Finite cell local system cohomology is finite and commutes with field extension

Work first on one component, with basepoint $x$, group $\pi ={\pi }_1(X,x)$ and fibre $M=L_x$. By [F1] the degree-$q$ term of the cellular local cochain complex is ${\mathrm{Hom}}_{k[\pi ]}(P_q,M)$ with $P_q$ free over $k[\pi ]$ on the finitely many relative $q$-cells of the component, so choosing that finite basis identifies it with the finite product $M^{n_q}$; since $M$ is finite-dimensional, the term is finite-dimensional, and it is zero for $q<0$ and for $q>d$. By [F2] the cohomology $H^q(X,A;L)$ is the cohomology of this complex, so it is finite-dimensional as a kernel, image or quotient of finite-dimensional spaces and vanishes outside $[0,d]$. For a disconnected pair the cochain complex is the degreewise product of the complexes of the finitely many components, and the same conclusions hold componentwise; the empty pair has the zero complex.

Let $k^{\mathrm{\prime }}$ be a field extension and give $M{\otimes }_k{k}^{\mathrm{\prime }}$ the induced $k^{\mathrm{\prime }}[\pi ]$-module structure; these fibres with the extended transport maps form the local system $L{\otimes }_k{k}^{\mathrm{\prime }}$. The termwise map ${\mathrm{Hom}}_{k[\pi ]}(P_q,M){\otimes }_k{k}^{\mathrm{\prime }}\to {\mathrm{Hom}}_{k^{\mathrm{\prime }}[\pi ]}(P_q{\otimes }_k{k}^{\mathrm{\prime }},M{\otimes }_k{k}^{\mathrm{\prime }})$, defined on a basis of the free module $P_q$, is an isomorphism, because both sides identify with $(M{\otimes }_k{k}^{\mathrm{\prime }}{)}^{n_q}$; it is natural in the second variable and therefore intertwines the differentials, which are obtained by dualizing the same $k[\pi ]$-linear cellular boundary of [F2]. Hence $C_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^{\ast }(X,A;L){\otimes }_k{k}^{\mathrm{\prime }}$ is isomorphic, as a complex, to $C_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^{\ast }(X,A;L{\otimes }_k{k}^{\mathrm{\prime }})$.

By [F3] the field $k^{\mathrm{\prime }}$, viewed as a $k$-vector space, is free and hence flat, so tensoring over $k$ with $k^{\mathrm{\prime }}$ preserves exact sequences. Applying this to the kernel-image sequences of the complex of step 1.1 gives a canonical isomorphism $H^q\left(C^{\ast }{\otimes }_k{k}^{\mathrm{\prime }}\right)\cong H^q(C^{\ast }){\otimes }_k{k}^{\mathrm{\prime }}$, which by step 2.1 and [F2] is the stated map $H^q(X,A;L){\otimes }_k{k}^{\mathrm{\prime }}\to H^q(X,A;L{\otimes }_k{k}^{\mathrm{\prime }})$, natural in the coefficients. The finiteness, the vanishing outside $[0,d]$, and the isomorphism all hold separately on each component and for the empty pair, where both complexes vanish.