Locally ringed space | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-29

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Ringed_space


00:01:15 1 Definition
00:01:57 2 Examples
00:03:31 3 Morphisms
00:05:15 4 Tangent spaces
00:06:52 5 iOsubX/sub/i modules



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SUMMARY
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In mathematics, a ringed space can be equivalently thought of as either

(a) a topological space together with a collection of commutative rings, the elements of which are "functions" on each open set of the space.
(b) a family of (commutative) rings parametrized by open subsets of a topological space, together with ring homomorphisms coming from the relationships between open sets; in short, the sheaf of rings.Ringed spaces appear in analysis as well as complex algebraic geometry and scheme theory of algebraic geometry. The point of view (b) is more amenable to generalization; one simply needs to cook up a different way of parametrizing rings (cf. ringed topos).
Note: Many expositions tend to restrict the rings to be commutative rings, including Hartshorne and Wikipedia, in the definition of a ringed space. "Éléments de géométrie algébrique", on the other hand, does not impose the commutativity assumption, although the book only considers the commutative case.