Lemma 38.25.3. Let $A_0$ be a local ring. If the lemma holds for every Situation 38.25.1 with $A = A_0$, with $B$ a localization of a polynomial algebra over $A$, and $N$ of finite presentation over $B$, then the lemma holds for every Situation 38.25.1 with $A = A_0$.

Proof. Let $A \to B$, $u : N \to M$ be as in Situation 38.25.1. Write $B = C/I$ where $C$ is the localization of a polynomial algebra over $A$ at a prime. If we can show that $N$ is finitely presented as a $C$-module, then a fortiori this shows that $N$ is finitely presented as a $B$-module (see Algebra, Lemma 10.6.4). Hence we may assume that $B$ is the localization of a polynomial algebra. Next, write $N = B^{\oplus n}/K$ for some submodule $K \subset B^{\oplus n}$. Since $B/\mathfrak m_ AB$ is Noetherian (as it is essentially of finite type over a field), there exist finitely many elements $k_1, \ldots , k_ s \in K$ such that for $K' = \sum Bk_ i$ and $N' = B^{\oplus n}/K'$ the canonical surjection $N' \to N$ induces an isomorphism $N'/\mathfrak m_ AN' \cong N/\mathfrak m_ AN$. Now, if the lemma holds for the composition $u' : N' \to M$, then $u'$ is injective, hence $N' = N$ and $u' = u$. Thus the lemma holds for the original situation. $\square$

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