# For an algebraic group $G$ acting on a scheme $X$, $H^0(X,F)$ of a $G$-linearized $\mathcal{O}_X$-module $F$ has the structure of a $G$-representation

Reading Huybrechts & Lehn's "The geometry of moduli spaces of sheaves" I am stuck with a particular statement that they make in the chapter on GIT without explanation. Namely, they say that for an algebraic $k$-group $G$ acting on a $k$-scheme of finite type $X$, the space of global sections of a $G$-linearized quasi-coherent $\mathcal{O}_X$-module $F$ naturally has the structure of a $G$-representation.

I give here the definition of $G$-linearization, it is essentialy the same one as given by Mumford in GIT: Let $G$ be an algebraic $k$-group, $X$ a $k$-scheme of finite type, and $\sigma : X \times G \rightarrow X$ a group action. Let $F$ be a quasi-coherent $\mathcal{O}_X$-module. Then a $G$-linearization of $F$ is an isomorphism of $\mathcal{O}_{G \times X}$-modules $\Phi : \sigma^* F \rightarrow p_1^*F$, where $p_1$ is the projection $X \times G \rightarrow X$, such that the following cocycle condition is sattisfied: $$(id_X \times \mu)^*\Phi = p^*_{12}\Phi \circ (\sigma \times id_G)^*\Phi$$ where $p_{12} : X \times G \times G \rightarrow X \times G$ is the projection onto the first two factors.

I don't see how to use the isomorphism $\Phi$ to define a group action of $G$ on $H^0(X,F)$, nor how to use the cocycle condition to show that this action must be linear. Any help would be great, thanks.

{Definition} Let $G$ be an algebraic group and $X$ be a $G$-variety. Let $L\in Pic(X)$ be an invertible sheaf. A $G$-linearization of $L$ consists of an isomorphism $$\phi:\sigma^{\ast}L\simeq p_{2}^{\ast}L$$ that satisfies the following co-cycle condition: There are three maps $$G\times G\times X\rightarrow G \times X$$ given by $1_{G}\times \sigma$, $p_{23}$ and $\mu\times 1_{X}.$ We then require $$(\mu\times 1_{X})^{\ast}\phi= (p_{23})^{\ast}\phi\circ(1_{G}\times \sigma)^{\ast}\phi.$$ \end{definition}

Let's establish G-module struture bit by bit.

Lemma: Let $X$ be a $G$-variety and $\mathscr{L}\in Pic(X)$ be a $G$-linearized invertible sheaf. Then there is a commutative diagram Firure 1 Moreover, $\alpha$ and $\beta$ are isomorphisms. \end{lemma}

Proof Given $a\in \Gamma(G)$ and $\tau\in\Gamma(X,\mathscr{L})$, define $$\alpha(a,\tau)=p_{1}^{\sharp}(a)\cdot p_{2}^{\ast}(\tau)$$ and given $a\otimes b\in \Gamma(G\times G)$ and $\tau^{\prime}\in \Gamma(X,\mathscr{L})$, define $$\beta(a\otimes b\otimes\tau^{\prime})=p_{12}^{\sharp}(a\otimes b)\cdot p_{3}^{\ast}(\tau^{\prime}).$$ Then observe

1. $p_{3}=p_{2}\circ(\mu\times 1_{X}):G\times G\times X\rightarrow X$
2. $p_{1}\circ(\mu\times 1_{X})=\mu\circ p_{12}:G\times G\times X\rightarrow G$
3. Give $f:X\rightarrow Y$ a morphism of schemes and $\mathscr{F}$ as sheaf of $\mathscr{O}_{Y}$-modules and a section $s\in\Gamma(Y,\mathscr{F})$, the pull back $f^{\ast}:\Gamma(Y,\mathscr{F})\rightarrow\Gamma(X,f^{\ast}\mathscr{F})$ is a morphism of $\Gamma(Y,\mathscr{O}_{Y})$-modules. Here the $\Gamma(Y,\mathscr{O}_{Y})$-module structure of $\Gamma(X,f^{\ast}\mathscr{F})$ is induced by the map $\mathscr{O}_{Y}\rightarrow f_{\ast}\mathscr{O}_{X}$. \end{enumerate} The commutativity of the diagram follows immediately from these three observations. That $\alpha$ and $\beta$ are isomorphisms follows from flat base change. \end{proof}

