Questions tagged [group-schemes]

Use this tag for scheme-theoretic and category-theoretic questions about group schemes, as well as those group schemes that are not algebraic groups. A group scheme G over a scheme S is simply a group object in the category of schemes over S. Finite type group schemes over a field are represented by varieties, and considered algebraic groups; for questions specific to algebraic groups use the [algebraic-groups] tag

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Question about a specific argument in the proof of Geometric Satake

Suppose $\tilde{G}_{\mathbb{Z}_p}$ is a flat affine group scheme over $\text{Spec}(\mathbb{Z}_p)$ (the p-adic integers) such that the fiber over the generic point $\tilde{G}_{\mathbb{Q}_p}$ is known ...
I'm Representable's user avatar
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Extensions and formal smoothness

Say $H$ and $K$ are formally smooth group schemes (or, even better, $p$-divisible groups). Can I deduce that that any extension of $H$ by $K$ is formally smooth? It seems like this should be well-...
asking's user avatar
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Translation morphism of Algebraic Groups .

My question is about a point on page 18 of J.S.Milne's "Algebraic Groups- The theory of group schemes of finite type over a field." and specifically about the fact that the translation map ...
Mouthfullofearth's user avatar
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Affine group schemes

I am trying to clarify the definition of affine group schemes. It can be seen as a representable functor from the category of algebras to the category of groups. My question Let $k$ be an ...
Tommk's user avatar
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1 answer
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Is a morphism $f:A\rightarrow B$ uniquely determined by the compositions $f\circ g$ for all $g:T\rightarrow A$?

Im asking in general but to give a little context, i'm studying group schemes and have been trying to prove that the composition morphism $m:G\times G \rightarrow G$ is uniquely determined by the ...
Camilo Gallardo's user avatar
1 vote
2 answers
103 views

Is there a "correct" $k$ scheme structure to put on $\coprod_{i=1}^n \operatorname{Spec}(k)$?

Let $k$ be an algebraically closed field, with $n\in\mathbb N\subset k$ invertible. I am trying to prove that if $\mathbb G_m=k[t,t^{-1}]$ is the multiplicative group scheme over $k$, and $\mu_n$ is ...
Chris's user avatar
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Grouplike Hopf algebras are group rings?

Let $H$ be a commutative and cocommutative Hopf algebra over an algebraically closed field $k$. I've read that if $H$ is grouplike in the sense that it has no nonzero primitive elements, then $H$ is ...
tcamps's user avatar
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Redundancy in the definition of a Toric Variety

So as I have it, a toric variety is a complex variety $X$ with an open embedding of a torus $T^n$ with dense image, and morphism: $$a:X\times_{\mathbb C}T^n\longrightarrow X$$ which extends the ...
Chris's user avatar
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Algebraic Torus is a group scheme

I am taking a course on toric varieties this semester, and I am a little confused by how the algebraic torus is a group scheme, as we didn't really define what a group scheme is. I was given the ...
Chris's user avatar
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Derivatives of morphisms of linear algebraic groups

I am currently trying to learn about linear algebraic groups and their lie algebra structure. However, I am struggling to explicitly calculate the derivatives of morphisms between algebraic groups, as ...
max_121's user avatar
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Is the automorphism group of an $R$-module always a group scheme?

Let $R$ be a commutative ring with identity and $M$ an $R$-module. A group scheme over $R$ is a group object in $\mathrm{Sch}/R$, or equivalently, a group valued functor $G : \mathrm{Sch}/R \to \...
Adelhart's user avatar
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Integrality for group schemes over a discrete valuation ring.

In this note, there is the following theorem: [Theorem 4.4] Let $\mathscr{G}$ be a separated $R$-group scheme of finite type such that its open relative identity component $\mathscr{G}^0$ is semi-...
Phanpu's user avatar
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Why is the Frobenius morphism for Witt rings epimorphism?

