locally ringed space

1 Definitions

A locally ringed space is a topological spaceMathworldPlanetmath X together with a sheaf of rings π’ͺX with the property that, for every point p∈X, the stalk (π’ͺX)p is a local ringMathworldPlanetmath11All rings mentioned in this article are required to be commutativePlanetmathPlanetmathPlanetmathPlanetmath..

A morphism of locally ringed spaces from (X,π’ͺX) to (Y,π’ͺY) is a continuous mapMathworldPlanetmath f:X⟢Y together with a morphism of sheaves Ο•:π’ͺY⟢π’ͺX with respect to f such that, for every point p∈X, the induced ring homomorphismMathworldPlanetmath on stalks Ο•p:(π’ͺY)f⁒(p)⟢(π’ͺX)p is a local homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. That is,

Ο•p⁒(y)∈π”ͺp⁒ for every ⁒y∈π”ͺf⁒(p),

where π”ͺp (respectively, π”ͺf⁒(p)) is the maximal idealMathworldPlanetmath of the ring (π’ͺX)p (respectively, (π’ͺY)f⁒(p)).

2 Applications

Locally ringed spaces are encountered in many natural contexts. Basically, every sheaf on the topological space X consisting of continuous functionsMathworldPlanetmath with values in some field is a locally ringed space. Indeed, any such function which is not zero at a point p∈X is nonzero and thus invertiblePlanetmathPlanetmath in some neighborhood of p, which implies that the only maximal ideal of the stalk at p is the set of germs of functions which vanish at p. The utility of this definition lies in the fact that one can then form constructions in familiar instances of locally ringed spaces which readily generalize in ways that would not necessarily be obvious without this framework. For example, given a manifold X and its locally ringed space π’ŸX of real–valued differentiable functions, one can show that the space of all tangent vectors to X at p is naturally isomorphic to the real vector space (π”ͺp/π”ͺp2)*, where the * indicates the dual vector space. We then see that, in general, for any locally ringed space X, the space of tangent vectors at p should be defined as the k–vector space (π”ͺp/π”ͺp2)*, where k is the residue fieldMathworldPlanetmath (π’ͺX)p/π”ͺp and * denotes dual with respect to k as before. It turns out that this definition is the correct definition even in esoteric contexts like algebraic geometryMathworldPlanetmathPlanetmath over finite fields which at first sight lack the differential structure needed for constructions such as tangent vector.

Another useful application of locally ringed spaces is in the construction of schemes. The forgetful functorMathworldPlanetmathPlanetmath assigning to each locally ringed space (X,π’ͺX) the ring π’ͺX⁒(X) is adjointPlanetmathPlanetmath to the β€œprime spectrum” functorMathworldPlanetmath taking each ring R to the locally ringed space Spec⁑(R), and this correspondence is essentially why the categoryMathworldPlanetmath of locally ringed spaces is the proper building block to use in the formulation of the notion of scheme.

Title locally ringed space
Canonical name LocallyRingedSpace
Date of creation 2013-03-22 12:37:41
Last modified on 2013-03-22 12:37:41
Owner djao (24)
Last modified by djao (24)
Numerical id 13
Author djao (24)
Entry type Definition
Classification msc 14A15
Classification msc 18F20
Related topic LocalRing
Related topic PrimeSpectrum
Related topic Scheme
Defines morphism of locally ringed spaces