How to show the open sets $X_{s_i}$ cover $X$?

Question

Let $X$ be a scheme and $\mathcal F$ an invertible sheaf on $X$ generated by $s_0,s_1,\cdots,s_n\in \mathcal F(X)$. How to show the open sets $X_{s_i}$ cover $X$?

Answer 1

The $(n+1)$-tuple $(s_0,s_1,\ldots,s_n)$ of global sections of $\mathscr{F}$ correspond to an $\mathscr{O}_X$-module homomorphism $\varphi:\mathscr{O}_X^{n+1}\to\mathscr{F}$ sending the "standard basis vectors" of the source to the $s_i$, and the assumption that the $s_i$ generate $\mathscr{F}$ means that $\varphi$ is surjective.

By definition, for any global section $s$ of the invertible $\mathscr{O}_X$-module $\mathscr{F}$,

$$X_s=\{x\in X:s_x\notin\mathfrak{m}_x\mathscr{F}_x\}\text{.}$$

By Nakayama's lemma, we can equivalently describe $X_s$ as the set of all $x\in X$ such that $s_x$ generates the $\mathscr{O}_{X,x}$-module $\mathscr{F}_x$.

We want to show that the sets $X_{s_i}$ cover $X$, so fix $x\in X$. Since $\varphi_x:\mathscr{O}_{X,x}^{n+1}\to\mathscr{F}_x$ is surjective, the induced $k(x)$-linear map $\overline{\varphi_x}:k(x)^{n+1}\to\mathscr{F}_x/\mathfrak{m}_x\mathscr{F}_x$ is surjective as well. By construction, this last map sends the standard basis vectors of $k(x)^{n+1}$ to the images of the $s_{i,x}$ in $\mathscr{F}_x/\mathfrak{m}_x\mathscr{F}_x$. Thus the $s_{i,x}$ span the one-dimensional $k(x)$-vector space $\mathscr{F}_x/\mathfrak{m}_x\mathscr{F}_x$, which means that for some $j$ with $0\leq j\leq n$, $s_{j,x}$ is nonzero modulo $\mathfrak{m}_x\mathscr{F}_x$. In other words, $s_{j,x}\notin\mathfrak{m}_x\mathscr{F}_x$, so $x\in X_{s_j}$, and we win.