All Questions
Tagged with hopf-algebras commutative-algebra
15
questions
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
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
2
votes
1
answer
142
views
Automorphism of tensor product of Hopf algebras
Let $A$ be a commutative Hopf algebra over the commutative ring $R$, that is we have comultiplication $\Delta:A\to A\otimes_{R}A$, counit $\epsilon: A\to R$ and coinverse (or antipode) $S: A\to A$. I ...
- 304
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
6
votes
1
answer
61
views
Commutativity up to scalar implies commutativity in an algebra
Let $A$ be a (not necessarily commutative) algebra over a field $k$. Suppose that for all $a,b\in A$, we have $kab=kba$, i.e. commutativity up to scalar. Show that then $A$ is commutative.
In the ...
- 1,279
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
0
answers
62
views
Prove that reduced symmetric algebra is graded
I've tried to prove that the reduced symmetric algebra is graded as an algebra and a coalgebra. $V$ is a vector space on a field of characteristic $p$. $T(V)$ is the tensor algebra.
$s(V):=\frac{T(V)}...
- 130
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
2
votes
0
answers
68
views
Why is the ideal in the definition of the derived subgroup of a linear algebraic group a Hopf ideal?
Let $G$ be a linear algebraic group over a field $k$, i. e. a functor from the category of $k$-algebras to the category of groups. Let $A = k[G]$ be the coordinate ring of $G$.
The definition of the ...
- 5,814
4
votes
1
answer
330
views
What are the natural surjections in the proof of Hopf's classification theorem?
I am currently reading Hatcher's book, trying to understand the proof of Hopf's classification Theorem on Hopf algebras that says the following:
Every Hopf algebra $A$ that is commutative and ...
- 2,555
9
votes
1
answer
681
views
$\mathbb{G}_m$ action on $\operatorname{Spec}A$ is equivalent to grading... what does coassociativity do?
I can interpret the other axioms for a grading via the definition of a group scheme action appropriately (I think):
Let $H = k[x,x^{-1}]$. We are given a coaction $ \rho : A \to A \otimes H$, which is ...
- 20.9k
4
votes
0
answers
306
views
The completion of the ring of Laurent polynomials with respect to the augmentation ideal.
Let $A = \langle a\rangle$ be an infinite cyclic group on one generator. I'm trying to understand the completion $\widehat{\mathbb{Q}}A$ of the group algebra $\mathbb{Q}A$ with respect to the ...
- 1,439