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.

Automorphisms, isomorphisms, and Tits classification

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Affine Group ◽  
Classification Theorem ◽  
Isomorphism Theorem ◽  
Reductive Groups ◽  
Semisimple Groups ◽  
Dynkin Diagrams ◽  
Weil Restriction ◽  
Characteristic 2 ◽  
Smooth Affine

This chapter considers automorphisms, isomorphisms, and Tits classification. It begins by establishing a version of the Isomorphism Theorem for pseudo-split pseudo-reductive groups, along with a pseudo-reductive variant of the Isogeny Theorem for split connected semisimple groups. The key to both proofs is a technique to construct pseudo-reductive subgroups of an ambient smooth affine group. Some instructive examples over imperfect fields k of characteristic 2 are given. The chapter goes on to discuss the behavior of the k-group ZG,C with respect to Weil restriction in the pseudoreductive case. It also describes automorphism schemes for pseudo-reductive groups, focusing only on the pseudo-semisimple case because commutative pseudo-reductive groups that are not tori generally have a non-representable automorphism functor. Finally, it examines Tits-style classification, using Dynkin diagrams to express the classification theorem.

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.

The Trace Formula for BunG(X)

2019 ◽  
pp. 239-308
Author(s):  
Dennis Gaitsgory ◽  
Jacob Lurie
Keyword(s):  
Trace Formula ◽  
Algebraic Curve ◽  
Group Scheme ◽  
Affine Group ◽  
Generic Fiber ◽  
Prove Theorem ◽  
Lefschetz Trace Formula ◽  
Moduli Stack ◽  
Smooth Affine

This chapter aims to prove Theorem 1.4.4.1, which is formulated as follows: Theorem 5.0.0.3, let X be an algebraic curve over F q and let G be a smooth affine group scheme over X. Suppose that the fibers of G are connected and that the generic fiber of G is semisimple. Then the moduli stack BunG(X) satisfies the Grothendieck–Lefschetz trace formula. However, Theorem 5.0.0.3 cannot be deduced directly from the Grothendieck–Lefschetz trace formula for global quotient stacks because the moduli stack BunG(X) is usually not quasi-compact. The strategy instead will be to decompose BunG (X) into locally closed substacks BunG(X)[P,ν‎] which are more directly amenable to analysis.

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 On the group of unit-valued polynomial functions

2021 ◽  
Author(s):  
Amr Ali Al-Maktry
Keyword(s):  
Commutative Ring ◽  
Semidirect Product ◽  
Polynomial Functions ◽  
Dual Numbers ◽  
Ring Of Integers ◽  
Group Of Units ◽  
Canonical Representations ◽  
Finite Commutative Ring

AbstractLet R be a finite commutative ring. The set $${{\mathcal{F}}}(R)$$ F ( R ) of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units $${{\mathcal{F}}}(R)^\times $$ F ( R ) × is just the set of all unit-valued polynomial functions. We investigate polynomial permutations on $$R[x]/(x^2)=R[\alpha ]$$ R [ x ] / ( x 2 ) = R [ α ] , the ring of dual numbers over R, and show that the group $${\mathcal{P}}_{R}(R[\alpha ])$$ P R ( R [ α ] ) , consisting of those polynomial permutations of $$R[\alpha ]$$ R [ α ] represented by polynomials in R[x], is embedded in a semidirect product of $${{\mathcal{F}}}(R)^\times $$ F ( R ) × by the group $${\mathcal{P}}(R)$$ P ( R ) of polynomial permutations on R. In particular, when $$R={\mathbb{F}}_q$$ R = F q , we prove that $${\mathcal{P}}_{{\mathbb{F}}_q}({\mathbb{F}}_q[\alpha ])\cong {\mathcal{P}}({\mathbb{F}}_q) \ltimes _\theta {{\mathcal{F}}}({\mathbb{F}}_q)^\times $$ P F q ( F q [ α ] ) ≅ P ( F q ) ⋉ θ F ( F q ) × . Furthermore, we count unit-valued polynomial functions on the ring of integers modulo $${p^n}$$ p n and obtain canonical representations for these functions.

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.

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