Ringed space

In mathematics, a ringed space is a special way to study shapes and numbers together. It connects a topological space with rings of numbers that change depending on where we look in that space. These rings are linked by special maps called ring homomorphisms.

A key idea in a ringed space is the structure sheaf. This tool assigns a ring of numbers to every open area of the space. It helps us understand how functions behave across the whole space. One important type of ringed space is called a locally ringed space.

Ringed spaces are useful in many areas of math, such as analysis, complex algebraic geometry, and scheme theory. Most books, like Hartshorne, assume these rings are commutative rings.

Definitions

A ringed space is a special way to study shapes and patterns in mathematics. It mixes a topological space — which tells us about points and areas around them — with sets of numbers (called rings) that change depending on where you look. These rings are organized in a structure called a sheaf, which helps us see how they connect in different parts of the space.

A locally ringed space is a type of ringed space. When you look closely at one point, the numbers there have a special property. This helps us study more detailed ideas, like how shapes can change up close.

Examples

In a ringed space, we can think of a space with extra information on its open areas. For example, if we have any space, we can create a locally ringed space by using the sheaf of real-valued or complex-valued continuous functions on its open parts. At each point, this creates a local ring of functions.

If the space is a manifold, we can use differentiable or holomorphic functions instead. For algebraic varieties with the Zariski topology, we can use rational mappings that stay finite within the open set. These ideas lead to more general structures like schemes, which are formed by gluing together spectra of commutative rings.

Main article: Spectrum

Main articles: Scheme, Locally ringed space

Morphisms

A morphism between two ringed spaces has two parts: a continuous map between the spaces and a special map between their structure sheaves. This lets us move from one space to another while keeping the ring structures the same.

For locally ringed spaces, there is an extra rule: the maps between the local rings must match the "maximal ideals," which are special subsets of the rings. This helps keep everything consistent when we move between spaces. These morphisms can be combined, creating categories of ringed spaces and locally ringed spaces.

Tangent spaces

See also: Zariski tangent space

Locally ringed spaces help us learn about changes at a point. Imagine you have a point on a shape and you want to know how things change around that point.

We use special numbers called a "field" and a space of values called a "vector space." The tangent space tells us about these changes. It is made using the dual of this vector space. This helps us understand how functions act near a point.

Modules over the structure sheaf

Main article: Sheaf of modules

When we study a special kind of space called a locally ringed space, we can look at collections of objects called sheaves of modules. These sheaves follow rules that link them to the rings on the space.

For each open part of the space, these sheaves behave like modules, which are groups with extra structure. At any single point, the collections also show this module-like property. This helps mathematicians see how different parts of the space are connected.