Exercise 7, § 4

Let $C$ be a complex of right $A$-modules and $M$ a left $A$-module. The torsion product of $C$ and $M$ is defined to be the graded $k$-module $\mathscr{T}or^A(C,M)=H(C\otimes_A L(M))$. If $f:C\to C'$ is a morphism of complexes and $g:M\to M'$ a homomorphism of $A$-modules, we denote by $\mathscr{T}or^A(f,g)=H(f\otimes L(g))$.

a) If $f:C\to C'$ is a homomorphism, $\mathscr{T}or^A(f,1)$ is an isomorphism; if $C$ is flat and right bounded, the graded $k$-module $\mathscr{T}or^A(C,M)$ is isomorphic to $H(C\otimes_A M)$.

b) If $0\to C'\to C\to C''\to0$ is an exact sequence of complexes of right $A$-modules and if $M$ is an $A$-module, prove that there exists an exact sequence

If $0\to M'\mathop{\longrightarrow}^{i}M\mathop{\longrightarrow}^{p}M''\to0$ is an exact sequence of $A$-modules, prove that one has an exact sequence:

(Notice that there exists a homotopism of $L(M'')$ onto the cone of $L(i)$.)

c) Prove that there exists a spectral sequence $'E$ (X, pp. 175-177, exercises 14 to 17) converging to $\mathscr{T}or^A(C,M)$, such that

If moreover the complex $C$ is right bounded, prove that there exists a spectral sequence $''E$, converging to $\mathscr{T}or^A(C,M)$, such that

(consider the double complex $C\otimes_A L(M)$).