The Stacks project

Proof. We are going to use that an $A$-module is flat if and only if it is torsion free, see Lemma 15.22.10. By Algebra, Lemma 10.57.10 we can find a graded $A$-algebra $S$ with $S_0 = A$ and generated by finitely many elements in degree $1$, an element $f \in S_1$ and a finite graded $S$-module $N$ such that $B \cong S_{(f)}$ and $M \cong N_{(f)}$. If $M$ is torsion free, then we can take $N$ torsion free by replacing it by $N/N_{tors}$, see Lemma 15.22.2. Similarly, if $B$ is torsion free, then we can take $S$ torsion free by replacing it by $S/S_{tors}$. Hence in case (1), we may apply Lemma 15.25.4 to see that $S$ is a finitely presented $A$-algebra, which implies that $B = S_{(f)}$ is a finitely presented $A$-algebra. To see (2) we may first replace $S$ by a graded polynomial ring, and then we may apply Lemma 15.25.3 to conclude. $\square$