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So I only started learning about group schemes this summer, and I found two approaches. For the record I am interested in affine group schemes $G$ of finite type over a field $k$ (algebraic groups).

Initially I saw their description as group objects in the category of affine schemes over $k$, which is also equivalent to a group object in the category of functors $\mathit{Alg}_k \to Set$ which is representable, which is also equivalent to a Hopf algebra.

When getting deeper into the theory, namely learning about parabolic and Borel subgroups, many texts started working with what I believe is $|G|$, the closed points of $G$ over the algebraic closure $\overline{k}$, sometimes denoted $\mathrm{spm}(G_{\overline{k}})$. This has the structure of a semi-topological group which seems useful in proving facts about $|G|$. My question is, are there general ways to lift results about $|G|$ to results about $G$?

For example, there is a neat result that for a group homomorphism $\phi: G \to H$ the corresponding map $|G| \to |H|$ has closed image, and I vaguely remember that $\mathrm{Im}(\phi)$ itself is closed in $H$ (correct me if I am wrong), and I wonder if you can prove the latter from the first.

What I tried

I became more interested in the question "how much is the functor of points of a scheme determined by its values on (algebraically closed) fields" but I could not find anything online even after a thorough search, so I produced the following result, which may still be wrong.

Proposition: Let $F,G: X \to Y$ be distinct morphisms of schemes with $X$ reduced with finitely many irreducible components. Then there is a field $L$ so that the functor of points morphisms $F(L),G(L): X(L) \to Y(L)$ are distinct.

Here we can obviously replace $L$ with an algebraically closed field. This would be helpful in lifting any result which can be stated in terms of a comparison between two morphisms.

Disclaimer: This may have been better to post on stackexchange, and I haven't posted any questions in years on either site. However I wanted the opinion of someone who has worked with algebraic groups for a while.

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    $\begingroup$ There is a comparison between finite type (sometimes assumed separated and/or integral) $k$-schemes and 'varieties' (in the classical sense, i.e. using only the closed points). See for example Hartshorne, Prop. II.4.10. This means that in finite type situations, considering $\bar k$-points is enough in some sense. However, you need more than just a topological language to describe the objects, e.g. $(X\times Y)(\bar k) = X(\bar k) \times Y(\bar k)$ does not have the product topology. Is this the kind of thing you're looking for? $\endgroup$ Jul 8, 2020 at 19:49
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    $\begingroup$ See Milne Algebraic Groups, CUP, 2017, which uses the modern approach and covers the deeper theory (parabolic, Borel subgroups...). $\endgroup$
    – anon
    Jul 8, 2020 at 23:47
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    $\begingroup$ No one should have to fuss with archaic language that, for example, Borel's Linear Algebraic Groups is written. Milne's book contains some hints for translating the archaic language into the language of modern algebraic geometry. $\endgroup$
    – anon
    Jul 9, 2020 at 6:57

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