Let $G$ be a group. I have two questions about the homology of $G$:


1. Consider a finite exact sequence
$$0 \rightarrow M\_1 \rightarrow \cdots \rightarrow M\_m \rightarrow 0$$
of $G$-modules. How are the homology groups $H\_k(G;M\_i)$ related?
2. Consider a filtration of $G$-modules
$$0 = N\_0 \subset N\_1 \subset \cdots \subset N\_n = N.$$
How can I relate $H\_k(G;N)$ to the homology groups $H\_k(G;N\_{i+1}/N\_i)$?


These feel related since the simplest cases of both of them reduce to the long exact sequence in group homology associated to a short exact sequence of coefficients. Maybe there is some kind of spectral sequence?



Groups homology with coefficients fitting into filtration or exact sequence
1. Think of your exact sequence as a resolution of $M\_m$. It's not necessarily a resolution by free $G$-modules, or by projective $G$-modules; it's just a resolution by $G$-modules. You get a spectral sequence whose $E\_1$-term is the direct sum of the $H\_\*(G; M\_i)$ for all $i<m$, and which converges to $H\_\*(G; M\_m)$.


When the $G$-modules $M\_i$ *are* projective for all $i<m$, then the $E\_1$-page degenerates on to a single line, and then running the spectral sequence reduces to the usual process for calculating $Tor$: take a projective resolution, apply the tensor product (at this point you have recovered the one nonzero line in the $E\_1$-page), then take homology (at this point you have both the $E\_2$-page and the final answer).


This "resolution spectral sequence" is a standard homological tool: see application 5.9.8 in Weibel's homological algebra textbook for its construction.
2. Having a filtration on the coefficient module $N$ yields a filtration on the bar complex, when you tensor $N$ with the bar resolution of $\mathbb{Z}$ by $G$-modules. The resulting filtered chain complex yields a spectral sequence whose input is homology of $G$ with coefficients in the associated graded $G$-module of your filtration of $N$, and whose output is the homology of $G$ with coefficients in $N$. This was suggested in the comments as well. The resulting spectral sequence is also a standard tool, and although I don't know a textbook that discusses it in exactly this level of generality, any treatment of the spectral sequence of a filtered chain complex will be applicable to this one.


