Ringed space

In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of rings called a structure sheaf. It is an abstraction of the concept of the rings of continuous (scalar-valued) functions on open subsets.

Among ringed spaces, especially important and prominent is a locally ringed space: a ringed space in which the analogy between the stalk at a point and the ring of germs of functions at a point is valid.

Ringed spaces appear in analysis as well as complex algebraic geometry and the scheme theory of algebraic geometry.

Note: In the definition of a ringed space, most expositions tend to restrict the rings to be commutative rings, including Hartshorne and Wikipedia. Éléments de géométrie algébrique, on the other hand, does not impose the commutativity assumption, although the book mostly considers the commutative case.

Definitions

A ringed space

(

X

,

O

X

)

{\displaystyle (X,{\mathcal {O}}_{X})}

is a topological space

X

{\displaystyle X}

together with a sheaf of rings

O

X

{\displaystyle {\mathcal {O}}_{X}}

on

X

{\displaystyle X}

. The sheaf

O

X

{\displaystyle {\mathcal {O}}_{X}}

is called the structure sheaf of

X

{\displaystyle X}

.

A locally ringed space is a ringed space

(

X

,

O

X

)

{\displaystyle (X,{\mathcal {O}}_{X})}

such that all stalks of

O

X

{\displaystyle {\mathcal {O}}_{X}}

are local rings (i.e.