Exercise 5, § 4

(a) Let H be a graded ring of type N, such that H_0 is a field and H is an H_0-algebra of finite type. Let M and N be two graded H-modules of finite type. Put T_i = Tor_i^H(M, N). Prove the equality

(b) Let A be a local noetherian ring, M and N two A-modules. Prove that if, for m \in \mathbf{Z} and i \geq 1, Tor_i^{grm(A)}(gr_m(M), gr_m(N)) is zero, then one has Tor_i^A(M, N) = 0 for i \geq 1 and $H_{M \otimes_A N, m} = \frac{H_{M,m} \cdot H_{N,m}}{H_{A,m}}$ for $m \in \mathbf{Z}$.