# Questions tagged [group-schemes]

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

128 questions
Filter by
Sorted by
Tagged with
61 views

### 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 $R$,...
20 views

21 views

### 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 ...
47 views

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 , we may and do assume $S = Spec \ k$, ...  refers ...
22 views

### 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 $U \ni p_1, p_2$....
17 views

### 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 ...
56 views

### 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.
47 views

### $p$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 $n$...
93 views

43 views

69 views

### 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 ...
35 views

### 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 ...
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 ...
166 views

64 views

48 views

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 ...
74 views

### 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 ...
67 views

### 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 ($k$...
165 views

### 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 ...
81 views

### 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 ...
325 views

### 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 ...
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....
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 ...