Exercise 11, § 3

Soit $A$ a local ring.

a) Let $n$ be an integer $\geq 1$, $u : A^{n-1} \to A^n$ an $A$-linear mapping, $\mathfrak{d}(u)$ the ideal generated by the minors of order $n-1$ of $u$ (Exercise 8). If $\mathrm{prof}_A(\mathfrak{d}(u); A) \geq 2$, prove that $\mathfrak{d}(u)$ admits the resolution

where $v$ is induced by the homomorphism $\wedge^{n-1}(t u) : A^n \to A$ (apply Exercise 9).

b) Conversely, let $J$ be an ideal of $A$ of projective dimension 1. Prove that there exist a cancellable element $a$ of $A$, an integer $n$ and an $A$-linear mapping $u : A^{n-1} \to A^n$ such that $J = a \mathfrak{d}(u)$; one has $\mathrm{prof}_A(J; A) = 2$ (“Hilbert-Burch theorem”): deduce from Exercise 9 that a minimal resolution of $J$ is of the form $0 \to A^{n-1} \xrightarrow{u} A^n \to J$, with $\mathrm{prof}_A(\mathfrak{d}(u); A) \geq 2$, then apply a) and Exercise 1 of § 1, observing that $\mathfrak{d}(u)$ contains a cancellable element of $A$.

EXERCISES

AC X.161