Exercise 5, § 2

Let $A$ be a noetherian local ring, $M$ a finitely generated a-module, $x = (x_1, \ldots, x_n)$ a sequence of elements of $\mathfrak{m}_A$, generating an ideal $x$; suppose that the a-module $M/xM$ is of finite length (which implies $n \geqslant \dim(M)$). We denote by gr(A) and gr(M) the associated graded modules of A and M for the x-adic topology, $\xi_i$ ($1 \leqslant i \leqslant n$) the image of $x_i$ in $x/x^2$, $\xi$ the sequence $(\xi_1, \ldots, \xi_n)$. a) For $p \in \mathbf{Z}$, put $F^p K_i(x, M) = K_i(x, x^{p-i}M)$ (with the convention $x^s = A$ for $s \leqslant 0$). Prove that $F^p K_\bullet(x, M)$ is a subcomplex of $K_\bullet(x, M)$, and that the complex $\bigoplus_p F^p K_\bullet(x, M)/F^{p+1} K_\bullet(x, M)$ is identified with $K_{\bullet}^{\operatorname{gr}(A)}(\xi, \operatorname{gr}(M))$. b) For every complex of A-modules C such that $H(C)$ is of finite length, put $\chi(C) = \sum_i (-1)^i \lg(H_i(C))$. Prove that there exists an integer $p_0$ such that the complex

$F^p K_\bullet(x, M)/F^{p+1} K_\bullet(x, M)$ has zero homology for $p \geqslant p_0$; deduce the equalities

for $p \geqslant p_0$.

c) Prove that the expression above is zero if $n > \dim(M)$, and equal to $c_x(M)$ if $n = \dim(M)$.

d) Let $h \geqslant 0$, and $p \geqslant p_0$; prove that the canonical injection of $F^p K_\bullet(x, M)$ into $F^{p+h} K_\bullet(x, M)$ is a homology equivalence. Deduce that $H_i(F^p K_\bullet(x, M))$ is discrete for the x-adic topology. Prove moreover that the filtration of this module defined by the images of the mappings $H_i(F^{p+h} K_\bullet(x, M)) \to H_i(F^p K_\bullet(x, M))$ is x-good, and deduce that $F^p K_\bullet(x, M)$ has zero homology. Conclude that $\chi(K_\bullet(x, M))$ is zero if $n > \dim(M)$, and equal to $e_x(M)$ if $n = \dim(M)$.