I have some questions about the steps in the proof of COROLLARY 1.35 from Milne's "Algebraic Groups : The theory of group schemes of
finite type over a field"(p. 17). Here the excerpt:
Let $G$ be a algebraic group over a field $k$. Denote by $G_{k^a}$ the base change (or in other words coefficient extension) of $G$ to the algebraic closure $k^a$ of $k$.
My questions:
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We assume that there is some point of $G$ lies in more than one irreducible component of $G$, why then the same is true for $G_{k^a}$?
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If I understood it correctly then the base change step from $G$ to $G_{k^a}$ is only done in order to apply the homogeneity, since $G$ is only homoegeneous if the base field is algebraically closed. Is this the only reason for it?
Best Answer
Consider a scheme $X$ of finite type over a field $k$. Define the intersection of two closed subschemes of $X$ to be their fibered product. The formation of the intersection commutes with base change. Therefore, if two irreducible components of $X$ have nonempty intersection over $k$, they will have nonempty intersection over its algebraic closure. Of course, they may no longer then be irreducible, but obviously two of their irreducible components will have nonempty intersection.
Yes.