In Demazure's book ''lectures on $p$-divisible groups'' page 57, he said the Frobenius morphism $F:W_k\to W_k$ is given by $$F(a_0,\cdots, a_n,\cdots)=(a_0^p,\cdots,a_n^p,\cdots)$$ and in the proof of ...
Phanpu's user avatar
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sequence of group schemes $0\to\underline{\mathbb{Z}/p\mathbb{Z}}\to G\to\mu_p\to0$

Let $p$ be a prime number and $G$ be a group scheme over a field $k$ of characteristic $0$. Assume that we have the following sequence of $0\to\underline{\mathbb{Z}/p\mathbb{Z}}\to G\to\mu_p\to0$. If ...
Jean's user avatar
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Complete resource of Ngô's course notes on Algebraic Groups and Group Schemes

I'm looking for Ngô's M2 course notes on "Groupes algébriques et schémas en groupes". The Wayback Machine has an incomplete capture here. However, it apparently lacks chapter 1, 3, and 5. ...
Modern_Hunter's user avatar
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equivalence of definitions of cartier duality

I am trying to understand the remark on page 20 of these notes. For $G=\operatorname{Spec} A$ a group scheme, Cartier duality is defined there as $G^\vee(B) = \operatorname{Hom}_{B-grp}(G_B, (\mathbb{...
user21560982's user avatar
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Closed subschemes of finite $k$-schemes

in another question (or in Wedhorn's Algebraic Geometry, page 88) it is shown that a subscheme of $k$-schemes of finite type are of finite type again. Does the same hold closed subschemes of finite $k$...
max_121's user avatar
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Hom exact sequence for group representations

in the proof of Proposition II, 3.3.7 (ii) -> (iv) in Demazure-Gabriel "Introduction to Algebraic Geometry and Algebraic Groups", it is stated as obvious that an exact sequence of kG-...
max_121's user avatar
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Flatness & surjectivity for Group Scheme Morphism

I am currently reading https://arxiv.org/abs/math/0703310 and I was wondering why the map $S \to B_SG''$ in proof of proposition 2.7 (c) is faithfully flat. This is as far as I already understood ...
max_121's user avatar
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1 answer
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Is the kernel of a morphism of finite group schemes finite?

is the kernel of homorphism of finite group schemes once again finite? (i.e. if you have $G = Spec A, H = Spec B$ over a field $k$, i.e. $A,B$ finite dimensional $k$-vector spaces, is ker$\varphi$ ...
max_121's user avatar
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1 answer
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Characterisation of Hochschild Cohomology and semi-simple representation

in Demazure-Gabriel "Introduction to Algebraic Geometry and Algebraic Groups" II, §3, 3.7 (Proposition), I do not understand why the implication (iii) -> (i) follows from the result 3.3: ...
max_121's user avatar
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Group schemes of multiplicative type

I am currently trying to understand the notion of a multiplicative group scheme. By Milne (https://www.jmilne.org/math/CourseNotes/AGS.pdf, 5.11) it is a group scheme that becomes diagonalizable over ...
max_121's user avatar
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When $G\subset E_{\overline{K}}$ is invariant by $\operatorname{Gal}(\overline{K}/K)$, can we construct $K$-isogeny s.t. the kernel is $G$?

Sorry for my bad English. In AEC Remark 4.13.2, there is the following proposition Let $K$ be a field, $E$ be an elliptic curve over $K$, and $G\subset E(\overline{K})$ be a finite subgroup. If $G$ ...
Yos's user avatar
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4 votes
1 answer
121 views

Is every finite-dimensional Lie algebra the Lie algebra of a group scheme?

Let $k$ be an algebraically closed field of characteristic $0$. It is a well known fact that there is a finite-dimensional Lie algebra over $k$ which is not the Lie algebra of an affine algebraic $k$-...
Thiago Brevidelli Garcia's user avatar
2 votes
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65 views

Connected affine $p$-divisible group (Cornell/ Silverman's Arithmetic Geometry)

Let $R$ be a commutative complete local ring with residue field of characteristic $p > 0$. A $p$-divisible group over $R$ of height $h$ is an inductive system $(G_{\nu}, i_{\nu})$ for which ...
user267839's user avatar
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Is Category of schemes locally small?

