Let $X := \mathbb{P}^1$, $S\subset X$ a finite set of points, $U := X - S$, and $j : U\rightarrow X$ the inclusion.

Let $F$ be a complex local system on $U$ of rank $r$, and let $F_0$ be a typical fiber, so $F_0$ is a complex vector space. On p14 of Katz's book *Rigid local systems*, he says that
$$\chi(X,j_*F) = r\cdot\chi(U,\mathbb{C}) + \sum_{s\in S}\text{dim}_{\mathbb{C}}F_0^{I(s)}$$
where $I(s)\cong\mathbb{Z}$ denotes the local monodromy group at $s$ (the fundamental group of a punctured neighborhood of $s$).

Why is this true? This is a very naive question, and I'm clearly just missing some basic points about the cohomology of local systems. I'm hoping someone here can help fill me in.

In the simple case $S = \{s\}$, let $D$ be a small neighborhood of $s$, $D^* := D - s$, then Mayer Vietoris would give: $$\chi(X,j_*F) = \chi(U,F) + \chi(D,j_*F) - \chi(D^*,F)$$ Since $h^0(D,j_*F) = h^0(D^*,F) = F_0^{I(s)}$, we have $$\chi(X,j_*F) = \chi(U,F) - h^1(D,j_*F) + h^1(D^*,F) + h^2(D,j_*F) - h^2(D^*,F)$$

- I think we should have $\chi(U,F) = r\cdot \chi(U,\mathbb{C})$ (why?).
- Assuming that local system cohomology is homotopy invariant, we should have $h^2(D^*,F) = 0$.
- By comparison with group cohomology, we have $h^1(D^*,F) = r$.

Thus the desired result would follow in this simple case as long as $\dim_{\mathbb{C}}F_0^{I(s)} = h^2(D,j_*F) - h^1(D,j_*F)$. Is this true?