Suppose I have a bounded chain complex $C_{\ast}$ over the group ring $\mathbb{Z}G$ for a finite group $G$. In topology we are usually in a situation when $C_{\ast}$ is a complex of projective modules. Now for any other $\mathbb{Z}G$-module $M$ homology of the complex $C_{\ast} \otimes_{\mathbb{Z}G} M$ can be computed using universal coefficient spectral sequence (UCSS). In fact UCSS is the hyperhomology spectral sequence and luckily in this case hyperhomology groups $\mathbb{H}_{\ast}(C_{\ast}, M)$ are equal to ordinary homology of $C_{\ast} \otimes_{\mathbb{Z}G} M$.
When $C_{\ast}$ is a chain complex of modules which are not flat, then we still have hyperhomology spectral sequence, however in this case hyperhomology is usually different from homology of $C_{\ast} \otimes_{\mathbb{Z}G} M$. So my question is the following: are there any theorems which give us any kind of direct relation between homology of $C_{\ast}$ and homology of $C_{\ast} \otimes_{\mathbb{Z}G} M$? For example is there any generalisation of UCSS, which allows to compute $H_{\ast}(C_{\ast} \otimes_{\mathbb{Z}G} M)$?
