The Conjugates of Algebraic Schemes

Details The Conjugates of Algebraic Schemes The Conjugates of Algebraic Schemes

2007-02-16

arxiv.org/abs/math/0702493v4

Mathematics

Algebraic Geometry

A reduced and refined version. 26 Pages

Scientific paper

Fixed an algebraic scheme $Y$. We suggest a definition for the conjugate of an algebraic scheme $X$ over $Y$ in an evident manner; then $X$ is said to be Galois closed over $Y$ if $X$ has a unique conjugate over $Y$. Now let $X$ and $Y$ both be integral and let $X$ be Galois closed over $Y$ by a surjective morphism $\phi$ of finite type. Then $\phi^{\sharp}(k(Y))$ is a subfield of $k(X)$ by $\phi$. The main theorem of this paper says that $k(X) /\phi^{\sharp}(k(Y)) $ is a Galois extension and the Galois group $Gal(k(X)/\phi^{\sharp}(k(Y))) $ is isomorphic to the group of $k-$automorphisms of $X$ over $Y$.

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