Germ

2020 Mathematics Subject Classification: Primary: 14-XX [MSN][ZBL]

A term signifying a "pointwise localization" of various mathematical objects (germs of functions, germs of mappings, germs of analytic sets, etc.). Let, for example, $x$ be a point in a topological space and let $F$ be some family of functions defined in a neighbourhood of $x$ (each in its own neighbourhood). Two functions $f,g\in F$ are said to be equivalent (at $x$) if they coincide in some neighbourhood of $x$. An equivalence class generated by this relation is called a germ of functions of class $F$ at $x$. In this way are defined the germs of continuous functions, of differentiable functions at the points of a differentiable manifold, of holomorphic functions at the points of a complex manifold, etc. If the family $F$ has some algebraic structure, then the set of germs of functions of the family $F$ inherits this structure (the operations are carried out on representatives of classes). In particular, the germs of holomorphic functions at a point $z$ form a ring. Elements of the quotient field of this ring are called germs of meromorphic functions at $z$.

Similarly one can define a germ of a family of subsets of a topological space. For instance, at the points of an analytic manifold there are germs of analytic sets (the equivalence class is defined by coincidence in a neighbourhood of a given point). On germs of families of subsets set-theoretic operations and relations are naturally defined. The notion of a germ is also meaningful in the case of other objects defined on open subsets of a topological space.

See also Analytic function; Meromorphic function; Sheaf.

Comments[edit]

The germs of analytic spaces or schemes are characterized by the stalks of their structure sheaves. These are local rings.

The study of germs of differentiable mappings is the subject of singularity theory (cf. Singularities of differentiable mappings).

An important theorem for the theory of germs of analytic sets is Weierstrass' preparation theorem (cf. Weierstrass theorem), see also [He].

References[edit]