Exercise 7, § 4

Let $A$ be a regular local ring of dimension 3, $f$ a nonzero element of $m_A$. Prove that the following conditions are equivalent:

(i) the ring $A/(f)$ is not factorial; (ii) there exists an integer $n \geqslant 2$ and a matrix $X \in M_n(A)$, with elements in $m_A$, such that $f = \det(X)$.

(Under assumption (ii), prove that the ideal of $A/(f)$ generated by the classes of the minors $X^{11}, \ldots, X^{1n}$ of $X$, where $X^{1p}$ is obtained by deleting the first row and the column of index $p$, has height 1 but is not principal. Under assumption (i), let $\mathfrak{p}$ be a prime ideal of height 2 of $A$ whose image in $A/(f)$ is not principal; deduce from Exercise 11 of § 3 that $\mathfrak{p}$ is generated by the minors of order $n-1$ of a matrix $Y \in M_{n,n-1}(A)$ with elements in $m_A$. Write $f$ as the determinant of a matrix $\tilde{Y}$ obtained by adding a column $Y_n$ to $Y$; if an element of $Y_n$ is invertible reduce to the case where $Y_n$ belongs to the canonical basis of $A^n$.)