Tensor product of two ample invertible sheaves is ample

Question

Why is the tensor product of two ample invertible sheaves ample when working over a noetherian scheme $X$?

This is what I have tried:

A sheaf $L$ is said to be ample if for every coherent sheaf $F$ there exist $n_0$ such that $F\otimes L^n$ is generated by global sections for all $n\geq n_0$ .

Let $L$ and $M$ be two ample invertible sheaves and let $F$ be a coherent sheaf on $X$. $L$ is ample so there exist $n_0$ such that $F\otimes L^n$ is generated by global sections for all $n\geq n_0$. Similarly there exist $m_0$ corresponding to $M$. I don’t know how to proceed further. Please someone help.

Answer 1

We can combine several bite-size claims to give the full result.

Proposition 1 (Hartshorne II.7.5): Let $L$ be an invertible sheaf on a noetherian scheme $X$. Then the following are equivalent:

1. $L$ is ample;
2. $L^{\otimes m}$ is ample for all $m>0$;
3. $L^{\otimes m}$ is ample for some $m>0$.

Proof: The implications 1 $\Rightarrow$ 2 $\Rightarrow$ 3 are all trivial, so we concentrate on 3 $\Rightarrow$ 1. Given a coherent sheaf $F$ on $X$ and an integer $0\leq a<m$, by the definition of ampleness we have that there exists some $n_a>0$ so that for all $n\geq n_a$, $(F\otimes L^{\otimes a})\otimes(L^{\otimes m})^{\otimes n}$ is generated by global sections. Taking $N=m\cdot\max_a(n_a)$, we have that $F\otimes L^{\otimes n}$ is generated by global sections for all $n\geq N$, and $L$ is ample. $\blacksquare$

Proposition 2: If $M_1$ and $M_2$ are two sheaves that are generated by a finite number of global sections, then their tensor product $M_1\otimes M_2$ is.

Proof: Consider the surjective composite map $\mathcal{O}_X^{\oplus n_1}\otimes \mathcal{O}_X^{\oplus n_2}\to \mathcal{O}_X^{\oplus n_1}\otimes M_2\to M_1\otimes M_2$. $\blacksquare$

Proposition 3: If $L$ is an ample invertible sheaf and $M$ is an invertible sheaf generated by global sections, then $L\otimes M$ is ample.

Proof: If $F$ is a quasi-coherent sheaf so that $F\otimes L^n$ is generated by global sections, then $F\otimes L^{\otimes n}\otimes M^{\otimes n} \cong F\otimes (L\otimes M)^{\otimes n}$ is generated by global sections. $\blacksquare$

Proposition 4: If $L$ and $M$ are ample invertible sheaves on a noetherian scheme $X$, then $L\otimes M$ is ample.

Proof: Find an $n>0$ so that $L\otimes M^{\otimes n}$ is globally generated by the definition of ampleness. Then $L^{\otimes n-1}\otimes (L\otimes M^{\otimes n}) \cong L^{\otimes n}\otimes M^{\otimes n}\cong (L\otimes M)^{\otimes n}$ is ample by proposition 3, as $L^{\otimes n-1}$ is ample by proposition 1 and $L\otimes M^{\otimes n}$ is globally generated. As a sheaf is ample iff some tensor power is by proposition 1, we're done. $\blacksquare$