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$\DeclareMathOperator\Hom{Hom}$I am reading Introduction to affine group schemes by Waterhouse. In the second chapter he defines a closed embedding in the following way. Let $G = \Hom_k(A,-)$ and $H'= \Hom_k(B',-)$ be affine group schemes and $\Psi : H' \to G $ be a homomorphism. $\Psi$ is called a closed embedding if the corresponding algebra map $\phi :A \to B'$ is surjective.

It is easy to see that when $\phi$ is surjective then for any $k$-algebra $R$ the homomorphism $\Psi_R : H'(R) \to G(R)$ is injective. But I am not sure about the converse.

A natural definition of "embedding" could have been the homomorphism $\Psi_R : H'(R) \to G(R)$ is injective for all $k$-algebras $R$ but the author has defined otherwise. Is there any example where each $\Psi_R$ is injective but the corresponding map $A \to B'$ is not surjective? May be this is the reason for the definition given in Waterhouse?

Note: Waterhouse does not consider $A$, $B'$ to be finitely generated $k$-algebras.

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    $\begingroup$ I think your algebra map is the wrong way around: a map of schemes from $H'$ to $G$ should be a map of algebras from $A$ to $B'$, not $B'$ to $A$. \\ I don't know an example off the top of my head, but note that a closed embedding is not just an injection on the functorial level, but an isomorphism of $H'$ with a closed subgroup of $G$. $\endgroup$
    – LSpice
    Commented Sep 17 at 12:03
  • $\begingroup$ Yes yes, I messed up while writing. Edited now $\endgroup$
    – Melon_Musk
    Commented Sep 17 at 12:44
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    $\begingroup$ Perhaps if $B' = S^{-1}A$ is a localization of $A$, then the canonical map $\phi: A \to B'$ is an epimorphism of commutative rings (but not a surjective ring homomorphism), and then the corresponding map of affine schemes is a monomorphism? At any rate, this question could be relevant: What do epimorphisms of (commutative) rings look like? $\endgroup$
    – Christopher Drupieski
    Commented Sep 18 at 2:07
  • $\begingroup$ The main reason for the terminology is that this is the definition for affine schemes. One would not want a different meaning for affine group schemes. $\endgroup$
    – Wilberd van der Kallen
    Commented Sep 18 at 7:35
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    $\begingroup$ @Melon_Musk Just delete 'group' and replace 'homomorphism' with 'morphism'. $\endgroup$
    – Wilberd van der Kallen
    Commented Sep 19 at 6:40

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