Exercise 7, § 9

Let A be a ring, M an A-module, $x = (x_1, ..., x_n)$ a sequence of elements of A. (a) Now prove that there exists a spectral sequence E (X, pp. 175-177, exercises 14 to 17), converging to H.(x, M), such that $E^2_{pq} = \operatorname{Tor}_p^A(H_q(x, A), M)$ (use X, p. 186, exercise 8). (b) Let p be an integer $\leq n$; set $x' = (x_1, ..., x_p)$ and $x'' = (x_{p+1}, ..., x_n)$. Now prove that there exists a spectral sequence ’E, such that ’$E^2_{pq} = H_p(x'', H_q(x', M))$, converging to H.(x, M).