Corrollary $10.30$ from Rotman's Homological Algebra.
Let $C$ be a positive complex of free $R$ modules. $A$ is a $R$-module, where $R$ is commutative hereditary. There's an exact sequence:
$0\to H_n(C)\otimes A \to H_n(C \otimes A) \to$ Tor$_1^R(H_{n-1}(C),A) \to 0$ for all $n \geq 0.$
In the proof, Rotman tells to apply $10.29$ which I put at the bottom.
I applied and found the corresponding term of Tor$_1^R(H_{n-1}(C),A)$ in the resultant sequence from $10.29$ is zero. So I concluded Tor$_1^R(H_{n-1}(C),A) = 0$ for all $n \geq 0.$
Any help would be appreciated!
Corollary $10.29$
Let M be a first quadrant bicomplex for which $E^2$ consists of two adjacent columns: $E^2_{p,q} = 0$ for all $p \neq 0,1.$ For each n, there is an exact sequence: $0 \to E^2_{0,n} \to H_n(Tot(M)) \to E^2_{1,n-1} \to 0$.