
Lemma 10.78.2. Let $R$ be a ring. Let $\varphi : M \to N$ be a map of $R$-modules with $M$ finite and $N$ finitely presented. Then

$U = \{ \mathfrak p \subset R \mid \varphi _{\mathfrak p} : M_{\mathfrak p} \to N_{\mathfrak p} \text{ is an isomorphism}\}$

is an open subset of $\mathop{\mathrm{Spec}}(R)$.

Proof. Let $\mathfrak p \in U$. Pick a presentation $N = R^{\oplus n}/\sum _{j = 1, \ldots , m} R k_ j$. Denote $e_ i$ the image in $N$ of the $i$th basis vector of $R^{\oplus n}$. For each $i \in \{ 1, \ldots , n\}$ choose an element $m_ i \in M_{\mathfrak p}$ such that $\varphi (m_ i) = f_ i e_ i$ for some $f_ i \in R$, $f_ i \not\in \mathfrak p$. This is possible as $\varphi _{\mathfrak p}$ is an isomorphism. Set $f = f_1 \ldots f_ n$ and let $\psi : R_ f^{\oplus n} \to M_ f$ be the map which maps the $i$th basis vector to $m_ i/f_ i$. Note that $\varphi _ f \circ \psi$ is the localization at $f$ of the given map $R^{\oplus n} \to N$. As $\varphi _{\mathfrak p}$ is an isomorphism we see that $\psi (k_ j)$ is an element of $M$ which maps to zero in $M_{\mathfrak p}$. Hence we see that there exist $g_ j \in R$, $g_ j \not\in \mathfrak p$ such that $g_ j \psi (k_ j) = 0$. Setting $g = g_1 \ldots g_ m$, we see that $\psi _ g$ factors through $N_{fg}$ to give a map $\chi : N_{fg} \to M_{fg}$. By construction $\chi$ is a right inverse to $\varphi _{fg}$. It follows that $\chi _\mathfrak p$ is an isomorphism. By Lemma 10.78.1 there is an $h \in R$, $h \not\in \mathfrak p$ such that $\chi _ h : N_{fgh} \to M_{fgh}$ is surjective. Hence $\varphi _{fgh}$ and $\chi _ h$ are mutually inverse maps, which implies that $D(fgh) \subset U$ as desired. $\square$

Comment #2909 by Dario Weißmann on

The target of the morphism $\psi$ should be $M_f$ instead of $M$.

Also I don't think we are quite finished with the proof. The $m_i$ don't necessarily generate $M_{fg}$. But as $M$ is finite and the $m_i$ generate $M_{\mathfrak{p}}$ we can find an $h\notin \mathfrak{p}$ such that the $m_i$ generate $M_{fgh}$. Then we have found an inverse to $\varphi_{fgh}$.

Seems like we don't need the assumption that $M$ is finitely generated, $M$ finite suffices.

Comment #2941 by on

Thanks for catching the oversight! The lemma is indeed more general than currently stated. If you look at Section 10.126 you'll find a lot more statements like this. I've fixed the argument here and I've weakened the hypotheses here.

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