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Unanswered Questions

248,985 questions with no upvoted or accepted answers
649
votes
0answers
35k views

A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
337
votes
0answers
12k views

The Ring Game on $K[x,y,z]$

I recently read about the Ring Game on MathOverflow, and have been trying to determine winning strategies for each player on various rings. The game has two players and begins with a commutative ...
324
votes
0answers
10k views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
185
votes
0answers
10k views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ $K_1=\left(\...
150
votes
0answers
3k views

Is there a categorical definition of submetry?

(Updated to include effective epimorphism.) This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use ...
112
votes
0answers
1k views

What functions can be made continuous by “mixing up their domain”?

Definition. A function $f:\Bbb R\to\Bbb R$ will be called potentially continuous if there is a bijection $\phi:\Bbb R\to\Bbb R$ so that $f\circ \phi$ is continuous. So one could say a potentially ...
110
votes
0answers
6k views

Is this really a categorical approach to integration?

Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A Categorical ...
99
votes
0answers
3k views

Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?

A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization of the groups in which ...
95
votes
0answers
2k views

Application of Hilbert's basis theorem in representation theory

In Smalo: Degenerations of Representations of Associative Algebras, Milan J. Math., 2008 there is an application of Hilbert's basis theorem that I don't understand: Two orders are defined on the set ...
87
votes
0answers
2k views

What is the Picard group of $z^3=y(y^2-x^2)(x-1)$?

I'm actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the ...
77
votes
0answers
1k views

Does every ring of integers sit inside a ring of integers that has a power basis?

Given a finite extension of the rationals, $K$, we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n,$$ ...
72
votes
1answer
1k views

Ring structure on the Galois group of a finite field

Let $F$ be a finite field. There is an isomorphism of topological groups $\left(\mathrm{Gal}(\overline{F}/F),\circ\right) \cong (\widehat{\mathbb{Z}},+)$. It follows that the Galois group carries the ...
66
votes
0answers
4k views

Pullback and Pushforward Isomorphism of Sheaves

Suppose we have two schemes $X, Y$ and a map $f\colon X\to Y$. Then we know that $\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where $\...
65
votes
0answers
1k views

Sorting of prime gaps

Let $g_i $ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we re-arrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$ the gaps are arranged from smallest to largest we have a new sequence ...
65
votes
0answers
1k views

Tate conjecture for Fermat varieties

I've been looking at Tate's Algebraic Cycles and Poles of Zeta Functions (hard to find online... Google books outline here) and have a question about his work on (conjecturing!) the Tate conjecture ...

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