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If $G$ is a $k-$ group scheme (seeing as a functor) exist a good definition of what is the derived group scheme? (Or a good reference for a good definition). Where derived I'm talking in the sense of group theory.

PS:We may assume that $G$ is of finite type and smooth.

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    $\begingroup$ Just to be extra clear, by derived group you mean the commutator subgroup? $\endgroup$ Oct 25, 2016 at 16:57
  • $\begingroup$ Yes, the group generated by the commutators in the group theory setting. I have read that if $G'$ is the derived group scheme and $R$ is a $k-$ algebra then $G'(R)\subset G(R)'$ (with $G(R)'$ the derived group in group theory setting) and we have the equality if $R$ is the algebraic closure of $k$. $\endgroup$
    – João Dias
    Oct 25, 2016 at 17:16
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    $\begingroup$ @JoãoDias I do not remember the details, but why can't you just take the scheme-theoretic image of the commutator morphism $G\times G\to G$ and endow it with a group structure? This should work at least when $k$ is of characteristic 0. $\endgroup$ Oct 25, 2016 at 17:28
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    $\begingroup$ Some standard references that you should consult are Demazure and Gabriel's "Groupes Algebriques," Jantzen's "Representations of Algebraic Groups," or Waterhouse's "Introduction to Affine Group Schemes." $\endgroup$ Oct 25, 2016 at 17:43
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    $\begingroup$ The representabiliy of the derived group functor is given in sections 2.3 and 2.4 of Borel's "Linear Algebraic Groups". $\endgroup$
    – Uri Bader
    Oct 25, 2016 at 18:23

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The following quote, from Pseudo-reductive groups by Conrad, Gabber and Prasad (Definition A.1.14, lightly edited) should answer your question.

The derived group D(G) of a smooth group G of finite type over a field k is the unique smooth closed k-subgroup such that (D(G))(K) is the commutator subgroup of G(K) for any algebraically closed extension K/k.

Note that the derived group exists without connectedness hypotheses on G; see [Bo2, I, 2.4] for the affine case and [SGA3, VI B , 7.2(vii), 7.10] for the general case. The formation of D(G) commutes with any extension of the base field, and the quotient map G → G/D(G) is initial among all k-homomorphisms from G to a commutative k-group scheme (see Lemma 5.3.4 for a generalization).

[Bo2] is the same reference to Borel's "Linear Algebraic Groups" that was provided in a comment by Uri Bader.

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  • $\begingroup$ One can also see Demazure-Gabriel's Groupes Algebriques Tome I, chapter II, 5.4.8 (though my French is a bit too weak to be sure what their precise assumptions are). $\endgroup$ Oct 26, 2016 at 8:32

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