pullback of global sections with respect to an automorphism of schemes

Question

Let $X$ be a projective scheme and $\sigma:X\to X$ an automorphism of $X$. Is there a natural pullback of global sections map $H^0(X,\mathcal{F}) \to H^0(X,\sigma^*\mathcal{F})$ for $\mathcal{O}_X$-modules $\mathcal{F}$? If it exist, then how to describe? Thanks a lot.

Answer 1

Yes, For any section $s\in H^0(X,\mathcal F)$ it associate the section $\sigma^*(s)=s\circ \sigma\in H^0(X,\sigma^*\mathcal F)$.

Answer 2

For any morphism $\sigma: Y\to X$ and any sheaf $\mathscr F$ there is a natural morphism of sheaves $\mathscr F\to \sigma_*\sigma^*\mathscr F$. (You can get this just by chasing the definition or see pretty much any book discussing sheaves, for instance [Hartshorne]). Taking global sections give you a natural morphism (using the fact that $\sigma^{-1}(X)=Y$): $H^0(X,\mathscr F)\to H^0(X,\sigma_*\sigma^*\mathscr F)=H^0(Y,\sigma^*\mathscr F)$. In your case $Y=X$.