# Property of representations of reductive group schemes over characteristic 0 field

I originally posted this on Maths SE, but then I thought it MO might be more fitting.

Let $$k$$ be a characteristic $$0$$ field and let $$G$$ be a linear algebraic group scheme over $$k$$. Then is it true that $$G$$ is reductive if and only if the category of representations of $$G$$, $$\mathsf{Rep}(G)$$ is semi-simple ?

P/s: $$\mathsf{Rep}(\mathcal{G})$$ is defined as the functor category $$[\mathcal{G}, \mathcal{C}]$$, where $$\mathcal{G}$$ is a groupoid and $$\mathcal{C}$$ is some category.

• Theorem 22.42 (jmilne.org/math/Books/iag.html) – Eoin Dec 14 '19 at 22:44
• @Eoin Would you mind expanding on the relevance of this theorem ? My background in representation theory isn't particularly strong. Thank you. – Dat Minh Ha Dec 14 '19 at 23:15
• Compare with 9.54. It answers your question in the affirmative. – Eoin Dec 15 '19 at 0:42
• @eoin Thank you. – Dat Minh Ha Dec 15 '19 at 0:55