scholarly journals Tannakization in derived algebraic geometry

2014 ◽  
Vol 14 (3) ◽  
pp. 642-700 ◽  
Author(s):  
Isamu Iwanari
Keyword(s):  
Algebraic Geometry ◽  
Galois Group ◽  
Monoidal Category ◽  
Group Scheme ◽  
Affine Group ◽  
Stable Category ◽  
Group Schemes ◽  
Symmetric Monoidal Category ◽  
Derived Algebraic Geometry ◽  
Mixed Motives

AbstractIn this paper we begin studying tannakian constructions in ∞-categories and combine them with the theory of motivic categories developed by Hanamura, Levine, and Voevodsky. This paper is the first in a series of papers. For the purposes above, we first construct a derived affine group scheme and its representation category from a symmetric monoidal ∞-category, which we shall call the tannakization of a symmetric monoidal ∞-category. It can be viewed as an ∞-categorical generalization of work of Joyal-Street and Nori. Next we apply it to the stable ∞-category of mixed motives equipped with the realization functor of a mixed Weil cohomology. We construct a derived motivic Galois group which represents the automorphism group of the realization functor, and whose representation category satisfies an appropriate universal property. As a consequence, we construct an underived motivic Galois group of mixed motives, which is a pro-algebraic group and has nice properties. Also, we present basic properties of derived affine group schemes in the Appendix.

scholarly journals Smooth affine group schemes over the dual numbers

2019 ◽  
Vol Volume 3 ◽  
Author(s):  
Matthieu ROMAGNY ◽  
Dajano Tossici
Keyword(s):  
Commutative Ring ◽  
Group Algebra ◽  
Group Scheme ◽  
Affine Group ◽  
Unipotent Group ◽  
Dual Numbers ◽  
Group Schemes ◽  
Weil Restriction ◽  
International Audience ◽  
Smooth Affine

International audience We provide an equivalence between the category of affine, smooth group schemes over the ring of generalized dual numbers $k[I]$, and the category of extensions of the form $1 \to \text{Lie}(G, I) \to E \to G \to 1$ where G is an affine, smooth group scheme over k. Here k is an arbitrary commutative ring and $k[I] = k \oplus I$ with $I^2 = 0$. The equivalence is given by Weil restriction, and we provide a quasi-inverse which we call Weil extension. It is compatible with the exact structures and the $\mathbb{O}_k$-module stack structures on both categories. Our constructions rely on the use of the group algebra scheme of an affine group scheme; we introduce this object and establish its main properties. As an application, we establish a Dieudonné classification for smooth, commutative, unipotent group schemes over $k[I]$. Nous construisons une équivalence entre la catégorie des schémas en groupes affines et lisses sur l'anneau des nombres duaux généralisés k[I], et la catégorie des extensions de la forme 1 → Lie(G, I) → E → G → 1 où G est un schéma en groupes affine, lisse sur k. Ici k est un anneau commutatif arbitraire et k[I] = k ⊕ I avec I 2 = 0. L'équivalence est donnée par la restriction de Weil, et nous construisons un foncteur quasi-inverse explicite que nous appelons extension de Weil. Ces foncteurs sont compatibles avec les structures exactes et avec les structures de champs en O k-modules des deux catégories. Nos constructions s'appuient sur le schéma en algèbres de groupe d'un schéma en groupes affines, que nous introduisons et dont nous donnons les propriétés principales. En application, nous donnons une classification de Dieudonné pour les schémas en groupes commutatifs, lisses, unipotents sur k[I] lorsque k est un corps parfait.

scholarly journals Local duality for representations of finite group schemes

2019 ◽  
Vol 155 (2) ◽  
pp. 424-453 ◽  
Author(s):  
Dave Benson ◽  
Srikanth B. Iyengar ◽  
Henning Krause ◽  
Julia Pevtsova
Keyword(s):  
Finite Group ◽  
Cohomology Ring ◽  
Group Scheme ◽  
Stable Category ◽  
Group Schemes ◽  
Stable Module Category ◽  
Serre Duality ◽  
Stable Module ◽  
A Finite Group ◽  
Finite Group Schemes

A duality theorem for the stable module category of representations of a finite group scheme is proved. One of its consequences is an analogue of Serre duality, and the existence of Auslander–Reiten triangles for the $\mathfrak{p}$-local and $\mathfrak{p}$-torsion subcategories of the stable category, for each homogeneous prime ideal $\mathfrak{p}$ in the cohomology ring of the group scheme.

