
> 
> Let $M$ be a finite CW-complex. Let $F$ be a finite rank local system over $M$ with coefficients in any field. Is it true that $\dim(H^k(M,F))$ is at most the number of $k$-cells times $\operatorname{rank}(F)$?
> 
> 
> 


If $F$ is the trivial local system then this result is proven in almost any standard textbook in topology (or at least immediately follows from there). I believe that it should be true in the above generality and **would be happy to have a reference**. 



Cohomogy of local systems over CW-complexes
I'm not sure of a reference off the top of my head, but the proof is straightforward. Let's say the number of $k$-cells of $M$ is $n\_k$. 


Because $M$ is given a CW structure, its universal cover $\tilde M$ is given a $\pi\_1(M)$-invariant CW structure; because $\pi\_1(M)$ acts freely, the cellular chain complex $C\_\*(\tilde M;R)$ is a free $R[\pi\_1(M)]$-module equipped with a canonical isomorphism $C\_\*(\tilde M;R) \otimes\_{R[\pi\_1(M)]} R \cong C\_\*(M;R)$. In particular, $C\_k(\tilde M;R)$ is a free $R[\pi\_1(M)]$-module of rank $n\_k$.


Suppose we are given a local system $\rho: \pi\_1(M) \to \text{Aut}\_R(A)$, where $A$ is a finitely generated module over some Noetherian ring $R$. The cellular chain complex with local system $\rho$ is given by $$C\_\*(M;\rho) = C\_\*(\tilde M;R) \otimes\_{R[\pi\_1(M)]} A,$$ with $\pi\_1(M)$ acting on $A$ by $\rho$. 


Then each $C\_k(M;\rho)$ is isomorphic to $A^{n\_k}$, which is finitely generated; it has a submodule $Z\_k \subset A^{n\_k}$ of $k$-cycles, and the homology group $H\_k(M;\rho)$ is the quotient $Z\_k/B\_k$, for the submodule $B\_k \subset Z\_k$ of boundaries. 


Every submodule of a finitely generated module over a Noetherian ring is finitely generated, and so $Z\_k$ is finitely generated, and hence so is its quotient $H\_k(M;\rho)$.


This level of generality includes $A$ any finite-dimensional vector space over a field and $A$ any finitely generated abelian group.


