Exercise 4, § 5

Let A be a noetherian local ring. Put $\delta(A) = [m_A / m_A^2 : \kappa_A] - \dim(A)$ (cf. § 4, No. 5). (a) Now prove that the dimension of the $\kappa_A$-vector space $H_i(z, A)$, where z is a minimal generating system of $m_A$, is independent of the choice of z; denote it by $h_i(A)$. (b) Exercise prove the equality $h_i(A) = h_i(\widehat{A})$ for all i. (c) Proving that one has $h_1(A) \geq \delta(A)$, and that equality holds if and only if A is a complete intersection ring (Remark 2 of No. 2; reduce to the case where A is complete and apply Exercise 4 by presenting A as a quotient $R/I$, where R is a regular local ring and $I \subset m_R^2$). (d) Let $(x_1, \ldots, x_p)$ be a completely secant sequence of elements of $m_A$, generating an ideal J. Exercise prove the equality $h_1(A/J) - \delta(A/J) = h_1(A) - \delta(A)$ (reduce to the case $p = 1$ and distinguish two cases, according as $x_1$ belongs or does not belong to $m_A^2$).