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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