On the structure of commutative affine group schemes over a non-perfect field

1975 ◽  
Vol 16 (2) ◽  
pp. 101-136 ◽  
Author(s):  
Mitsuhiro Takeuchi
Keyword(s):  
Affine Group ◽  
Perfect Field ◽  
Group Schemes

scholarly journals Affine group schemes with integrals

1972 ◽  
Vol 22 (3) ◽  
pp. 546-558 ◽  
Author(s):  
John Brendan Sullivan
Keyword(s):  
Affine Group ◽  
Group Schemes

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 One-dimensional affine group schemes

1980 ◽  
Vol 66 (2) ◽  
pp. 550-568 ◽  
Author(s):  
William C Waterhouse ◽  
Boris Weisfeiler
Keyword(s):  
Affine Group ◽  
One Dimensional ◽  
Group Schemes

scholarly journals On the Structure of Affine Flat Group Schemes Over Discrete Valuation Rings, II

2020 ◽  
Author(s):  
Phùng Hô Hai ◽  
João Pedro dos Santos
Keyword(s):  
Discrete Valuation Ring ◽  
Free Module ◽  
Galois Groups ◽  
Affine Group ◽  
Group Schemes ◽  
Infinite Type ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
D Modules ◽  
Tannakian Categories

Abstract In the first part of this work [ 12], we studied affine group schemes over a discrete valuation ring (DVR) by means of Neron blowups. We also showed how to apply these findings to throw light on the group schemes coming from Tannakian categories of $\mathcal{D}$-modules. In the present work, we follow up this theme. We show that a certain class of affine group schemes of “infinite type,” Neron blowups of formal subgroups, are quite typical. We also explain how these group schemes appear naturally in Tannakian categories of $\mathcal{D}$-modules. To conclude, we isolate a Tannakian property of affine group schemes, named prudence, which allows one to verify if the underlying ring of functions is a free module over the base ring. This is then successfully applied to obtain a general result on the structure of differential Galois groups over complete DVRs.

scholarly journals HOPF PAIRINGS AND (CO)INDUCTION FUNCTORS OVER COMMUTATIVE RINGS

2005 ◽  
Vol 04 (04) ◽  
pp. 369-404 ◽  
Author(s):  
JAWAD Y. ABUHLAIL
Keyword(s):  
Algebraic Groups ◽  
Commutative Rings ◽  
Affine Group ◽  
Group Schemes

(Co)induction functors appear in several areas of Algebra in different forms. Interesting examples are the so called induction functors in the Theory of Affine Algebraic Groups. In this paper we investigate Hopf pairings (bialgebra pairings) and use them to study (co)induction functors for affine group schemes over arbitrary commutative ground rings. We present also a special type of Hopf pairings (bialgebra pairings) satisfying the so called α-condition. For those pairings the coinduction functor is studied and nice descriptions of it are obtained. Along the paper several interesting results are generalized from the case of base fields to the case of arbitrary commutative (Noetherian) ground rings.

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.

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