The Stacks project

Lemma 38.4.9. Let $S$, $X$, $\mathcal{F}$, $s$ be as in Definition 38.4.1. Let $(Z, Y, i, \pi , \mathcal{G})$ be a one step dévissage of $\mathcal{F}/X/S$ over $s$. Let $\xi \in Y_ s$ be the (unique) generic point. Then there exists an integer $r > 0$ and an $\mathcal{O}_ Y$-module map

\[ \alpha : \mathcal{O}_ Y^{\oplus r} \longrightarrow \pi _*\mathcal{G} \]

such that

\[ \alpha : \kappa (\xi )^{\oplus r} \longrightarrow (\pi _*\mathcal{G})_\xi \otimes _{\mathcal{O}_{Y, \xi }} \kappa (\xi ) \]

is an isomorphism. Moreover, in this case we have

\[ \dim (\text{Supp}(\mathop{\mathrm{Coker}}(\alpha )_ s)) < \dim (\text{Supp}(\mathcal{F}_ s)). \]

Proof. By assumption the schemes $S$ and $Y$ are affine. Write $S = \mathop{\mathrm{Spec}}(A)$ and $Y = \mathop{\mathrm{Spec}}(B)$. As $\pi $ is finite the $\mathcal{O}_ Y$-module $\pi _*\mathcal{G}$ is a finite type quasi-coherent $\mathcal{O}_ Y$-module. Hence $\pi _*\mathcal{G} = \widetilde{N}$ for some finite $B$-module $N$. Let $\mathfrak p \subset B$ be the prime ideal corresponding to $\xi $. To obtain $\alpha $ set $r = \dim _{\kappa (\mathfrak p)} N \otimes _ B \kappa (\mathfrak p)$ and pick $x_1, \ldots , x_ r \in N$ which form a basis of $N \otimes _ B \kappa (\mathfrak p)$. Take $\alpha : B^{\oplus r} \to N$ to be the map given by the formula $\alpha (b_1, \ldots , b_ r) = \sum b_ ix_ i$. It is clear that $\alpha : \kappa (\mathfrak p)^{\oplus r} \to N \otimes _ B \kappa (\mathfrak p)$ is an isomorphism as desired. Finally, suppose $\alpha $ is any map with this property. Then $N' = \mathop{\mathrm{Coker}}(\alpha )$ is a finite $B$-module such that $N' \otimes \kappa (\mathfrak p) = 0$. By Nakayama's lemma (Algebra, Lemma 10.20.1) we see that $N'_{\mathfrak p} = 0$. Since the fibre $Y_ s$ is geometrically irreducible of dimension $n$ with generic point $\xi $ and since we have just seen that $\xi $ is not in the support of $\mathop{\mathrm{Coker}}(\alpha )$ the last assertion of the lemma holds. $\square$

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