The Stacks project

Lemma 115.5.4. Let $R$ be a ring. Let $F(X, Y) \in R[X, Y]$ be homogeneous of degree $d$. Assume that for every prime $\mathfrak p$ of $R$ at least one coefficient of $F$ is not in $\mathfrak p$. Let $S = R[X, Y]/(F)$ as a graded ring. Then for all $n \geq d$ the $R$-module $S_ n$ is finite locally free of rank $d$.

Proof. The $R$-module $S_ n$ has a presentation

\[ R[X, Y]_{n-d} \to R[X, Y]_ n \to S_ n \to 0. \]

Thus by Algebra, Lemma 10.79.4 it is enough to show that multiplication by $F$ induces an injective map $\kappa (\mathfrak p)[X, Y] \to \kappa (\mathfrak p)[X, Y]$ for all primes $\mathfrak p$. This is clear from the assumption that $F$ does not map to the zero polynomial mod $\mathfrak p$. The assertion on ranks is clear from this as well. $\square$

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