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Let $G$ be an algebraic group over a field $k$, and $\mathbb{P}(V)$ is a projective space. Then Mumford said in his book Geometric Invariant Theory that there's a equivalence between the set of all dual actions of $G$ on $H^0(\mathbb{P}(V), \mathcal{O}_{\mathbb{P}(V)} (1))$ and the set of all actions of $G$ on $\mathbb{P}(V)$ plus a $G$-linearization of $\mathcal{O}_{\mathbb{P}(V)}(1)$.

This appears in the proof of Proposition 1.7 at the page 35 .

I have no idea on how to prove this statement...

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  • $\begingroup$ What is a coaction? The term does not appear in Mumford's book. $\endgroup$
    – abx
    Commented Nov 3, 2020 at 9:15
  • $\begingroup$ @abx Sorry, I mean the dual action. $\endgroup$
    – Kim
    Commented Nov 4, 2020 at 2:02
  • $\begingroup$ Are you aware that $H^0(\mathbb{P}(V),\mathscr{O}_{\mathbb{P}}(1))=V^*$? That helps... $\endgroup$
    – abx
    Commented Nov 4, 2020 at 7:30
  • $\begingroup$ @abx It seems that $H^0(\mathbb{P}(V), \mathcal{O}(1))=V$ in this book... I add the picture of the proof of Proposition 1.7 in question. $\endgroup$
    – Kim
    Commented Nov 4, 2020 at 9:02
  • $\begingroup$ For any line bundle $L$ on a variety $X$, a $G$-linearization induces a linear action of $G$ on $H^0(X,L)$. Conversely, a linear action of $G$ on $V$ induces an action on $\mathbb{P}(V)$ with a $G$-linearization of $\mathscr{O}_{\mathbb{P}}(1)$. $\endgroup$
    – abx
    Commented Nov 4, 2020 at 12:58

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