One can use Yoneda lemma in the context of locally small category. due to yoneda lemma, we can consider a scheme X as a functor of points. (like SpecA $\mapsto$ X(A)=Hom(SpecA, X) ). but I don't know ...
Yong's user avatar
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conditions on glueing group scheme

Say let $S=\mathbb{P}^1_k=\mathop{\mathrm{Proj}}k[x_0,x_1]$ be the base scheme where $k$ is a field. Let $\pi:G\to S$ be a morphism of $S$-schemes s.t. $G_i:=G|_{D_+(x_i)}$ is a group scheme over $D_+(...
Z Wu's user avatar
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1 answer
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Is $\mathbb{P}^1$ a group scheme?

It's known that the set of meromorphic functions (functions to $\mathbb{C}\cup\{\infty\}$) on a complex variety $X$ forms a field, called the function field of $X$. Edit: Thanks to the comment by @...
Z Wu's user avatar
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2 votes
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Lifting the connected-étale sequence of a finite group scheme over a residue field

Let $R$ be a complete DVR with fraction field $K$, characteristic $0$ and algebraically closed residue field $k$ of characteristic $p>0$. Suppose $G_{0}$ is a finite flat group scheme over $k$ so ...
David Hubbard's user avatar
9 votes
2 answers
368 views

Is $\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)\cong\mathrm{Spec}(\overline{\mathbb Q}\otimes\overline{\mathbb Q})$ a group scheme

This question gives a homeomorphism $\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)\cong\mathrm{Spec}(\overline{\mathbb Q}\otimes_\mathbb Q\overline{\mathbb Q})$, sending $\sigma\in\mathrm{Gal}(\...
Kenta S's user avatar
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16 votes
1 answer
574 views

Are the roots of unity the only algebraic subgroups of the multiplicative group?

$\newcommand{\G}{\mathbb{G}}$ Let $k$ a field (or maybe more generally an arbitrary ring with connected spectrum), $\G_m$ the multiplicative group over $k$. Are the $\mu_n = \{x^n = 1\}$ the only $k$-...
C.D.'s user avatar
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1 answer
45 views

The names of $\Bbb{G}_a$ and $\Bbb{G}_m$

By wikipedia, over a field $k$, $\Bbb{G}_a=k$ and $\Bbb{G}_m=k^*$ as scheme over $k$. My question is why these group schemes calles "the additive group scheme" and "the multiplicative ...
Or Shahar's user avatar
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1 answer
70 views

Surjection of Group Schemes

In Bjorn Poonen's book, Rational Points on Varieties, page 125, in ``Warning 5.1.19", my understanding of what is stated is that over a field of characteristic $p$, the homomorphism from the ...
user940160's user avatar
3 votes
1 answer
66 views

Two affine group schemes over $k$ with isomorphic group for each $k$-algebra $S$ but not isomorphic as affine group schemes

I'm looking for two affine group schemes $G_1$ and $G_2$ (i.e. functors from (affine group schemes over $k$) to (Group) ) such that for every $k$-algebra $S$, $G_1(S)\cong G_2(S)$, but $G_1$ and $G_2$ ...
Menezio's user avatar
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1 vote
0 answers
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Dieudonné module of $\alpha_{p^2}$

Let $k$ be a perfect field of characteristic $p>0$. I'm trying to calculate the Dieudonné module $D(\alpha_{p^2,k})$ "explicitly". Let $M:=D(\alpha_{p^2})$. Since $\alpha_{p^2}$ is of ...
Nico's user avatar
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1 vote
0 answers
94 views

Does an exact sequence of commutative group schemes define an exact sequence of sheaves?

Suppose we are given an exact sequence $$ 0\to A\to B\to C\to 0$$ of commutative group scheme over a field $k$. Then does the sequence define an exact sequence of étale sheaves on a $k$-scheme $X$? I ...
user393795's user avatar
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0 answers
145 views

Criteria for splitting of exact sequences of flat $R$-algebras

Let $R$ be a complete local integral domain with fraction field $K= \operatorname{Frac}(R)$ and residue field $k=R/m$. Let $$ 0 \to A_1 \to A \to A_2 \to 0 $$ a short exact sequence of finite, flat $R$...
user267839's user avatar
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3 votes
2 answers
125 views

Example of linear algebraic group which has unique rational point

Sorry for my bad English. For a field $k$, I want to find examples of linear algebraic groups over $k$, which has a unique rational point (it is just unit element $e$). Of course trivial group is one ...
Yos's user avatar
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2 votes
1 answer
78 views

Question on isogenies of degree d.

