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. they have unique maximal ideals). Note that it is not required that O X ( U ) {\displaystyle {\mathcal {O}}_{X}(U)} be a local ring for every open set U {\displaystyle U} ; in fact, this is almost never the case.
Examples An arbitrary topological space X {\displaystyle X} can be considered a locally ringed space by taking O X {\displaystyle {\mathcal {O}}_{X}} to be the sheaf of real-valued (or complex-valued) continuous functions on open subsets of X {\displaystyle X} . The stalk at a point x {\displaystyle x} can be thought of as the set of all germs of continuous functions at x {\displaystyle x} ; this is a local ring with the unique maximal ideal consisting of those germs whose value at x {\displaystyle x} is 0 {\displaystyle 0} . If X {\displaystyle X} is a manifold with some extra structure, we can also take the sheaf of differentiable, or holomorphic functions. Both of these give rise to locally ringed spaces. If X {\displaystyle X} is an algebraic variety carrying the Zariski topology, we can define a locally ringed space by taking O X ( U ) {\displaystyle {\mathcal {O}}_{X}(U)} to be the ring of rational mappings defined on the Zariski-open set U {\displaystyle U} that do not blow up (become infinite) within U {\displaystyle U} . The important generalization of this example is that of the spectrum of any commutative ring; these spectra are also locally ringed spaces. Schemes are locally ringed spaces obtained by "gluing together" spectra of commutative rings.
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