Lemma 10.94 from Goertz' and Wedhorn's Algebraic Geometry I

Question

I have a question about the proof of Lemma 10.94 from Goertz' & Wedhorn's Algebraic Geometry I:

Lemma 10.94. Let $S$ = $\operatorname{Spec}A$ be an integral noetherian scheme with generic point $\eta$, let $X = \operatorname{Spec}B$ be an affine $S$-scheme of finite type, and let $\mathcal{F}$ be a coherent $\mathcal{O}_X$-module. Assume that for all $s \in S$, the fiber $X_s$ is integral, and that $\mathcal{F}_{\eta}$ is a torsion-free $\mathcal{O}_{X_{\eta}}$-module. Then for all $s$ in a non-empty open subset of $S$, the $\mathcal{O}_{X_s}$-module $\mathcal{F}_{s}$ is torsion-free.

Proof. Let $M = \Gamma(\mathcal{F},X)$, and $M_s = \Gamma(X_s,\mathcal{F}_{s}) = M \otimes_A \kappa(s)$ for $s \in S$, and similarly define $B_s, s \in S$. Since $M_{\eta}$ is a torsion-free $B_{\eta}$-module, the natural homomorphism $M \to M \otimes_{B_{\eta}} \operatorname{Frac}(B_{\eta}) \cong \operatorname{Frac}(B_{\eta})^n$ is injective. The latter is the union of free $B_{\eta}$- modules, and because $M$ is finitely generated, we can embed $M_{\eta}$ into a free module $B_{\eta}^n$.
By Theorem 10.58, the homomorphism $M_{\eta} \to B_{\eta}^n$ can be extended to an open subset of $S$, and by Proposition 10.93, it is still injective on the fibers over a possibly smaller open subset of $S$.

Theorem 10.58 uses a collection of following notations (10.13):

1. Let $\Lambda$ be a filtered partially ordered set with a unique minimal element $0$.

2. Let $(R_{\lambda})_{\lambda}$ be an inductive system of rings indexed by Λ and with transition maps $\sigma_{\lambda \mu} : R_{\lambda} \to R_{\mu}$. Set $R := \varinjlim R_{\lambda}$ and let $\sigma_{\lambda} : R_{\lambda} \to R$ be the natural maps.

3. Set $S_{\lambda} := \operatorname{Spec} R_{\lambda}, S := \operatorname{Spec} R$, and let $s_{\lambda \mu} : S_{\mu} \to S_{\lambda}, s_{\lambda} : S \to S_{\lambda}$ be the morphisms associated with the $\sigma$’s.

4. Let $\mathcal{F}_0$ and $\mathcal{G}_0$ be $\mathcal{O}_{S_0}$-modules, and for each $\lambda , \mathcal{F}_{\lambda}:= s^*_{0 \lambda} \mathcal{F}_0$. Let $\mathcal{F} := s^*_{\lambda} \mathcal{F}_{\lambda}$, and similary for $\mathcal{F}$.

The functoriality of the pull-back $s^*_{\lambda \mu}$ gives us a natural homomorphism

$$ u_{\mathcal{F}, \mathcal{G}}: \varinjlim \operatorname{Hom}_{S_{\lambda}}(\mathcal{F}_{\lambda}, \mathcal{G}_{\lambda}) \to \operatorname{Hom}_S(\mathcal{F}, \mathcal{G}). $$

Theorem 10.58. Suppose that for some $\lambda, \mathcal{F}_{\lambda}$ is quasi-coherent and of finite type, and that $\mathcal{G}_{\lambda}$ is quasi-coherent. Then the homomorphism $u_{\mathcal{F}, \mathcal{G}}$ is injective. If furthermore $\mathcal{F}_{\lambda}$ is of finite presentation, then $u_{\mathcal{F}, \mathcal{G}}$ is bijective.

Question: How Theorem 10.58 is precisely applied in the proof of Lemma 10.94? What are the filtered partially ordered set $\Lambda$ and the $0$ in this case?

Answer 1

Let $\Lambda$ be the set of all affine open neighborhoods of $\eta\in S$, which is partially ordered by reverse inclusion. Then $\Lambda$ has a unique minimal element ($S$ itself), for which we write $0$. We set $R_U=\Gamma(X_U,\mathcal{O}_X)\;(X_U=X\times_SU)$ for $U\in \Lambda$, $\mathcal{G}_0=\mathcal{F}$ and $\mathcal{H}_0=\mathcal{O}_X^n$.

By the first paragraph of the proof, we have a monomorphism $\varphi\colon \mathcal{G}=\mathcal{F}_\eta\to \mathcal{O}_{X_\eta}^n=\mathcal{H}$, so (the second part of) Theorem 10.58 implies that there is some $U\in \Lambda$ and a morphism $\varphi_U\colon \mathcal{G}_U=\mathcal{F}|_{X_U}\to \mathcal{O}_{X_U}^n=\mathcal{H}_U$ that extends $\varphi$.