- Ringed space
In

mathematics , a**ringed space**is, intuitively speaking, a space together with a collection ofcommutative ring s, the elements of which are "functions" on eachopen set of the space. Ringed spaces appear throughout analysis and are also used to define the schemes ofalgebraic geometry .**Definition**Formally, a

**ringed space**is atopological space "X" together with a sheaf ofcommutative ring s "O"_{"X"}on "X". The sheaf "O"_{"X"}is called the**structure sheaf**of "X".A

**locally ringed space**is a ringed space ("X", "O"_{"X"}) such that all stalks of "O"_{"X"}arelocal ring s (i.e. they have uniquemaximal ideal s). Note that it is "not" required that "O"_{"X"}("U") be a local ring for every open set "U" — in fact, that is almost never going to be the case.**Examples**An arbitrary topological space "X" can be considered a locally ringed space by taking "O

_{X}" to be the sheaf of real-valued (or complex-valued)continuous function s on open subsets of "X" (there may exist continuous functions over open subsets of X which are not the restriction of any continuous function over X). The stalk at a point "x" can be thought of as the set of all germs of continuous functions at "x"; this is a local ring withmaximal ideal consisting of those germs whose value at "x" is 0.If "X" is a

manifold with some extra structure, we can also take the sheaf ofdifferentiable , or complex-analytic functions. Both of these give rise to locally ringed spaces.If "X" is an

algebraic variety carrying theZariski topology , we can define a locally ringed space by taking "O_{X}"("U") to be the ring ofrational function s defined on the Zariski-open set "U" which do not blow up (become infinite) within 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.**Morphisms**A

morphism of ringed spaces is simply amorphism of sheaves . Explicitly, a morphism from ("X", "O_{X}") to ("Y", "O_{Y}") is given by the following data:* a continuous map "f" : "X" → "Y"

* a family ofring homomorphism s φ_{"V"}: "O_{Y}"("V") → "O_{X}"("f"^{ -1}("V")) for everyopen set "V" of "Y" which commute with the restriction maps. That is, if "V"_{1}⊂ "V"_{2}are two open subsets of "Y", then the following diagram must commute (the vertical maps are the restriction homomorphisms):There is an additional requirement for morphisms between "locally" ringed spaces:

*the ring homomorphisms induced by φ between the stalks of "Y" and the stalks of "X" must be "local homomorphisms", i.e. for every "x" ∈ "X" the maximal ideal of the local ring (stalk) at "f"("x") ∈ "Y" is mapped to the maximal ideal of the local ring at "x" ∈ "X".

Two morphisms can be composed to form a new morphism, and we obtain the category of ringed spaces and the category of locally ringed spaces.

Isomorphism s in these categories are defined as usual.**Tangent spaces**Locally ringed spaces have just enough structure to allow the meaningful definition of

tangent space s. Let "X" be locally ringed space with structure sheaf "O_{X}"; we want to define the tangent space "T_{x}" at the point "x" ∈ "X". Take the local ring (stalk) "R_{x}" at the point "x", with maximal ideal "m"_{"x"}. Then "k"_{"x"}:= "R_{x}"/"m_{x}" is a field and "m_{x}"/"m_{x}^{2}" is avector space over that field (thecotangent space ). The tangent space "T_{x}" is defined as the dual of this vector space.The idea is the following: a tangent vector at "x" should tell you how to "differentiate" "functions" at "x", i.e. the elements of "R

_{x}". Now it is enough to know how to differentiate functions whose value at "x" is zero, since all other functions differ from these only by a constant, and we know how to differentiate constants. So we only need to worry about "m"_{"x"}. Furthermore, if two functions are given with value zero at "x", then their product has derivative 0 at "x", by theproduct rule . So we only need to know how to assign "numbers" to the elements of "m_{x}"/"m_{x}^{2}", and this is what the dual space does.**"O**_{X}" modulesGiven a locally ringed space ("X", "O

_{X}"), certain sheaves of modules on "X" occur in the applications, the "O_{X}"-modules. To define them, consider a sheaf "F" ofabelian group s on "X". If "F"("U") is a module over the ring "O_{X}"("U") for every open set "U" in "X", and the restriction maps are compatible with the module structure, then we call "F" an "O_{X}"-module. In this case, the stalk of "F" at "x" will be a module over the local ring (stalk) "R"_{"x"}, for every "x"∈"X".A morphism between two such "O

_{X}"-modules is amorphism of sheaves which is compatible with the given module structures. The category of "O_{X}"-modules over a fixed locally ringed space ("X", "O_{X}") is anabelian category .An important subcategory of the category of "O"

_{"X"}-modules is the category of "quasi-coherent" sheaves on "X". A sheaf of "O"_{"X"}-modules is called quasi-coherent if it is, locally, isomorphic to the cokernel of a map between free "O"_{"X"}-modules. A "coherent" sheaf "F" is a quasi-coherent sheaf which is, locally, of finite type and for every open subset "U" of "X" the kernel of any morphism from a free "O"_{"U"}-modules of finite rank to "F"_{"U"}is also of finite type.

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