Questions tagged [group-schemes]

A group scheme over a scheme S is simply a group object in the category of schemes over S.

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When is $(\mathbf{Z}/n\mathbf{Z})_R$ self-dual?

Let $R$ be a Noetherian ring, and consider the constant $R$-group scheme $G = (\mathbf{Z}/n\mathbf{Z})_R$. I know that the Cartier dual of $G$ is the corresponding diagonalizable group scheme over $R$,...
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Why is $\mathbb{Z}/n\mathbb{Z}$ étale?

Let $S=\operatorname{Spec}(R)$ a scheme with $R$ un commutative ring. Let $G:= (\mathbb{Z}/n\mathbb{Z})_S$ the constant group scheme associated to the abstract group $\mathbb{Z}/n\mathbb{Z}$. We have $...
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Neutral element product group scheme

Let $G=Spec(A)$ be an affine group scheme over $Spec(k)$ with $k$ a field. The neutral of $G$ is an element $e\in |G|$, which is the image of the zero ideal by the continuous map: $|Spec(k)|\...
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Group schemes for beginners

Which book/notes would you advise for someone who knows (a little) scheme theory but not group scheme theory ? I am looking for some notes where everything is explained, and where there are lots of ...
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finite free group scheme of prime power order unramified

Let $p \in \mathbb{P}$ be a prime and $k$ a field where $\operatorname{char} k\neq p$. equivalently, $p \in k^*$. Let $G:= \operatorname{Spec}A$ be an affine finite group scheme over $k$ of order $p^...
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Dimension of algebraic group and morphism

Let $G$ and $G'$ be two affine connected algebraic groups. Let $f: G\rightarrow G'$ be an epimorphisme, etale and finite morphism of algebraic groups. Why do we have $dim(G)=dim(G')$?
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Classification of the algebraic affine smooth group schemes of dimension 1

How to prove that all algebraic affine smooth group schemes of dimension 1 are the additive group and the multiplicative one ?
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44 views

Simply connected algebraic group and quotient

If everything is well define, does it exist a result like : (maybe some hypothesis are missing) "Let $G$ be a simply connected algebraic group over a field $k$ and $N$ a normal algebraic subgroup of $...
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52 views

Result on étaleness of Group schemes

I have a question about a proof from Arithmetic Geometry (edited by Cornell & Silverman) on page 51: The proof starts with "According to [6], we may and do assume $S = Spec \ k$, ... [6] refers ...
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Affine open for arbitrarily chosen 2 points on a scheme [duplicate]

Let $X$ be a reduced scheme. If there are arbitrarily chosen two points $p_1, p_2 \in X$, does the following always hold? Q. There exists some affine open neighbourhood $U$ such that $U \ni p_1, p_2$....
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The property of infinitesimally flat for a scheme

There is a little calculus in the book "Representaions of Algebraic Groups" of Jantzen (p 98) that I don't really understand. Let $X=Spec(A)$, $X'=Spec(B)$ be two affine schemes over $Spec(k)$, where ...
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Why is a $k$-group scheme separated? [closed]

Let $G$ be a group scheme over a field $k$. Why is the unit section $e: Spec(k) \rightarrow G$ a closed immersion ? I know someone asked that on this forum but it's still not clear for me.
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$p$th roots of unity in a characteristic $p$ field

Let $\mu_n$ denote the group scheme of $n$-th roots of unity over a field $k$. Let $p$ be the characteristic of $k$. I've read that if $(n,p) = 1$, $\mu_n$ is the discrete group isomorphic to the $n$...
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Proof verification: The symplectic group scheme is smooth over $\mathbb Z$

Edit: I finally found the missing steps in my proof and understood my mistakes, see the comments and answer below. This question is now solved. Let $n\geq 1$ and consider $G$ the group scheme over $\...
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Group schemes, group law and base change

The following is a confusion I can't get rid of concerning group-schemes. Let $G$ be a group scheme over some base $S$: for any scheme $T$ over $S$, the set of $T$-valued points $G(T)$ has a group ...
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When $\phi_{\mathcal L}=0$ for $\mathcal L$ a line bundle over an abelian scheme $X/S$

Let $X\rightarrow S$ be a projective abelian scheme. To a line bundle $\mathcal L$ on $X$, we associate its Mumford line bundle $\Lambda(\mathcal L):= \mu^{\star}\mathcal L\otimes p_1^{\star}\mathcal ...
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Is the Poincaré sheaf symmetric?

