Sorry if this is sort of a soft question. I added an example at the end to mitigate. So I hope you can bear with me here.
Let $\phi : X \rightarrow Y$ be a morphism of noetherian schemes, $\mathscr F$ a coherent sheaf on $X$. I am looking to better understand $\phi^*\phi_*\mathscr F$, or the natural map $\phi^*\phi_*\mathscr F \rightarrow \mathscr F$.
I want to collect interesting or useful facts, that one ought to be aware of. No matter if trivial(-ish) or rather specific, I want to collect 'em all. Provide either proofs or references, both are fine.
Assumptions might, among others, include:
- $\phi_*O_X = O_Y$
- $\phi$ is birational
- $\mathscr F$ is generated by global sections (gbgs)
- ...
Answers might, among others, concern any of the following:
- the morphism being injective / surjective / an isomorphism
- the $\phi$-movable part of a divisor $D$, which is defined by $M=$ Im$\{\phi^*\phi_*\mathscr L(D) \rightarrow \mathscr L(D)\}$ (e.g. the motivation behind the definition)
- ...
One example statement: [Hartshorne III.8.8]: $f:X\rightarrow Y$ a projective morphism of noetherin schemes, with $O_X(1)$ very ample, and $\mathscr F$ coherent on $X$. Then $\forall n>>0, f^*f_*\mathscr F (n) \rightarrow \mathscr F(n)$ is surjective. (Because $\mathscr F(n)$ is gbgs.)
Now the promised concrete example question: consider two birational morphisms $f:X\rightarrow Y$, $f': X'\rightarrow Y$ of normal projective varieties. Let $V$ be a common resolution of $X$ and $X'$, with morphisms $s$ and $s'$ respectively. Suppose we have an $l$ such that $\mathscr L(lK_{X'})$ is very ample, and $\mathscr L (lK_X)$ is Cartier. (E.g. take $f'$ to be the flip of $f$, in which case we assume both morphisms to be small, $-K_X$ and $K_{X'}$ to be $f$- and $f'$-ample, respectively, and $\rho(X/Y)=\rho(X'/Y)=1$.) Then we write $s^*\mathscr L (lK_X) = \mathscr L (M) \otimes \mathscr L (F)$, where $$\mathscr L (M) = Im\{(f\circ s)^*(f\circ s)_*s^*\mathscr L (lK_X)\rightarrow s^* \mathscr L (lK_X)\}$$ is the $(f\circ s)$-movable part.
We have $$ \mathscr L (M) = Im\{(f'\circ s')^* f'_* \mathscr L (lK_{X'}) \rightarrow s^*\mathscr L (lK_X)\}, \tag{1}$$
where I would want to show: $$= Im\{(f'\circ s')^* f'_* \mathscr L (lK_{X'}) \rightarrow s'^*\mathscr L (lK_{X'})\} ( = s'^*\mathscr L (lK_{X'}) \subset s^*\mathscr L (lK_{X})), \tag{2}$$ but I am not certain why the images of the two maps are the same.
Related questions:
[1]https://mathoverflow.net/questions/110866/interpreting-ff
