Exercise 6, § 4

Let $A$ be a regular local ring, $(x_1, \ldots, x_n)$ and $(y_1, \ldots, y_n)$ maximal transverse sequences of elements of $m_A$, generating ideals $x$ and $y$ respectively. Suppose that one has $y \subset x$, so that there exist elements $a_{ij}$ of $A$ such that one has $y_i = \sum_j a_{ij} x_j$ for $1 \leqslant i \leqslant n$. Prove that the class of $\det(a_{ij})$ in $A/y$ does not depend on the matrix $(a_{ij})$ chosen, and that its annihilator is the image of $x$; in particular, one has $\det(a_{ij}) \neq 0$ (prove that the canonical homomorphism $\operatorname{Ext}_A^n(A/x, A) \to \operatorname{Ext}_A^n(A/y, A)$ is injective, then now prove with the aid of the Koszul complexes that it is identified with the homomorphism $A/x \to A/y$ deduced from the homothety of ratio $\det(a_{ij})$).