The following discussion is based on the content of FGA explained about the Picard scheme. This is mostly formal: I am trying to find a good way to think about the Poincaré sheaf. Let us consider $\...
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Quotients of $\mathbb{P}^1 \times \mathbb{P}^1$

It is known that $\mathbb{P}^1 \times \mathbb{P}^1 \not \cong \mathbb{P}^2$. One way to see this is working with their respective class groups, $\mathbb{Z}^2$ and $\mathbb{Z}$, which in this case, ...
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Is multiplication by $n$ always an isogeny on an abelian scheme?

Given $n$ a non-zero integer, it is known that multiplication by $n$ on an abelian variety (defined over any field $k$) is an isogeny. The proof of this fact uses the existence of an ample symmetric ...
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Are open $\mathbb{G}_m$-invariant subschemes of an affine scheme precisely the homogeneous radical ideals of its coordinate ring?

I have a bunch of questions regarding actions of a group scheme $G$ on a scheme $X$. I'm fine with assuming $G$ affine. $\newcommand{\IG}{\mathbb{G}} \newcommand{\pmo}{{\pm 1}} \newcommand{\IZ}{\...
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About the connected-étale sequence for finite group schemes over a complete noetherian local ring

Let $R$ be a complete noetherian local ring and $G$ be a finite group scheme over $R$ (that is a group scheme over $\operatorname{Spec}(R)$ whose structure morphism is finite locally free, or ...
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1answer
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Confused about the axioms of $p$-divisible groups applied to ordinary abelian groups

Edit: Actually, I just mixed up indices in my thinking. This was not a deep question/problem at all. I have started reading Tate's paper about $p$-divisible group. The definition given is the ...
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90 views

Reference for functorial point of view in algebraic geometry

I've studied a little bit of scheme theory and category theory. I'd like to understand better what is the functorial point of view in modern algebraic geometry and how it is related to the theory of ...
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166 views

Quotient of group schemes

Is the following sequence of affine group schemes over $k \in \mathsf{CRing}$ exact, or even for that matter, is there a meaningful notion of quotient of (group) schemes: $$0 \rightarrow H \rightarrow ...
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Proof of “A geometric quotient is categorical”

I'm reading Geometric Invariant Theory by Mumford-Fogarty, but I can't understand some details in the proof that any geometric quotient is categorical. Let $\sigma$ be an action of $G/S$ on $X/S$ ...
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Quotient of abelian variety by finite group scheme

Let $X$ be an abelian variety over a field $k$. Let $G$ be a finite group scheme over $k$ acting on $X$ and denote $X/G$ to be the geometric quotient of $X$ by $G$, which always exists in this case. ...
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Classification of finite affine group schemes of order $2$

Assume $G=\text{Spec}A$ is a group scheme of order $2$ over a ring $R$, I think it is well-known that these group schemes are $G_{a,b}=\text{Spec}\frac{R[T]}{T^2+aT}$ where $ab=2, a,b\in R$, and the ...
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Example of commutative Hopf algebra over the integers

I'm looking for an example of a commutative Hopf algebra $H$ such that $H$ is a torsion free $\mathbb{Z}$ module of finite rank $H$ is not isomorphic to the dual of a group algebra $\mathbb{Z}G$ $H ...
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Mumford's Geometric Invariant Theory proof of proposition 0.1

I'm reading now D. Mumford's Geometric Invariant Theorem and have faced an argument in the proof of proposition 0.1 on page 4 that I don't understand: main setting: let $X/S$ a $S$-scheme and a group ...
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if an Affine group scheme G isn't solvable then G(R) isn't solvable for some R

I want to do exercise 10.1 of Waterhouse book which says that if an affine group scheme over a field isn't solvable then for some ring R it's R-point isn't solvable. Waterhouse defines commutator ...
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Is every $\mathcal O$-structure an $\mathcal O$-group scheme?

Let $\mathcal O$ be a principal ideal domain with field of fractions $F$. Let $G = \operatorname{Spec} A$ be a linear algebraic group over $F$, with comultiplication map $d: A \rightarrow A \otimes_F ...
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Question on isogeny of abelian varieties and principal homogeneous spaces

I know that for an abelian variety $A$ over an algebraically closed field $k=\overline k$ and an ample line bundle $L$ on $A$, one can define an isogeny $\phi_L \colon A \rightarrow \mathrm{Pic}^0(A)$ ...
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1answer
52 views

Morphism of affine group schemes gives a morphism of hopf-algebras? (But not by Yoneda I guess?)

