Punkteringer

Punkteringer, or "point rings," is a term occasionally used in Danish mathematical writing to denote the local ring of a point on a geometric object. In English-language algebraic geometry, the corresponding object is usually called the local ring at a point. A punktering captures the algebraic structure of functions defined in a neighborhood of that point.

Definition. Let X be a scheme (or a variety) over a field and p a point of X. The punktering at p is the local ring OX,p, consisting of regular functions defined near p. It has a unique maximal ideal mp, consisting of those functions vanishing at p. The residue field is k(p) = OX,p/mp.

Properties. OX,p is a local ring with a single maximal ideal. Its Krull dimension equals the dimension of the space at p. The ring is regular at p precisely when the minimal number of generators of mp equals dim OX,p. The cotangent space at p is dual to mp/(mp)2, and the tangent space is obtained from that dual space.

Examples. On a smooth variety of dimension d, the local ring at any point is a regular local ring of dimension d. At a singular point, the local ring is non-regular and reflects the nature of the singularity (for instance, increased embedding dimension or more complicated mp-structure).

Applications. Punkteringer are used to study local properties of schemes, including singularities, dimension theory, and deformation problems. They provide a framework for defining and analyzing invariants such as multiplicity, embedding dimension, and tangent spaces, which inform geometric and algebraic behavior near p.

See also. Local ring, stalk, germ, tangent space, residue field, deformation theory. Notes. The term appears in some Danish texts; standard English terminology is local ring.