Exercise 3, § 5

Let $R$ be a regular local ring, $I$ an ideal of $R$, $A$ the ring $R/I$. Let $x = (x_1, \ldots, x_d)$ be a system of coordinates of $R$, and $y = (y_1, \ldots, y_m)$ a minimal generator system of $I$, such that $m = [I/m_R I : \kappa_R]$. (a) When I is contained in $m_R^2$, define a canonical isomorphism of $H_1(x, A)$ onto $I/m_R I$ (consider an exact sequence of Koszul complexes). (b) For the images of the $x_i$ in A to form a minimal system of generators of $m_A$, it is necessary and sufficient that I be contained in $m_R^2$. (c) Prove that the dimension of the $\kappa_A$-vector space $H_1(x, A)$ is equal to $m - d + [m_A / m_A^2 : \kappa_A]$.