locally ringed space

locally ringed space (plural locally ringed spaces)

1. (sheaf theory, scheme theory) An abstract mathematical object generalizing the relationship between spaces of functions and the germs of those functions at particular points (i.e. a ringed space) where the analogy with the motivating case is particularly strong. Formally, a topological space equipped with a sheaf of rings (called the structure sheaf) such that the stalk at each point is a local ring.

   Hypernym: ringed space

   Hyponym: scheme

Properly, a locally ringed space is a pair ${\displaystyle (X,{\mathcal {O}}_{X})}$ where ${\displaystyle X}$ is the space and ${\displaystyle O_{X}}$ is the structure sheaf. By abuse of notation, ${\displaystyle X}$ alone may also be referred to as a locally ringed space when ${\displaystyle {\mathcal {O}}_{X}}$ is understood to be present in some context.