Exercise 23, § 4

Let A be a ring, E a finitely generated A-module, r the least of the integers k such that $\bigwedge^{k+1} E = 0$ and e(E) the annihilator of $\bigwedge^r E$. Show that, for every ideal a $\supseteq e(E)$ in A, the (A/a)-module E/aE is flat (reduce it to the case where A is a local ring and use Exercise 22 (b)). Deduce that, if c is the Jacobson radical of A and if, for every maximal ideal m of A, the rank of the vector (A/m)-space E/mE is the same, then E/rE is a flat (A/r)-module.