An affine group scheme $G$ over the field $k$ is a functor $$G:Alg_k\to Grp$$ that is isomorphic (as a functor) to a functor $$h^A=hom_{alg_k}(A,-),$$ where $A$ is a hopf-algebra (which gives the ...
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Normality of group schemes

I am interested in calculating etale cohomology of group schemes. It would be useful to know when certain group schemes are normal, i.e., when their local rings at every point are integrally closed ...
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Grading equivalent to action of multiplicative group scheme

Consider the group scheme $\mathbb{G}_m = \mathrm{Spec}(\mathbb{Z}[t, t^{-1}])$. An action of $\mathbb{G}_m$ on an affine scheme $X = \mathrm{Spec}(A)$ is a morphism of schemes $$ \sigma: \mathbb{G}_m ...
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1answer
64 views

finite group scheme over char p, question about proof in Shatz book

Let $k$ be a field of characteristic $p>0$, and let $G$ be a finite connected group scheme over $k$. Let $FG$ be the frobenius twist of $G$, i.e. $FG=G \times_{\mathrm{Spec}\ k, Fr} \mathrm{Spec}\ ...
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146 views

Kernel of an algebra map and module of Kahler Differentials

Let $A$ be a $k$-algebra, $f:A\rightarrow k$ an algebra map with kernel $I$. I'd like to prove that $\Omega_A\otimes_f k $ is canonically isomorphic to $I/I^2$. This is from W.C.Waterhouse Intro to ...
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142 views

Kahler differentials of a Hopf Algebra

Let $A$ be a $k$-Hopf Algebra over some ring $k$, with augmentation ideal $J_A=$ ker $(\epsilon:A\rightarrow k)$ I would like to prove that the module of Khaler Differentials $\Omega_A$ of $A$ over $...
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Geometric invariant theory Affine Quotients

Let $V=Spec(R[V])$ an affine variey and $G$ a linear reductive group action on $V$. (I'm not sure if the linear reductive condition is neccessary for the question). According to GIT the "quotient" $V/...
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1answer
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Affine Group Scheme Definition

From what I understand, a affine group scheme $G$ should be an affine scheme on which there exists a group structure in the sense that $$ \phi: G \times_k G \to G $$ is also a morphism of groups. ...
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76 views

Representations of $\mathbb{G}_m$

I know that the multiplicative affine group scheme $\mathbb{G}_m$ is diagonalizable, since the algebra that represents it is $k[X,X^{-1}]$, which is isomorphic to the group algebra $k[\mathbb{Z}]$. ...
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Characters of $\mathbb{G}_m$

Fix a field $k$. Let $\mathbb{G}_m$ be the multiplicative affine group scheme over $k$. A $k-$character $\chi$ of $\mathbb{G}_m$ is an endomorphism of affine group schemes $\mathbb{G}_m \to \mathbb{G}...
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1answer
120 views

Galois Action on underlying Topological space of a Group Scheme

I have a question arising from the answer of following thread: https://mathoverflow.net/questions/324887/why-is-for-a-group-scheme-of-finite-type-smooth-resp-irreducible-equivale Let $G/K$ be a group ...
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Action on Group Scheme

I have question which arises from the answer of my former thread: https://mathoverflow.net/questions/324887/why-is-for-a-group-scheme-of-finite-type-smooth-resp-irreducible-equivale Let $G/K$ be a ...
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1answer
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Definition of Functor in Waterhouse

On page 21 of Waterhouse's Introduction to Affine Group Schemes he defines (on objects, which are $k$-algebras $R$) a functor $GL_V(R) = \mathrm{Aut}_R(V\otimes R)$ where $V$ is a fixed $k$-module ($k$...
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If $G$ is a smooth scheme over $S$ of characteristic $p$, is the relative Frobenius morphism $F_{G/S}$ faithfully flat?

Let $G$ be a smooth scheme over $S$ of characteristic $p$, do we have that the reltaive Frobenius morphism $F_{G/S}$ is faithfully flat? There is an excersice in Liu's book saying that this is true ...
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1answer
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closed embedding is open iff ideal sheaf idempotent

Let $f: \mathrm{Spec}(R) \to \mathrm{Spec}(A)$ be a closed embedding of affine Noetherian schemes, given by the ideal $I = \mathrm{ker}(A \twoheadrightarrow R)$. If $I = I^2$, then it's not too hard ...
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Different notions of torsors in algebraic geometry

In what follows $X$ will be a scheme and $G$ a group scheme. In the examples I will take $X=\mathbb{P}^1_k$ and $G=\mathbb{G}_{m}$. When reading about "the torsor..." I found many ...
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Weil group and conjugacy classes of cocharacters

We have the following setup: $G$ a compact reductive group over $\mathbb{R}$, $T$ a maximal Torus, $U^1$ the real algebraic Torus and $W$ the Weil group of $T$ In Variétiés de Shimura Lemma 1.2.4 p....
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Base Change of Algebraic Group

I have some questions about the steps in the proof of COROLLARY 1.35 from Milne's "Algebraic Groups : The theory of group schemes of finite type over a field"(p. 17). Here the excerpt: Let $G$ be a ...