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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Vanishing of Ext involving unipotent group schemes and $\mathbb{G}_m$
I am currently reading Serre's Algebraic Groups and Class Fields. In chapter 8.1.6, Prop. 7 says that if $A$ and $B$ are linear algebraic groups then $H^2_{reg}(A,B)_s=Ext(A,B)$. The definition of $H^...
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Local Action by Group Scheme (Milne's Algebraic Groups)
I have a question about following proof from Milne's book "Algebraic Groups: the theory of group schemes of finite type over a field", Chapter 8, proposition 8.9:
PROPOSITION 8.9. Let $G \...
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How does base change affect group schemes?
I am reading through Waterhouse's Introduction to Affine Group Schemes and am having trouble understanding how I should think of base change as it relates to the way a groups scheme "looks".
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Is there a scheme-theoretic definition for the center of an algebraic group (or group scheme)?
Since this is the case for other usual group-theoretic concepts like normal subgroups or derived subgroup (i.e. commutator), I was wondering if there exists a similar construction for an object ...
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Exercise 3.14b of Waterhouse Affine Group Schemes
I have been working on Exercise 3.14b of Waterhouse Introduction to Affine Group Schemes and want to make sure I am on the right track.
Suppose G is represented by A. Write down the map $\varphi: A \...
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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 ...
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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-...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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