I'm reading through Szamuely's Galois Groups and Fundamental Groups.
Szamuely defines complex local systems as locally constant sheafs of finite dimensional complex vector spaces. I'm stuck on the very first example he gives:
Let $D \subset \mathbb C$ be a connected open subset. Consider over $D$ a homogeneous linear differential equation: $$y^{(n)} + a_1y^{(n-1)} + \ldots + a_ny = 0, $$ with $a_i$ holomorphic functions on $D$.
For open subset $U \subset D$ we define $S(U)$ as local solutions of the equation over $U$. Szamuely claims $S$ is a local system.
For me it's clear that each $S(U)$ is a finite dimensional complex vector space, and it's clear that $S$ really defines a sheaf. The problem is I can't see why it's locally constant?
By definition, I must show that every point $x \in D$ has an open neighborhood $V$ such that $S|_V$ ($S$ restricted to V) is isomorphic to a constant sheaf.
Consider a point $x \in D$ and $V$ is a disk around $x$. On $V$, a solution $g$ is uniquely determined by $(g(x), g'(x), \dots, g^{(n-1)}(x))$, so sending $g$ to its $n-1$ first derivatives defines an isomorphism $S(V) \cong \mathbb C^n_V$, where $\mathbb C^n_V$ is the constant sheaf on $V$ with stalk $\mathbb C^n$.