 1521-1567 ◽  
Author(s):  
Dario Beraldo
Keyword(s):  
Differential Operators ◽  
Algebraic Stacks ◽  
Finiteness Conditions ◽  
Coherent Sheaves ◽  
Singular Support ◽  
Mild Conditions ◽  
Geometric Langlands ◽  
Category Of Modules ◽  
Langlands Programme

The notion of Hochschild cochains induces an assignment from $\mathsf{Aff}$ , affine DG schemes, to monoidal DG categories. We show that this assignment extends, under appropriate finiteness conditions, to a functor $\mathbb{H}:\mathsf{Aff}\rightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$ , where the latter denotes the category of monoidal DG categories and bimodules. Any functor $\mathbb{A}:\mathsf{Aff}\rightarrow \mathsf{Alg}^{\text{bimod}}(\mathsf{DGCat})$ gives rise, by taking modules, to a theory of sheaves of categories $\mathsf{ShvCat}^{\mathbb{A}}$ . In this paper, we study $\mathsf{ShvCat}^{\mathbb{H}}$ . Loosely speaking, this theory categorifies the theory of $\mathfrak{D}$ -modules, in the same way as Gaitsgory’s original $\mathsf{ShvCat}$ categorifies the theory of quasi-coherent sheaves. We develop the functoriality of $\mathsf{ShvCat}^{\mathbb{H}}$ , its descent properties and the notion of $\mathbb{H}$ -affineness. We then prove the $\mathbb{H}$ -affineness of algebraic stacks: for ${\mathcal{Y}}$ a stack satisfying some mild conditions, the $\infty$ -category $\mathsf{ShvCat}^{\mathbb{H}}({\mathcal{Y}})$ is equivalent to the $\infty$ -category of modules for $\mathbb{H}({\mathcal{Y}})$ , the monoidal DG category of higher differential operators. The main consequence, for ${\mathcal{Y}}$ quasi-smooth, is the following: if ${\mathcal{C}}$ is a DG category acted on by $\mathbb{H}({\mathcal{Y}})$ , then ${\mathcal{C}}$ admits a theory of singular support in $\operatorname{Sing}({\mathcal{Y}})$ , where $\operatorname{Sing}({\mathcal{Y}})$ is the space of singularities of ${\mathcal{Y}}$ . As an application to the geometric Langlands programme, we indicate how derived Satake yields an action of $\mathbb{H}(\operatorname{LS}_{{\check{G}}})$ on $\mathfrak{D}(\operatorname{Bun}_{G})$ , thereby equipping objects of $\mathfrak{D}(\operatorname{Bun}_{G})$ with singular support in $\operatorname{Sing}(\operatorname{LS}_{{\check{G}}})$ .

scholarly journals Group actions on algebraic stacks via butterflies

2017 ◽  
Vol 486 ◽  
pp. 36-63
Author(s):  
Behrang Noohi
Keyword(s):  
Group Actions ◽  
Algebraic Stacks

scholarly journals Introduction to the Ekedahl Invariants

2017 ◽  
Vol 120 (2) ◽  
pp. 211 ◽  
Author(s):  
Ivan Martino
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Algebraic Stacks ◽  
Cohomological Invariants ◽  
Equivalent Definition ◽  
A Finite Group

In 2009, T. Ekedahl introduced certain cohomological invariants for finite groups. In this work we present these invariants and we give an equivalent definition that does not involve the notion of algebraic stacks. Moreover we show certain properties for the class of the classifying stack of a finite group in the Kontsevich value ring.

Sign in / Sign up

Export Citation Format

Share Document