Global generation by the evaluation map

Let $X$ be a scheme and let $F$ be an ${\mathcal{O}}_X$-module, for instance a quasi-coherent sheaf (Quasi-coherent module on a scheme). The evaluation map of $F$ is the morphism of ${\mathcal{O}}_X$-modules ${\mathrm{ev}}_F:\mathrm{\Gamma }(X,F){\otimes }_{\mathbb{Z}}{\mathcal{O}}_X⟶F,f\otimes g⟼g\cdot f{\mathrm{\mid }}_U\text{ on }U\subseteq X,$ where the tensor product is taken over $\mathbb{Z}$ because the global sections $\mathrm{\Gamma }(X,F)$ are only an abelian group in general and the sheaf ${\mathcal{O}}_X$ carries the ${\mathcal{O}}_X$-module structure. Concretely, on an open $U$ the map sends a finite sum ${\sum }_i{f}_i\otimes g_i$ with $f_i\in \mathrm{\Gamma }(X,F)$, $g_i\in \mathrm{\Gamma }(U,{\mathcal{O}}_X)$ to ${\sum }_i{g}_i\cdot f_i{\mathrm{\mid }}_U$. The sheaf $F$ is globally generated when ${\mathrm{ev}}_F$ is surjective. Equivalently, for every point $x\in X$ the images of the global sections generate the stalk $F_x$ as an ${\mathcal{O}}_{X,x}$-module: passing to stalks, $({\mathrm{ev}}_F{)}_x$ is the map $\mathrm{\Gamma }(X,F){\otimes }_{\mathbb{Z}}{\mathcal{O}}_{X,x}\to F_x$, and a morphism of sheaves is surjective exactly when it is surjective on every stalk. For an invertible sheaf $L$ the condition takes the following form (Invertible sheaves): if there are finitely many global sections $s_0,\dots ,s_n\in \mathrm{\Gamma }(X,L)$ such that the induced morphism ${\mathcal{O}}_X^{n+1}⟶L,(g_0,\dots ,g_n)⟼{\sum }_i{g}_i{s}_i,$ is surjective, then $L$ is globally generated, since this morphism factors the evaluation map. The converse holds when $X$ is quasi-compact: see the Remarks.

• Finiteness on a quasi-compact scheme. Let $L$ be invertible on a quasi-compact scheme $X$ and suppose $L$ is globally generated. Then there exist finitely many global sections $s_0,\dots ,s_n$ of $L$ with ${\mathcal{O}}_X^{n+1}\to L$ surjective. For each global section $t$, let $X_t$ be the locus where its germ generates $L$. In a local frame, $t$ has coefficient $f$, and this locus is where $f_x$ is a unit. A germ inverse extends to a neighbourhood and its product with $f$ equals $1$ after shrinking, so $X_t$ is open. These opens, indexed by all global sections, cover $X$: if every section had coefficient in the maximal ideal at some $x$, their germs could not generate the free rank-one stalk $L_x$. Quasi-compactness gives a finite subcover $X_{s_0},\dots ,X_{s_n}$, and the corresponding sections generate at every stalk. This proves the asserted surjectivity. If $X$ is empty, take the single zero section. The argument uses the local freeness of $L$, not a general openness assertion for surjectivity loci.
• Terminology. The phrase generated by global sections is used interchangeably with globally generated.
• Relation to ampleness. Ampleness, defined at Absolute ampleness by affine section opens, is not the same condition as global generation, in either direction. The interaction is a theorem, not a convention: under the Axiom of Choice (The Axiom of Choice), on a Noetherian scheme an invertible sheaf is ample exactly when every coherent sheaf twisted by a sufficiently high power is globally generated, so in particular high powers of an ample invertible sheaf are globally generated; the precise statement is Serre global-generation criterion for ampleness.