All Questions
Tagged with hopf-algebras group-schemes
25
questions
0
votes
0
answers
50
views
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".
...
0
votes
0
answers
27
views
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 \...
1
vote
0
answers
43
views
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 ...
- 11
4
votes
2
answers
195
views
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 ...
- 6,033
1
vote
0
answers
55
views
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 ...
2
votes
0
answers
55
views
For an algebraic group $G$, a $G$-module induces a $Dist(G)$-module
I'm reading Representation of Algebraic Groups of Jantzen about Distribution of Algebra. In chapter 7, page 103, 7.11, the author is stating that if $G$ is a group scheme over $k$, then any $G$-module ...
- 4,845
1
vote
1
answer
173
views
Character group of diagonalizable group scheme
Suppose $k$ is an integral domain, $\Lambda$ is a commutative group, $k[\Lambda]$ is the corresponding group ring and $\text{Diag}(\Lambda) = \text{Spec}(k[\Lambda])$ is the diagonalizable group ...
- 4,845
2
votes
1
answer
157
views
Lie operator is left exact
In page 190, part (c) of the book "Algebraic groups, the theory of group schemes of finite type over a field" of Milne, there's a part stating that:
An exact sequence of algebraic groups $e ...
- 4,845
0
votes
1
answer
46
views
Is the unit section of a finite flat commutative group scheme determined by sending group-like elements to 1?
Let $R$ be a ring and $\mathop{\mathrm{Spec}}A$ be a finite flat commutative group scheme over $\mathop{\mathrm{Spec}}R$ so that the theory of Cartier duality applies.
Denote $s:R\to A,\ m:A\to A\...
- 1,785
1
vote
1
answer
115
views
Exercise 2.11 of Waterhouse about Hopf algebra
I asked part (c) of this exercise already in this link Categorization of Group Scheme of rank 2. But I still have some difficulty solving next parts of this exercise. So I hope anyone can help me to ...
- 4,845
2
votes
1
answer
101
views
Categorization of Group Scheme of rank 2
I got stuck on this Exercise 2.11 in the book Introduction to Affine Group Scheme of Waterhouse. I really appreciate if anyone can give me a hint.
Let $A$ be a Hopf algebra over $k$ (a base ring) ...
- 4,845
2
votes
1
answer
130
views
Prove that $\Delta(x) = x \otimes1 + 1 \otimes x$ mod $I \otimes I $ for all $x \in I$.
I got stuck on this problem, so if anyone can give me a hint on this, I really appreciate.
Let $I$ be the augmentation ideal in Hopf algebra $A$. Prove that $\Delta(x) = x \otimes1 + 1 \otimes x$
mod $...
- 4,845
1
vote
0
answers
100
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 $R$,...
- 573
3
votes
0
answers
70
views
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 ...
- 2,161
1
vote
1
answer
194
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 ...
- 651