Exercise 27, § 3

Let $A$ be a ring, $B$ a commutative $A$-algebra which is a Zariski ring and $N$ a finitely generated $B$-module.

(a) Suppose that, for an ideal $\mathfrak{g}$ of $B$ contained in the Jacobson radical of $B$, the $A$-modules $N/\mathfrak{g}^{n+1}N$ are flat for $n \geq 0$. Show that $N$ is a flat $A$-module. (If $v : M \to M'$ is an injective homomorphism of finitely generated $A$-modules, it is necessary to prove that $u = v \otimes 1 : M \otimes_A N \to M' \otimes_A N$ is injective. Reduce it to proving that, if we write $N_n = N/\mathfrak{g}^{n+1}N$ and $u_n = v \otimes 1_{N_n}$, the homomorphism $\lim u_n$ is injective, using the fact that the 3-adic topologies on $M \otimes_A N$ and $M' \otimes_A N$ are Hausdorff.)

(b) Let $b$ be an element contained in the Jacobson radical of $B$ and such that the homothety with ratio $b$ on $N$ is injective. Show that, if $N/bN$ is a flat $A$-module, $N$ is a flat $A$-module (reduce it to the case in (a)).

(c) Suppose further that $A$ is a local ring with maximal ideal $m$ and residue field $k = A/m$ and that $mB$ is contained in the Jacobson radical of $B$. Let $P$ be a $B$-module which is a flat $A$-module and $u : N \to P$ a homomorphism such that $u \otimes 1_k : N \otimes_A k \to P \otimes_A k$ is injective. Then show that $N$ is a flat $A$-module and that $u$ is injective. (Reduce it to showing that, for every finitely generated $A$-module $M$, the homomorphism $u \otimes 1_M : N \otimes_A M \to P \otimes_A M$ is injective; observe that the $m$-adic topology on $N \otimes_A M$ is Hausdorff; then use § 2, no. 8, Corollary 1 to Theorem 1, considering the commutative diagram

and using the flatness of P, the vertical arrows being the canonical homomorphisms of § 2, Exercise 6.)