Lemma 15.25.2. Let $R \to S$ be a ring homomorphism. Assume

1. there exist finitely many primes $\mathfrak p_1, \ldots , \mathfrak p_ m$ of $R$ such that the map $R \to \prod R_{\mathfrak p_ j}$ is injective,

2. $R \to S$ is of finite type,

3. $S$ flat over $R$, and

4. for every prime $\mathfrak p$ of $R$ the ring $S_{\mathfrak p}$ is of finite presentation over $R_{\mathfrak p}$.

Then $S$ is of finite presentation over $R$.

Proof. By assumption $S$ is a quotient of a polynomial ring over $R$. Thus the result follows directly from Lemma 15.25.1. $\square$

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