All Questions
Tagged with hopf-algebras algebraic-geometry
25
questions
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
1
answer
74
views
Which algebraic subvarieties of a group variety have a group structure?
Let $G$ be an algebraic group. Given an algebraic subvariety $X\subseteq G$, is there any way simple criterion to determine whether $X$ has a group structure or not?
For example, when $X$ and $G$ are ...
- 1,260
2
votes
1
answer
146
views
Are there exotic group laws on affine space in characteristic zero?
Let $k$ be a field of characteristic zero, and let $X = \mathbb{A}^n_k$ be affine $n$-space. Up to $k$-rational translation, we may assume that a map $e : \operatorname{Spec} k \to X$ is the inclusion ...
- 1,601
3
votes
0
answers
69
views
When is an image of a Hopf algebra a Hopf algebra?
Suppose $R\subseteq S$ is a flat extension of rings and $A/R$, $B/S$ are flat Hopf algebras. Let $\varphi:A\otimes_R S\to B$ be a surjective $S$-Hopf algebra homomorphism. When is it the case that $\...
- 593
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
5
votes
2
answers
396
views
Distributions of a group scheme as differential operators.
I'm trying to understand the distributions on an affine group scheme act as differential operators. Let $X$ be a scheme and $G$ a group scheme both affine over some commutative ring $k$. Suppose $\...
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
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
2
votes
2
answers
322
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 ...
user364213
4
votes
2
answers
299
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 $...
user364213
1
vote
1
answer
327
views
Antipode of Cocommutative Hopf Algebra
I’m reading about affine group schemes by Waterhouse and in the proof of showing the (Jordan) decomposition of Abelian affine group scheme (equivalently cocommutative Hopf algebra), I came across the ...
4
votes
1
answer
986
views
Hopf Ideal and Normal Hopf ideal
I am trying to understand how to obtain the axioms of an Hopf ideal, which is an ideal inducing a closed subgroup $H$ of an affine group sceme $G$. Let $G$ be represented by the Hopf algebra $A$. ...
- 409