0
votes
1answer
32 views

For a finite etale map $X\rightarrow T$ of degree $d$, and a $U$-point $t\in T(U)$, are there at most $d$ points in $X(U)$ lying over $t$?

If $f : X\rightarrow T$ is a finite etale morphism of connected schemes, and $U$ is another connected scheme and we're given a map $t : U\rightarrow T$, then must it be true that there are at most $d$ ...
1
vote
1answer
59 views

Action of $\mathbb Z_2$

Is there a connection between Artin-Schreier theorem on finite groups which can be absolute Galois groups and the classification of finite groups freely acting on even-dimensional sphere? The former ...
1
vote
1answer
16 views

Square of normal covering splits

Concerning Galois theory, let $A/B$ be a separable extension. Then $$A/B - \text{normal} \Leftrightarrow A \otimes_B A=A \oplus \cdots \oplus A,$$ where the sum has $n$ summands. Is the same correct ...
0
votes
1answer
27 views

Finite coverings are closed.

I'm working on solving as many of the exercises in Lenstra's Galois Theory for Schemes as possible, but there is one problem I'm partially stuck on. The statement of the problem is: ...
1
vote
1answer
55 views

Monomorphisms and epimorphisms in the category of finite coverings of a topological space

I'm working my way through Lenstra's Galois Theory for Schemes, and I've run into a bit of a conundrum with Exercise 3.14(b). In this exercise, we consider the category $\textbf{FC}_X$ of finite ...
2
votes
0answers
51 views

Are there Galois covers of curves branched at 1 point?

Let $G$ be a finite group, not necessarily abelian. Is there any smooth algebraic curve $C$, with an action of $G$ on $C$, such that the natural quotient map $C \to C/G$ is branched at precisely one ...
3
votes
1answer
115 views

Are these two notions of Galois morphism the same

Let $f:X\to Y$ be a finite morphism of integral schemes. Let $G$ be the automorphism group of $X$ over $Y$. Are the following two conditions equivalent? The function field extension $K(Y)\subset ...