I am trying to understand the following question (Proposition 5.12. in ABELIAN VARIETIES, Bas Edixhoven, Gerard van der Geer, and Ben Moonen) If $f: X \to Y$ is an isogeny of degree $d$ between ...
Khainq's user avatar
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1 vote
1 answer
100 views

Does Repf($G$) have enough injectives?

Let $G$ be a group scheme over a field $k$. My question is Does Repf(G), the category of finite-dimensional linear representations of $G$, have enough injectives? It is well-known that Rep(G), the ...
Khainq's user avatar
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3 votes
0 answers
53 views

Torsion at base change of group schemes

I am slightly confused about how torsion behaves with base change of group schemes. These notes (https://www.math.uni-bonn.de/ag/alggeom/veranstaltungen/Vorlesungen/2015ws_vl_morrow/2015ws_vl_p-div%...
Sofía Marlasca Aparicio's user avatar
2 votes
1 answer
112 views

A subscheme closed under algebraic group operations is an algebraic subgroup

At the very beginning of his book on Algebraic Groups, Milne defines an algebraic group as a group object in the category of finite type $k$-schemes, and an algebraic subgroup as a (locally closed) $k$...
C.D.'s user avatar
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1 vote
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Is Spec $\mathbb{C}[-] $ exact?

I am struggling to find a reference to understand the following fact. Let $0 \to A \to B \to C \to 0$ be a short exact sequence of abelian groups. I first apply the functor $\mathbb{C}[-]$ taking ...
LeaderLasagne's user avatar
3 votes
2 answers
155 views

Structure ring of constant group scheme.

For the finite abelian group $G$, the group scheme $G_{\mathrm{Spec}\,{\Bbb Z}} = {\mathrm{Spec}}\,{\cal O}_G$ over ${\mathrm{Spec}}\,{\Bbb Z}$ is defined as follows$\colon$ $$ {\cal O}_G = {\Bbb Z}...
Pierre MATSUMI's user avatar
2 votes
1 answer
174 views

Recovering an affine group scheme from its representations

I want to understand the proof of recovering an affine group scheme from its category of representations which is presented as proposition 2.8 in Milne's notes here. Firstly I want to understand the ...
bluebird's user avatar
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0 answers
42 views

Why Néron model is a separated scheme?

Let $K$ be a local field and $R$ be ring of integers. Néron model is defined as smooth group scheme over $\operatorname{Spec}R$, whose generic fiber is isomorphic to $E$, and satisfies Néron mapping ...
user avatar
1 vote
1 answer
168 views

Neron model of multiplicative group scheme

Let $R$ be a discrete valuation ring and $K$ be its fraction field. Consider the multiplicative group scheme $\mathbb{G}_m$, as scheme $\mathbb{G}_m=Spec K[T,T^{-1}]$. How can we construct the Néron ...
Desunkid's user avatar
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1 answer
113 views

Can we construct reduced group scheme which is same group structure given an abstract group?

Sorry for my bad English. Let $G$ be an abstract group (if necessary finite), and $k$ be an algebraically closed field. Now is there a group scheme $X$ over $k$ such that group of $k$-valued point $X(...
Yos's user avatar
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1 vote
0 answers
21 views

Why is it better to define $ SO(q) $ using Clifford algebras?

http://math.stanford.edu/~conrad/papers/luminysga3.pdf, appendix C.2 defines $ \mathrm{SO}(q) $ as the kernel of the algebraic group morphism $ D_q: \mathrm{O}(q) \to (\mathbb Z / 2 \mathbb Z)_S $, ...
Tempestas Ludi's user avatar
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28 views

For an algebraic group $G$, elements in $[G,G](R)$ lies in $[(G(R')),(G(R'))]$ for a faithful flat $R'$ over $R$.

Let $G$ be an algebraic group over $k$ which is either smooth or affine. Then the derived group can be defined by the algebraic subgroup generated by the image of the morphism $$G\times G \rightarrow ...
XT Chen's user avatar
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