(Local and infinitesimal criteria for flatness)
Let \(A\) be a Noetherian ring, \(I \subseteq A\) an ideal, \(B\) a Noetherian \(A\)-algebra with \(IB\) contained in the Jacobson radical of \(B\), and \(M\) a finitely generated \(B\)-module. The following are equivalent.
\(M\) is flat over \(A\).
\(M/IM\) is flat over \(A/I\) and \(\Tor_1^A(A/I, M) = 0\).
\(M/I^{n}M\) is flat over \(A/I^{n}\) for every \(n \geq 1\).
In particular, if \(A \to B\) is a local homomorphism of Noetherian local rings with residue field \(k\) of \(A\) and \(M\) is a finitely generated \(B\)-module, then \(M\) is flat over \(A\) if and only if \(\Tor_1^A(k, M) = 0\).
Condition 2 is the local criterion for flatness; with \(I = \mathfrak{m}_A\) the hypothesis on \(M/\mathfrak{m}_A M\) is automatic, because every module over the field \(A/\mathfrak{m}_A\) is flat. Condition 3 is the infinitesimal criterion for flatness: flatness over \(A\) is detected on the infinitesimal thickenings \(\Spec A/I^n\) of \(\Spec A/I\). Geometrically, for a morphism \(f \colon X \to Y\) of locally Noetherian schemes, a point \(x \in X\) and \(y = f(x)\), a coherent sheaf \(\mathcal{F}\) on \(X\) is flat over \(Y\) at \(x\) if and only if \(\Tor_1^{\OO_{Y,y}}(\kappa(y), \mathcal{F}_x) = 0\).
Let \(A = k[t]_{(t)}\) and \(B = (A[x]/(tx))_{(t, x)}\). Then \(\Tor_1^A(k, B)\) is the \(t\)-torsion of \(B\), computed from \(0 \to A \xrightarrow{t} A \to k \to 0\), and it contains \(x \neq 0\). So \(B\) is not flat over \(A\): the family \(V(tx) \to \Spec k[t]\) has fibre the point \(x = 0\) over \(t \neq 0\) and the line \(\AA^1\) over \(t = 0\).