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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:

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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:

  1. 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 $G_{k^a}$?

  2. 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?

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  1. 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.

  2. Yes.

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