# 6.1: The Ring of Germs

- Page ID
- 74244

Let \(p\) be a point in a topological space \(X\). Let \(Y\) be a set and \(U, V \subset X\) be open neighborhoods of \(p\). We say that two functions \(f \colon U \to Y\) and \(g \colon V \to Y\) are equivalent if there exists a neighborhood \(W\) of \(p\) such that \(f|_W = g|_W\).

An equivalence class of functions defined in a neighborhood of \(p\) is called a *germ of a function*. Usually it is denoted by \((f,p)\), but we simply say \(f\) when the context is clear.

The set of germs of complex-valued functions forms a commutative ring, see exercise below to check the details. For example, to multiply \((f,p)\) and \((g,p)\), take two representatives \(f\) and \(g\) defined on a common neighborhood multiply them and then consider the germ \((fg,p)\). Similarly, \((f,p) + (g,p)\) is defined as \((f+g,p)\). It is easy to check that these operations are well-defined.

Let \(X\) be a topological space and \(p \in X\). Let \(\mathcal{F}\) be a class of complex-valued functions defined on open subsets of such that whenever \(f \colon U \to \mathbb{C}\) is in \(\mathcal{F}\) and \(W \subset U\) is open, then \(f|_W \in \mathcal{F}\), and such that whenever \(f\) and \(g\) are two functions in \(\mathcal{F}\), and \(W\) is an open set where both are defined, then \(fg|_W\) and \((f+g)|_W\) are also in \(\mathcal{F}\). Assume that all constant functions are in \(\mathcal{F}\). Show that the ring operations defined above on a set of germs at \(p\) of functions from \(\mathcal{F}\) are well-defined, and that the set of germs at \(p\) of functions from \(\mathcal{F}\) is a commutative ring.

Let \(X=Y=\mathbb{R}\) and \(p=0\). Consider the ring of germs of continuous functions (or smooth functions if you wish). Show that for every continuous \(f \colon \mathbb{R} \to \mathbb{R}\) and every neighborhood \(W\) of \(0\), there exists a \(g \colon \mathbb{R} \to \mathbb{R}\) such that \((f,0) = (g,0)\), but \(g|_W \not= f|_W\).

Germs are particularly useful for holomorphic functions because of the identity theorem. In particular, the behavior of the exercise above does not happen for holomorphic functions. Furthermore, for holomorphic functions, the ring of germs is the same as the ring of convergent power series, see exercise below. No similar result is true for only smooth functions.

Let \(p \in \mathbb{C}^n\). Write \({}_n\mathcal{O}_p = \mathcal{O}_p\) as the ring of germs at \(p\) of holomorphic functions.

The ring of germs \(\mathcal{O}_p\) has many nice properties, and it is generally a “nicer” ring than the ring \(\mathcal{O}(U)\) for some open \(U\), and so it is easier to work with if we are interested in local properties.

- Show that \(\mathcal{O}_p\) is an (has no zero divisors).
- Prove the ring of germs at \(0 \in \mathbb{R}\) of smooth real-valued functions is not an integral domain.

Show that the units (elements with multiplicative inverse) of \(\mathcal{O}_p\) are the germs of functions which do not vanish at \(p\).

- (easy) Show that given a germ \((f,p) \in \mathcal{O}_p\), there exists a fixed open neighborhood \(U\) of \(p\) and a representative \(f \colon U \to \mathbb{C}\) such that any other representative \(g\) can be analytically continued from \(p\) to a holomorphic function \(U\).
- (easy) Given two representatives \(f \colon U \to \mathbb{C}\) and \(g \colon V \to \mathbb{C}\) of a germ \((f,p) \in \mathcal{O}_p\), let \(W\) be the connected component of \(U \cap V\) that contains \(p\). Then \(f|_W = g|_W\).
- There exists a germ \((f,p) \in \mathcal{O}_p\), such that for any open neighborhood \(U\) of \(p\), and any representative \(f \colon U \to \mathbb{C}\) we can find another representative of \(g \colon V \to \mathbb{C}\) of that same germ such that \(g|_{U \cap V} \not= f|_{U \cap V}\). Hint: \(n=1\) is sufficient.

Show that \(\mathcal{O}_p\) is isomorphic to the ring of convergent power series.

Let \(p\) be a point in a topological space \(X\). We say that sets \(A, B \subset X\) are equivalent if there exists a neighborhood \(W\) of \(p\) such that \(A \cap W = B \cap W\). An equivalence class of sets is called a *germ of a set* at \(p\). It is denoted by \((A,p)\), but we may write \(A\) when the context is clear.

The concept of \((X,p) \subset (Y,p)\) is defined in an obvious manner, that is, there exist representatives \(X\) and \(Y\), and a neighborhood \(W\) of \(p\) such that \(X \cap W \subset Y \cap W\). Similarly, if \((X,p)\), \((Y,p)\) are germs and \(X\), \(Y\) are any representatives of these germs, then the intersection \((X,p) \cap (Y,p)\), is the germ \((X \cap Y,p)\) and the union \((X,p) \cup (Y,p)\) is the germ \((X \cup Y,p)\).

Check that the definition of subset, union, and intersection of germs of sets is well-defined.

Let \(R\) be some ring of germs of complex-valued functions at \(p \in X\) for some topological space \(X\). If \(f\) is a complex-valued function, let \(Z_f\) be the zero set of \(f\), that is \(f^{-1}(0)\). When \((f,p) \in R\) is a germ of a function it makes sense to talk about the germ \((Z_f,p)\). We take the zero set of some representative and look at its germ at \(p\).

Suppose \(f\) and \(g\) are two representatives of a germ \((f,p)\) show that the germs \((Z_f,p)\) and \((Z_g,p)\) are the same.

Show that if \((f,p)\) and \((g,p)\) are in \(R\) and \(f\) and \(g\) are some representatives, then \((Z_f,p) \cup (Z_g,p) = (Z_{fg},p)\).