Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Galois Closure of Essentially Finite Morphisms   Let $X$ be a reduced connected $k$-scheme pointed at a rational point $x \in
X(k)$. By using tannakian techniques we construct the Galois closure of an
essentially finite $k$-morphism $f:Y\to X$ satisfying the condition
$H^0(Y,\mathcal{O}_Y)=k$; this Galois closure is a torsor $p:\hat{X}_Y\to X$
dominating $f$ by an $X$-morphism $\lambda:\hat{X}_Y\to Y$ and universal for
this property. Moreover we show that $\lambda:\hat{X}_Y\to Y$ is a torsor under
some finite group scheme we describe. Furthermore we prove that the direct
image of an essentially finite vector bundle over $Y$ is still an essentially
finite vector bundle over $X$. We develop for torsors and essentially finite
morphisms a Galois correspondence similar to the usual one. As an application
we show that for any pointed torsor $f:Y \to X$ under a finite group scheme
satisfying the condition $H^0(Y,\mathcal{O}_Y)=k$, $Y$ has a fundamental group
scheme $\pi_1 (Y,y)$ fitting in a short exact sequence with $\pi_1 (X,x)$.

31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
30
The categories of this paper are: q-alg
35
The categories of this paper are: mtrl-th
37
The categories of this paper are: supr-con
24
The categories of this paper are: dg-ga
20
The categories of this paper are: acc-phys
14
The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom
32
The categories of this paper are: atom-ph
25
The categories of this paper are: patt-sol
29
The categories of this paper are: funct-an
33
The categories of this paper are: plasm-ph
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg