Ringed space

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. ^[1]

Definitions

A ringed space${\displaystyle (X,{\mathcal {O}}_{X})}$ is a topological space ${\displaystyle X}$ together with a sheaf of rings ${\displaystyle {\mathcal {O}}_{X}}$ on ${\displaystyle X}$. The sheaf ${\displaystyle {\mathcal {O}}_{X}}$ is called the structure sheaf of ${\displaystyle X}$.

A locally ringed space is a ringed space ${\displaystyle (X,{\mathcal {O}}_{X})}$ such that all stalks of ${\displaystyle {\mathcal {O}}_{X}}$ are local rings (i.e. they have unique maximal ideals). Note that it is not required that ${\displaystyle {\mathcal {O}}_{X}(U)}$ be a local ring for every open set ${\displaystyle U}$; in fact, this is almost never the case.

Examples

An arbitrary topological space ${\displaystyle X}$ can be considered a locally ringed space by taking ${\displaystyle {\mathcal {O}}_{X}}$ to be the sheaf of real-valued (or complex-valued) continuous functions on open subsets of ${\displaystyle X}$. The stalk at a point ${\displaystyle x}$ can be thought of as the set of all germs of continuous functions at ${\displaystyle x}$; this is a local ring with the unique maximal ideal consisting of those germs whose value at ${\displaystyle x}$ is ${\displaystyle 0}$.

If ${\displaystyle X}$ 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 ${\displaystyle X}$ is an algebraic variety carrying the Zariski topology, we can define a locally ringed space by taking ${\displaystyle {\mathcal {O}}_{X}(U)}$ to be the ring of rational mappings defined on the Zariski-open set ${\displaystyle U}$ that do not blow up (become infinite) within ${\displaystyle U}$. 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.

Morphisms

A morphism from ${\displaystyle (X,{\mathcal {O}}_{X})}$ to ${\displaystyle (Y,{\mathcal {O}}_{Y})}$ is a pair ${\displaystyle (f,\varphi )}$, where ${\displaystyle f:X\to Y}$ is a continuous map between the underlying topological spaces, and ${\displaystyle \varphi$ :{\mathcal {O}}_{Y}\to f_{*}{\mathcal {O}}_{X}} is a morphism from the structure sheaf of ${\displaystyle Y}$ to the direct image of the structure sheaf of X. In other words, a morphism from ${\displaystyle (X,{\mathcal {O}}_{X})}$ to ${\displaystyle (Y,{\mathcal {O}}_{Y})}$ is given by the following data:

• a continuous map ${\displaystyle f:X\to Y}$
• a family of ring homomorphisms ${\displaystyle \varphi _{V}:{\mathcal {O}}_{Y}(V)\to {\mathcal {O}}_{X}(f^{-1}(V))}$ for every open set ${\displaystyle V}$ of ${\displaystyle Y}$ that commute with the restriction maps. That is, if ${\displaystyle V_{1}\subseteq V_{2}}$ are two open subsets of ${\displaystyle Y}$, 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 ${\displaystyle \varphi }$ between the stalks of ${\displaystyle Y}$ and the stalks of ${\displaystyle X}$ must be local homomorphisms , i.e. for every ${\displaystyle x\in X}$ the maximal ideal of the local ring (stalk) at ${\displaystyle f(x)\in Y}$ is mapped into the maximal ideal of the local ring at ${\displaystyle x\in X}$.

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. Isomorphisms in these categories are defined as usual.

Tangent spaces

See also: Zariski tangent space

Locally ringed spaces have just enough structure to allow the meaningful definition of tangent spaces. Let ${\displaystyle X}$ be a locally ringed space with structure sheaf ${\displaystyle {\mathcal {O}}_{X}}$; we want to define the tangent space ${\displaystyle T_{x}(X)}$ at the point ${\displaystyle x\in X}$. Take the local ring (stalk) ${\displaystyle R_{x}}$ at the point ${\displaystyle x}$, with maximal ideal ${\displaystyle {\mathfrak {m}}_{x}}$. Then ${\displaystyle k_{x}:=R_{x}/{\mathfrak {m}}_{x}}$ is a field and ${\displaystyle {\mathfrak {m}}_{x}/{\mathfrak {m}}_{x}^{2}}$ is a vector space over that field (the cotangent space). The tangent space ${\displaystyle T_{x}(X)}$ is defined as the dual of this vector space.

The idea is the following: a tangent vector at ${\displaystyle x}$ should tell you how to "differentiate" "functions" at ${\displaystyle x}$, i.e. the elements of ${\displaystyle R_{x}}$. Now it is enough to know how to differentiate functions whose value at ${\displaystyle x}$ 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 consider ${\displaystyle {\mathfrak {m}}_{x}}$. Furthermore, if two functions are given with value zero at ${\displaystyle x}$, then their product has derivative 0 at ${\displaystyle x}$, by the product rule. So we only need to know how to assign "numbers" to the elements of ${\displaystyle {\mathfrak {m}}_{x}/{\mathfrak {m}}_{x}^{2}}$, and this is what the dual space does.

Modules over the structure sheaf

Main article: Sheaf of modules

Given a locally ringed space ${\displaystyle (X,{\mathcal {O}}_{X})}$, certain sheaves of modules on ${\displaystyle X}$ occur in the applications, the ${\displaystyle {\mathcal {O}}_{X}}$-modules. To define them, consider a sheaf F of abelian groups on ${\displaystyle X}$. If F(U) is a module over the ring ${\displaystyle {\mathcal {O}}_{X}(U)}$ for every open set ${\displaystyle U}$ in ${\displaystyle X}$, and the restriction maps are compatible with the module structure, then we call ${\displaystyle F}$ an ${\displaystyle {\mathcal {O}}_{X}}$-module. In this case, the stalk of ${\displaystyle F}$ at ${\displaystyle x}$ will be a module over the local ring (stalk) ${\displaystyle R_{x}}$, for every ${\displaystyle x\in X}$.

A morphism between two such ${\displaystyle {\mathcal {O}}_{X}}$-modules is a morphism of sheaves that is compatible with the given module structures. The category of ${\displaystyle {\mathcal {O}}_{X}}$-modules over a fixed locally ringed space ${\displaystyle (X,{\mathcal {O}}_{X})}$ is an abelian category.

An important subcategory of the category of ${\displaystyle {\mathcal {O}}_{X}}$-modules is the category of quasi-coherent sheaves on ${\displaystyle X}$. A sheaf of ${\displaystyle {\mathcal {O}}_{X}}$-modules is called quasi-coherent if it is, locally, isomorphic to the cokernel of a map between free ${\displaystyle {\mathcal {O}}_{X}}$-modules. A coherent sheaf ${\displaystyle F}$ is a quasi-coherent sheaf that is, locally, of finite type and for every open subset ${\displaystyle U}$ of ${\displaystyle X}$ the kernel of any morphism from a free ${\displaystyle {\mathcal {O}}_{U}}$-module of finite rank to ${\displaystyle F_{U}}$ is also of finite type.

Citations

1. Éléments de géométrie algébrique, Ch 0, 4.1.1.

References

• Section 0.4 of Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS . 4. doi:10.1007/bf02684778. MR   0217083.
• Hartshorne, Robin (1977), Algebraic Geometry , Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN   978-0-387-90244-9, MR   0463157

External links

• Onishchik, A.L. (2001) [1994], "Ringed space", Encyclopedia of Mathematics , EMS Press