All Questions
33
questions
1
vote
0
answers
126
views
A Weierstrass product theorem for invertible formal Laurent series over local Artinian rings?
Let $(A,\mathfrak{m},\kappa)$ denote a commutative local Artinian ring. Somewhat by accident, I've stumbled across the following interesting decomposition:
$$
A(\!(t)\!)^\times = t^\mathbb{Z} \cdot (1 ...
- 6,996
3
votes
1
answer
238
views
(non)reduced stabilizer scheme
A well known open question is whether the scheme of commuting pairs in a complex reductive group $G$, for example in $G=GL(n)$, is reduced. The variety of commuting pairs is a special case of a more ...
- 1,526
0
votes
0
answers
111
views
Krull dimension of ring of invariants
Let $A$ be a $K$-algebra for some local number field $K$, and denote by $\dim A$ its Krull dimension. Let $G$ be an algebraic group defined over $\text{Spec}K$, and assume $G$ acts on $A$ by $K$-...
- 2,113
1
vote
0
answers
121
views
Confusion regarding change of variable and irreducibility
Let $\mathbb{K}$ be an algebraically closed field of characteristics zero. Let $X$ be an irreducible affine variety, with a rational action of a linearly reductive algebraic group $G$. Also, assume ...
- 767
1
vote
0
answers
146
views
Software for computing invariant rings
I have an linearly reductive algebraic group $G$ acting regularly on an affine variety $X$(over an algebraically closed field of characteristic 0). I want to compute the invariant ring $\mathbb{K}[X]^{...
- 767
6
votes
0
answers
176
views
Computing the automorphism scheme of projective space
$\newcommand{\Spec}{\operatorname{Spec}}$I'm trying to understand why $PGL_{n}$ is the automorphism scheme of $\mathbb{P}^{n-1}_{\mathbb{Z}}$.
In Conrad's Reductive Group Schemes, the following ...
- 565
4
votes
2
answers
532
views
Are algebraic groups over algebraically closed fields Cohen–Macaulay?
$\DeclareMathOperator\CM{CM}\DeclareMathOperator\Spec{Spec}$Let $k$ be an algebraically closed field and let $G$ be an algebraic group over $k$.
My question: is $G$ Cohen–Macaulay? If not, are there ...
- 119
5
votes
1
answer
460
views
Is there a non-split algebraic torus (over a finite field) satisfying the following properties?
Is there a non-split algebraic torus $T$ (over a finite field $\mathbb{F}_{\!q}$) satisfying the following properties?
$T$ is not $\mathbb{F}_{\!q}$-isomorphic to the direct product of algebraic tori ...
- 1,813
3
votes
0
answers
148
views
Finite commutative group schemes whose exponent coincides with its rank
In group theory, a finite commutative group $G$ contains an element whose order is the exponent of $G$. Thus, If the exponent of $G$ is the same as the order of $G$, it must be that $G$ is cyclic. ...
- 329
1
vote
1
answer
123
views
Koszul complex of equations defining a stabilizer
Very specific question. We work over $\mathbb{C}$, although really just want alg. closed of char. 0.
Suppose that $G$ is an algebraic group and $V$ is a finite-dimensional $G$-module, meaning that we ...
- 1,077
14
votes
0
answers
812
views
What goes wrong with this alternate proof of Dirichlet's Theorem?
I had an idea for an alternate proof of Dirichlet's theorem, but something goes wrong. Dirichlet's theorem on primes in arithmetic progression says that for $ m,a \in \mathbb{N} $ which are ...
- 782
3
votes
1
answer
228
views
Why is this $ \mathbb{G}_{a} $ bundle trivial
Please tell me why the following example of a principal $ \mathbb{G}_{a} $-bundle over an affine ring is trivial. Let $ \{x_{1},x_{2},x_{3}\} $ a basis of $ \mathbf{V}^{\ast} $, $ c_{1}(t),c_{2}(t) $ ...
- 782
2
votes
0
answers
80
views
Reference request: additive basis of $\mathbb{C}[N]$
Let $N$ be the maximal unipotent subgroup of $SL_k$. I think that the following is an additive basis of $\mathbb{C}[N]$:
$$\{ e_T: T \text{ is a semi-standard Young tableau with at most $k-1$ rows and ...
- 6,141
2
votes
1
answer
241
views
Proving that finite, connected group schemes in characteristic 0 are trivial
I have a question about a specific proof that all finite group schemes in characteristic 0 are etale. The proof is here, Proposition 8 in lecture notes by Andrew Snowden.
In his notation, let $A = k\...
- 7,716
1
vote
0
answers
147
views
Formal group as a limit of its finite subgroups
I'm reading Manin's article on formal groups and I have a problem with Lemma 1.1.
Consider $k$ a prefect ring of characteristic $p$ and $(A,m,k)$ a noetherian complete local ring of the same ...
- 1,053