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I need a reference for the following statement:

Let $G$ be a linear algebraic group over algebraically closed field $k.$ Let $V$ be a finite dimensional $G$-module. Then $V$ is subrepresentation of $k[G]^n$ for some $n$ where $k[G]$ is coordinate ring of $G.$

I could find this statement in Steinberg's lecture notes on "Conjugacy Classes in Algebraic groups" but am not happy with the proof there.

Thanks in advance.

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    $\begingroup$ Do you like better pp. 1 and 2 of this seminar talk by Grothendieck (séminaire Chevalley, 1956)? The proof seems very clear to me. $\endgroup$
    – abx
    Commented Aug 21, 2015 at 8:58
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    $\begingroup$ The proof is short enough to fit in a comment: For any $v \in V$ and $w \in V^{\ast}$, the function $\langle w, gv \rangle$ is a regular function on $G$. Letting $v$ vary, this gives a map $V \to k[G]$, which is a map of $G$-reps for the right action of $G$ on itself. This map might not be injective but, taking $w_1$, ..., $w_n$ a basis of $V^{\ast}$, we get an injection $V \to k[G]^n$. $\endgroup$
    – David E Speyer
    Commented Aug 21, 2015 at 13:51
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    $\begingroup$ As abx and David point out, this is a basic (early) lemma in the Chevalley development of affine algebraic groups. It's found in textbooks by Borel et al., and generalizes well to affine group schemes as Scott Carnahan indicates. The traditional language involves "representative functions" as developed by Chevalley, Hochschild, and others. (The notes of Steinberg's Tata lectures were written up by Deodhar, then a student there, but could have used some editing.) $\endgroup$
    – Jim Humphreys
    Commented Aug 21, 2015 at 17:26

1 Answer 1

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This is the unique Lemma in section 3.5 of Waterhouse's "Introduction to Affine Group Schemes". It only requires that $G$ be an affine group scheme over a field.

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