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In "Théorie des Faisceaux", Godement states the following theorem due to Leray (Theorem 5.2.5, page 209).

Let ${\mathcal M}=(M_i )_{i\in I}$ be a locally finite closed covering of a topological space, and $\mathcal A$ a sheaf of abelian groups on $X$. Let $M$ be the nerve of the covering $\mathcal M$. Then there exists a spectral sequence whose $E_2$ term is $$ E_2^{pq}=H^p (M, {\mathcal H}^q({\mathcal A})) $$ and whose $E_{\infty}$ term is the bigraduated group attached to a certain filtration of $H^* (X,{\mathcal A})$.

Here the ${\mathcal H}^q({\mathcal A})$ are certain coefficient systems on the simplicial complex $M$ that Godement describes explicitely.

In fact I would like to know whether there exists a version of this theorem in the case where there is a group $G$ acting on $X$, $\mathcal A$ is $G$-equivariant, and cohomology is replaced by cohomology with compact support. In fact the case where $\mathcal A$ is the constant sheaf with values in complex numbers would be sufficient. But I do need equivariance and compactly supported cohomology.

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