Lemma 5.11. (Detection by local coefficients)
Let \(f\colon X\to Y\) be a map of animae. If
\[H^0(Y,A)\longrightarrow H^0(X,f^*A)\]
is an isomorphism for every local system \(A\) of abelian groups on \(Y\), then \(\pi_0(f)\) is an isomorphism and \(\pi_1(X,x)\to\pi_1(Y,f(x))\) is surjective for every \(x\in X\).
Proof
Taking \(A\) to be the constant local system \(\mathbb Z/2\) shows that pullback along \(\pi_0(f)\) induces a bijection on functions to \(\mathbb Z/2\). Hence \(\pi_0(f)\) is a bijection.We may therefore work on one connected component. Fix \(x\in X\), write \(G=\pi_1(Y,f(x))\), and let \(H\subseteq G\) be the image of \(\pi_1(X,x)\). If \(H\neq G\), consider the \(G\)-module
\[M:=(\mathbb Z/2)^{G/H}\]
with the permutation action. The characteristic function of the coset \(H\) is fixed by \(H\) but not by \(G\). Thus the inclusion \(M^G\to M^H\) is not surjective. This contradicts the assumed isomorphism on \(H^0\), since \(H^0(Y,M)=M^G\) and \(H^0(X,f^*M)=M^H\). Therefore \(H=G\).