I am reading the proof of Theorem II.7.6 in Hartshorne's "Algebraic Geometry", which states:
Let $X$ be a scheme of finite type over a noetherian ring $A$, and $\mathscr{L}$ be an invertible sheaf on $X$. Then $\mathscr{L}$ is ample if and only if $\mathscr{L}^m$ is very ample over $\text{Spec} \; A$ for some $m > 0$.
In the proof of the direction $(\Rightarrow)$, they let $Y$ be a certain closed subscheme of $X$ and $\mathscr{I}_Y$ be its ideal sheaf, which is a coherent sheaf. Then by the ampleness of $\mathscr{L}$, we conclude that $\mathscr{I}_Y \otimes \mathscr{L}^n$ is globally generated, for some $n > 0$. Then the proof states:
Now $\mathscr{I}_Y \otimes \mathscr{L}^n$ is a subsheaf of $\mathscr{L}^n$, so we can think of $s$ as an element of $\Gamma(X,\mathscr{L}^n)$
where $s \in \Gamma(X,\mathscr{I}_Y \otimes \mathscr{L}^n)$.
I don't understand how this statement is true. If you had a tensor product of sheaves $\mathscr{F} \otimes \mathscr{G}$, is there some way to see it as a subsheaf of its factors $\mathscr{F}$ and $\mathscr{G}$? Or we are in some special situation involving $\mathscr{I}_Y$?
Best Answer
You're correct, this is something unique going on here - this remark relies on a couple special features of our situation.
$\mathcal{I}_Y$ injects in to $\mathcal{O}_X$ and $\mathcal{L}$ is flat (being a line bundle), so we get an injection $\mathcal{I}_Y\otimes\mathcal{L}^n\to\mathcal{O}_X\otimes\mathcal{L}^n$. As $\mathcal{O}_X\otimes_{\mathcal{O}_X}\mathcal{L}^n\cong\mathcal{L}^n$, we actually have an injection $\mathcal{I}_Y\otimes\mathcal{L}^n\to\mathcal{L}^n$, and taking global sections we get an injection $\Gamma(X,\mathcal{I}_Y\otimes\mathcal{L}^n)\to\Gamma(X,\mathcal{L}^n)$. That's how we're thinking of $s$ as an element of $\Gamma(X,\mathcal{L}^n)$.