The Sheaf of Germs of Holomorphic Functions

Abstract

Let X be a Riemann surface, and let a ∈ X. We consider pairs (U, f), where U is an open neighbourhood of a and f is a holomorphic function on U. Two such pairs (U, f) and (V, g) are said to be equivalent, and define the same germ of holomorphic function at a, if there exists an open neighbourhood W of a, W ⊂ U ∩ V, such that f|W = g|W. An equivalence class is called a germ of holomorphic function at a; the class of a pair (U, f) is called the germ of f at a and denoted by f a. The value at a of f a is defined by f a(a) = f(a) for any pair (U, f) defining f a.