All Questions
16 questions with no upvoted or accepted answers
14
votes
0
answers
821
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
6
votes
0
answers
190
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 ...
- 605
3
votes
0
answers
150
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
3
votes
0
answers
119
views
Finite generation of the module of invariant vector fields
Let $G$ be a linear algebraic group (not necessarily reductive) and let
$X$ be an affine variety with a regular $G$-action (everything defined over the field of complex numbers $\mathbb{C}$). Denote ...
- 413
3
votes
1
answer
230
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 $\mathbb{C}[N]$:
$$\{ e_T: T \text{ is a semi-standard Young tableau with at most $k-1$ rows and ...
- 6,201
2
votes
0
answers
290
views
tangent space to a (not necessarily algebraic/Lie/..) group
Are there some standard ways to define the tangent space to a group $G$ at its unit element, $e$, when the group is not (pro)algebraic/(pro)Lie, not necessarily over a field, does not have the ``...
- 2,232
2
votes
0
answers
97
views
Can we write an element in a super Grassmannian as a pair of matrices?
Super Grassmannians are introduced by Manin, see for example.
Elements in a grassmannian can be written as matrices, see for example.
Can we write an element in a super Grassmannian as a pair of ...
- 6,201
1
vote
0
answers
127
views
A Weierstrass product theorem for invertible formal Laurent series over local Artinian rings?
Let $(A,\mathfrak{m},\kappa)$ denote a commutative local Artinian ring. Somewhat by accident, I've stumbled across the following interesting decomposition:
$$
A(\!(t)\!)^\times = t^\mathbb{Z} \cdot (1 ...
- 7,127
1
vote
0
answers
125
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 ...
- 839
1
vote
0
answers
154
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]^{...
- 839
1
vote
0
answers
150
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 ...
- 1,093
1
vote
0
answers
189
views
Bijective correspondence between $\mathbb G_a$ actions on affine varieties and exponential maps on affine $k$-domains
Let $A$ be an integral domain which is a finitely generated algebra over an algebraically closed field $k$. Let $\phi :A \to A^{[1]}$ be a $k$-algebra homomorphism and let us write $\phi_t : A \to A[...
user111524
1
vote
0
answers
151
views
Generators of the same degree in a graded ring and GIT quotient
Let $S$ be a graded ring i.e. $S_d\cdot S_e \subset S_{d+e}$ where $S_d$ is the degree $d$ part of $S$. Assume $S_{<0}$ vanishes. For simplicity, you may think of $S$ as a graded subring of $\...
- 2,789
1
vote
0
answers
218
views
How to check that an ideal of $\mathbb{C}[GL_n]$ is a coideal or not?
Let $I$ be an ideal of $\mathbb{C}[GL_n]$. Are there effective methods or software to check whether $I$ is a coideal or not? Thank you very much.
For example, let I be the ideal of $\mathbb{C}[GL_3]$ ...
- 6,201
0
votes
0
answers
124
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 $K$-...
- 2,907