Ample Cartier divisors and coherent sheaves

Question

my question regards the definition of ample Cartier divisors.

An invertible sheaf $\mathcal{L}$ on a noetherian scheme $X$ is said to be ample if for every coherent sheaf $\mathcal{F}$ on $X$, there exists an integer $n_{0}> 0$ such that for every integer $n\geq n_{0}$, the sheaf $\mathcal{F} \otimes \mathcal{L}^{\otimes n}$ is generated by its global sections. We say that a Cartier divisor $D$ is ample if the corresponding invertible sheaf $\mathcal{L}(D)$ is ample.

In my paper I only work with schemes that correspond to projective varieties over an algebraically closed field $k$. I would like to formulate the definitons of ample invertible sheaves and ample Cartier divisors without using coherent sheaves. Is that possible? If so, another question arises. Consider the following theorem due to Serre:

Let X be a projective scheme over a notherian Ring $A$, let $\mathcal{O}_{X}(1)$ be a very ample invertible sheaf on $X$, and let $\mathcal{F}$ be a coherent $\mathcal{O}_{X}$-module. Then there is an integer $n_{0}$ such that for all $n\geq n_{0}$, the sheaf $\mathcal{F}(n):= \mathcal{F}\otimes \mathcal{O}_{X}(n)$ can be generated by a finite number of global sections.

If $X$ is a scheme corresponding to a projective variety over an algebraically closed field $k$, can we rewrite Serre's theorem in a way that we'll be able to show that very ample sheaves are ample, but without using coherent sheaves? Bibliographical references are also welcome.

Thank you.

Answer 1

For a noetherian scheme $X$, an invertible sheaf $\mathcal L$ is ample if there exist $n\ge 1$ and sections $s_1, \dots, s_m\in \mathcal L^{\otimes n}(X)$ such that

• $X_{s_i}:=\{ x\in X \mid \mathcal L_{x}=(s_i)_x\mathcal O_{X,x} \}$ is affine for all $i\le m$;

• $X=\cup_i X_{s_i}$.

See EGA II.4.5.2(a') or the definition in Stacks projects. They do not use coherent sheaves.

For a projective scheme $X$ over $A$, $\mathcal O_X(1)$ is given by a closed immersion $$i: X\to P=\mathrm{Proj}(A[T_O, \dots, T_N])$$ with $\mathcal O_X(1)=i^*\mathcal O_P(1)$. For any $i\le N$, let $s_i$ be the image in $H^0(X, \mathcal O_X(1))$ of $T_i\in H^0(P, \mathcal O_P(1))$, then $$X_{s_i}=i^{-1}(P_{T_i})=i^{-1}(D_+(T_i))$$ is affine and $X=\cup_i X_{s_i}$. Thus $\mathcal O_X(1)$ is ample with the above definition.