Similarly,

There is a commutative diagram Figure 2

Let me also note that pull back of sheaf is a functor, hence if $$(\mu\times 1_{X})^{\ast}_{p_{2}}:=\Gamma(G\times X,\mathscr{p_{2}}^{\ast}\mathscr{L})\rightarrow \Gamma(G\times G\times X,(\mu\times 1_{X})^{\ast}p_{2}^{\ast}\mathscr{L})$$ and $$(\mu\times 1_{X})^{\ast}_{\sigma}:=\Gamma(G\times X,\sigma^{\ast}\mathscr{L})\rightarrow\Gamma(G\times G\times X,(\mu\times 1_{X})^{\ast}\sigma^{\ast}\mathscr{L}),$$ then $$\phi\circ(\mu\times 1_{X})^{\ast}_{p_{2}}=(\mu\times 1_{X})^{\ast}(\phi)\circ(\mu\times 1_{X})^{\ast}_{\sigma}.$$ Same property holds for $(1_{G}\times\sigma).$ With the notations, we get the Theorem: Let $X$ be a $G$-variety. Then for any $G$-lineraization $\mathscr{L}\in Pic(X)$, $\Gamma(X,\mathscr{L})$ is a $G$-module. \end{theorem}

Proof} Define $\hat{\sigma}:\Gamma(X,\mathscr{L})\rightarrow\Gamma(G)\otimes\Gamma(X,\mathscr{L})$ by the composition $$\hat{\sigma}:=\alpha^{-1}\phi\sigma^{\ast}.$$ Note that the bottom row in Figure 2 is just $(1\otimes\hat{\sigma})$. Then \begin{equation*} \begin{split} (1\otimes \hat{\sigma})\circ\hat{\sigma} =&(\beta^{-1}p_{23}^{\ast}(\phi)(1_{G}\times\sigma)^{\ast}_{p_{2}}\alpha)\circ(\alpha^{-1}\phi\sigma^{\ast})\\ =&\beta^{-1}p_{23}^{\ast}(\phi)(1_{G}\times \sigma)^{\ast}_{p_{2}}\phi\sigma^{\ast}\\ =&\beta^{-1}p_{23}^{\ast}(\phi)(1_{G}\times \sigma)^{\ast}(\phi)(1_{G}\times \sigma)^{\ast}_{\sigma}\phi^{-1}\phi\sigma^{\ast}\\ =&\beta^{-1}(\mu\times 1_{X})^{\ast}(\phi)(1_{G}\times \sigma)^{\ast}_{\sigma}\sigma^{\ast}\\ =&\beta^{-1}(\mu\times 1_{X})^{\ast}(\phi)(\mu\times 1_{X})^{\ast}_{\sigma}\sigma^{\ast} \end{split} \end{equation*}

On the other hand, we have

\begin{equation*} \begin{split} (\mu^{\sharp}\otimes 1)\circ\hat{\sigma}=& (\beta^{-1}(\mu\times 1_{X})^{\ast}_{p_{2}}\alpha)\alpha^{-1}\phi\sigma^{\ast}\\ =&\beta^{-1}(\mu\times 1_{X})^{\ast}_{p_{2}}\phi\sigma^{\ast}\\ =&\beta^{-1}(\mu\times 1_{X})^{\ast}(\phi)(\mu\times 1_{X})^{\ast}_{\sigma}\phi^{-1}\phi\sigma^{\ast}\\ =&\beta^{-1}(\mu\times 1_{X})^{\ast}(\phi)(\mu\times 1_{X})^{\ast}_{\sigma}\sigma^{\ast} \end{split} \end{equation*} We see that $(1\otimes\hat{\sigma})\circ\hat{\sigma}=(\mu^{\sharp}\otimes 1)\circ\hat{\sigma}$. \end{proof}