Ringed space

 ( Sheaf Theory ) 

In mathematics, a ringed space is a family of (Commutative ring) rings parametrized by 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 , 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. Éléments de géométrie algébrique, Ch 0, 4.1.1.

---

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

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

---

Examples An arbitrary topological space $X$ can be considered a locally ringed space by taking $\backslash mathcal\{O\}\_X$ to be the sheaf of Real number (or Complex number) continuous functions on open subsets of $X$. 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 with the unique maximal 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 of differentiable, or holomorphic functions. Both of these give rise to locally ringed spaces.

If $X$ is an algebraic variety carrying the Zariski topology, we can define a locally ringed space by taking $\backslash mathcal\{O\}\_X(U)$ to be the ring of defined on the Zariski-open set $U$ that 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 from $(X,\backslash mathcal\{O\}\_X)$ to $(Y,\backslash mathcal\{O\}\_Y)$ is a pair $(f,\backslash varphi)$, where $f:X\backslash to\; Y$ is a continuous map between the underlying topological spaces, and $\backslash varphi:\backslash mathcal\{O\}\_Y\backslash to\; f\_*\backslash mathcal\{O\}\_X$ is a morphism from the structure sheaf of $Y$ to the direct image of the structure sheaf of . In other words, a morphism from $(X,\backslash mathcal\{O\}\_X)$ to $(Y,\backslash mathcal\{O\}\_Y)$ is given by the following data:

• a continuous map $f:X\backslash to\; Y$
• a family of ring homomorphisms $\backslash varphi\_V\; :\; \backslash mathcal\{O\}\_Y(V)\backslash to\backslash mathcal\{O\}\_X(f^\{-1\}(V))$ for every open set $V$ of $Y$ that commute with the restriction maps. That is, if $V\_1\backslash subseteq\; V\_2$ 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 $\backslash varphi$ between the stalks of $Y$ and the stalks of $X$ must be local homomorphisms, i.e. for every $x\backslash in\; X$ the maximal ideal of the local ring (stalk) at $f(x)\backslash in\; Y$ is mapped into the maximal ideal of the local ring at $x\backslash in\; X$.

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. in these categories are defined as usual.

---

Tangent spaces Locally ringed spaces have just enough structure to allow the meaningful definition of . Let $X$ be a locally ringed space with structure sheaf $\backslash mathcal\{O\}\_X$; we want to define the tangent space $T\_x(X)$ at the point $x\backslash in\; X$. Take the local ring (stalk) $R\_x$ at the point $x$, with maximal ideal $\backslash mathfrak\{m\}\_x$. Then $k\_x\; :=\; R\_x/\backslash mathfrak\{m\}\_x$ is a field and $\backslash mathfrak\{m\}\_x/\backslash mathfrak\{m\}\_x^2$ is a vector space over that field (the cotangent space). The tangent space $T\_x(X)$ is defined as the dual space 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 consider $\backslash mathfrak\{m\}\_x$. Furthermore, if two functions are given with value zero at $x$, then their product has derivative 0 at $x$, by the product rule. So we only need to know how to assign "numbers" to the elements of $\backslash mathfrak\{m\}\_x/\backslash mathfrak\{m\}\_x^2$, and this is what the dual space does.

---

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

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

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

---

Citations

• Section 0.4 of

---

External links

Post Comment