Ringed space

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a ringed space is a family of (<a href="/facts/Commutative_ring/QS1r2ovR">commutative</a>) <a href="/facts/Ring_(mathematics)/JnCUXz63">rings</a> parametrized by <a href="/facts/Open_subset/QfTCIqMu">open subsets</a> of a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> together with <a href="/facts/Ring_homomorphism/UptcMpFk">ring homomorphisms</a> that play roles of <a href="/facts/Restriction_(mathematics)/JV59DhKy">restrictions</a>. Precisely, it is a topological space equipped with a <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">sheaf</a> of rings called a structure sheaf. It is an abstraction of the concept of the rings of <a href="/facts/Continuous_function/sKbl02pB">continuous</a> (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 <a href="/facts/Stalk_of_a_sheaf/qAD2etdW">stalk</a> at a point and the ring of <a href="/facts/Germ_of_a_function/g28mtBu5">germs of functions</a> at a point is valid.
Ringed spaces appear in <a href="/facts/Mathematical_analysis/XXeR26Ih">analysis</a> as well as <a href="/facts/Complex_algebraic_geometry/rh7RHVp3">complex algebraic geometry</a> and the <a href="/facts/Scheme_theory/c7uSqGqL">scheme theory</a> of <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>.
Note: In the definition of a ringed space, most expositions tend to restrict the rings to be <a href="/facts/Commutative_ring/QS1r2ovR">commutative rings</a>, including <a href="/facts/Algebraic_Geometry_(book)/YcAhHWBj">Hartshorne</a> and Wikipedia. <a href="/facts/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique/SDtmhwWH">Éléments de géométrie algébrique</a>, on the other hand, does not impose the commutativity assumption, although the book mostly considers the commutative case.

Ringed space open-in-new

Ringed space