E∞-Structures on l-Adic Cohomology

2019 ◽  
pp. 129-203
Author(s):  
Dennis Gaitsgory ◽  
Jacob Lurie
Keyword(s):  
Algebraic Geometry ◽  
Algebraic Curve ◽  
Group Scheme ◽  
Affine Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Smooth Affine

For applications to Weil's conjecture, a version of (3.1) is formulated in the setting of algebraic geometry, where M is replaced by an algebraic curve X (defined over an algebraically closed field k) and E by the classifying stack BG of a smooth affine group scheme over X. This chapter lays the groundwork by constructing an analogue of the functor B.

scholarly journals Noetherian schemes over abelian symmetric monoidal categories

2017 ◽  
Vol 28 (07) ◽  
pp. 1750051 ◽  
Author(s):  
Abhishek Banerjee
Keyword(s):  
Algebraic Geometry ◽  
Local Ring ◽  
Function Field ◽  
Monoidal Category ◽  
Commutative Monoid ◽  
Monoidal Categories ◽  
Compactness Property ◽  
Generic Point ◽  
Symmetric Monoidal Category ◽  
Symmetric Monoidal Categories

In this paper, we develop basic results of algebraic geometry over abelian symmetric monoidal categories. Let [Formula: see text] be a commutative monoid object in an abelian symmetric monoidal category [Formula: see text] satisfying certain conditions and let [Formula: see text]. If the subobjects of [Formula: see text] satisfy a certain compactness property, we say that [Formula: see text] is Noetherian. We study the localization of [Formula: see text] with respect to any [Formula: see text] and define the quotient [Formula: see text] of [Formula: see text] with respect to any ideal [Formula: see text]. We use this to develop appropriate analogues of the basic notions from usual algebraic geometry (such as Noetherian schemes, irreducible, integral and reduced schemes, function field, the local ring at the generic point of a closed subscheme, etc.) for schemes over [Formula: see text]. Our notion of a scheme over a symmetric monoidal category [Formula: see text] is that of Toën and Vaquié.

Finite group schemes of p-rank ≤ 1

2017 ◽  
Vol 166 (2) ◽  
pp. 297-323
Author(s):  
HAO CHANG ◽  
ROLF FARNSTEINER
Keyword(s):  
Finite Group ◽  
Lie Algebras ◽  
Group Scheme ◽  
Closed Field ◽  
Difficult Case ◽  
Modular Representation Theory ◽  
Group Schemes ◽  
Restricted Lie Algebras ◽  
A Finite Group ◽  
Finite Group Schemes

AbstractLet be a finite group scheme over an algebraically closed field k of characteristic char(k) = p ≥ 3. In generalisation of the familiar notion from the modular representation theory of finite groups, we define the p-rank rkp() of and determine the structure of those group schemes of p-rank 1, whose linearly reductive radical is trivial. The most difficult case concerns infinitesimal groups of height 1, which correspond to restricted Lie algebras. Our results show that group schemes of p-rank ≤ 1 are closely related to those being of finite or domestic representation type.

scholarly journals Twin TQFTs and Frobenius Algebras

2013 ◽  
Vol 2013 ◽  
pp. 1-25
Author(s):  
Carmen Caprau
Keyword(s):  
Monoidal Category ◽  
Topological Quantum Field Theory ◽  
Algebra Homomorphism ◽  
Frobenius Algebra ◽  
Quantum Field ◽  
Generators And Relations ◽  
Symmetric Monoidal Category ◽  
Frobenius Algebras ◽  
Topological Quantum ◽  
Algebraic Description

We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on atwin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra(C,W,z,z∗)consists of a commutative Frobenius algebraC, a symmetric Frobenius algebraW, and an algebra homomorphismz:C→Wwith dualz∗:W→C, satisfying some extra conditions. We also introduce a generalized 2-dimensional Topological Quantum Field Theory defined on singular 2-dimensional cobordisms and show that it is equivalent to a twin Frobenius algebra in a symmetric monoidal category.

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