I'm reading the proof of the following theorem from Hartshorne (Theorem II.7.17), which says the following:
Let $Z$ be a variety and let $X$ be a quasi projective variety, both over $k$. Suppose $f:Z\to X$ is a projective birational map. Then the pair $(Z,f)$ is isomorphic to a blowup of $X$.
In step 5 of the proof, my understanding is that we want to show (following notation from the book) $\mathscr{M}^{-dn}(f_*\mathscr{L})^d\cong \mathscr{M}^{-dn}f_*\mathscr{L}^d$. I think I understand why we have a surjection from the left to the right - the way everything has been set up yields$f_*\mathscr{L}$ generates $\mathscr{S}:= \mathscr{O}_X \oplus \bigoplus_{d\ge 1}f_*\mathscr{L}^d$ (i.e. the $\mathscr{O}_X$ algebra is generated in degree $1$). Thus there's a surjection from $(f_*\mathscr{L})^d \to f_*\mathscr{L}^d$, which gives the required surjection. However, I don't quite see why it is obvious that the resulting map $\mathscr{M}^{-dn}(f_*\mathscr{L})^d\to \mathscr{M}^{-dn}f_*\mathscr{L}^d$ is an injection too. Hartshorne writes this follows from the fact the fact that these are both thought of as subsheaves of $\mathscr{K}_X$, but I don't quite see why.
To conclude, my questions are:
- Is my reasoning for the surjectivity of the map $\mathscr{M}^{-dn}(f_*\mathscr{L})^d\to \mathscr{M}^{-dn}f_*\mathscr{L}^d$ correct?
- Why is the above map also an injection?
Thanks.
Edit: I think I see what this means. While considering $(f_*\mathscr{L})^d$, we're locally doing the "multiplication" of sections of $f_*\mathscr{L}$ in $K_X$. The fact that $\mathscr{S}$ is generated by $f_*\mathscr{L}$ means that $f_*\mathscr{L}^d$ is just $(f_*\mathscr{L})^d$, in $\mathscr{K}_X$. Is this what's going on?