Lemma 37.45.6. Let $f : X \to S$ be a morphism of schemes which is flat, locally of finite presentation, and locally quasi-finite. Let $g \in \Gamma (X, \mathcal{O}_ X)$ nonzero. Then there exist an open $V \subset X$ such that $g|_ V \not= 0$, an open $U \subset S$ fitting into a commutative diagram
\[ \xymatrix{ V \ar[r] \ar[d]_\pi & X \ar[d]^ f \\ U \ar[r] & S, } \]
a quasi-coherent subsheaf $\mathcal{F} \subset \mathcal{O}_ U$, an integer $r > 0$, and an injective $\mathcal{O}_ U$-module map $\mathcal{F}^{\oplus r} \to \pi _*\mathcal{O}_ V$ whose image contains $g|_ V$.
Proof.
We may assume $X$ and $S$ affine. We obtain a filtration $\emptyset = Z_{-1} \subset Z_0 \subset Z_1 \subset Z_2 \subset \ldots \subset Z_ n = S$ as in Lemmas 37.45.3 and 37.45.4. Let $T \subset X$ be the scheme theoretic support of the finite $\mathcal{O}_ X$-module $\mathop{\mathrm{Im}}(g : \mathcal{O}_ X \to \mathcal{O}_ X)$. Note that $T$ is the support of $g$ as a section of $\mathcal{O}_ X$ (Modules, Definition 17.5.1) and for any open $V \subset X$ we have $g|_ V \not= 0$ if and only if $V \cap T \not= \emptyset $. Let $r$ be the smallest integer such that $f(T) \subset Z_ r$ set theoretically. Let $\xi \in T$ be a generic point of an irreducible component of $T$ such that $f(\xi ) \not\in Z_{r - 1}$ (and hence $f(\xi ) \in Z_ r$). We may replace $S$ by an affine neighbourhood of $f(\xi )$ contained in $S \setminus Z_{r - 1}$. Write $S = \mathop{\mathrm{Spec}}(A)$ and let $I = (a_1, \ldots , a_ m) \subset A$ be a finitely generated ideal such that $V(I) = Z_ r$ (set theoretically, see Algebra, Lemma 10.29.1). Since the support of $g$ is contained in $f^{-1}V(I)$ by our choice of $r$ we see that there exists an integer $N$ such that $a_ j^ N g = 0$ for $j = 1, \ldots , m$. Replacing $a_ j$ by $a_ j^ r$ we may assume that $Ig = 0$. For any $A$-module $M$ write $M[I]$ for the $I$-torsion of $M$, i.e., $M[I] = \{ m \in M \mid Im = 0\} $. Write $X = \mathop{\mathrm{Spec}}(B)$, so $g \in B[I]$. Since $A \to B$ is flat we see that
\[ B[I] = A[I] \otimes _ A B \cong A[I] \otimes _{A/I} B/IB \]
By our choice of $Z_ r$, the $A/I$-module $B/IB$ is finite locally free of rank $r$. Hence after replacing $S$ by a smaller affine open neighbourhood of $f(\xi )$ we may assume that $B/IB \cong (A/IA)^{\oplus r}$ as $A/I$-modules. Choose a map $\psi : A^{\oplus r} \to B$ which reduces modulo $I$ to the isomorphism of the previous sentence. Then we see that the induced map
\[ A[I]^{\oplus r} \longrightarrow B[I] \]
is an isomorphism. The lemma follows by taking $\mathcal{F}$ the quasi-coherent sheaf associated to the $A$-module $A[I]$ and the map $\mathcal{F}^{\oplus r} \to \pi _*\mathcal{O}_ V$ the one corresponding to $A[I]^{\oplus r} \subset A^{\oplus r} \to B$.
$\square$
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