Scheme (mathematics)

In mathematics, specifically algebraic geometry, a scheme is a special structure that builds on the idea of an algebraic variety. Schemes let mathematicians study shapes and patterns more deeply. They look at details like how often equations repeat and allow these shapes to use different kinds of numbers, such as the integers.

Scheme theory began with Alexander Grothendieck in 1960. It gives strong tools for solving big problems in algebraic geometry. It also links this part of math to number theory.

Schemes use ideas from commutative algebra, topology, and homological algebra. They study algebraic shapes by looking at the functions that can be defined on them. This helps connect many areas of mathematics, making schemes very important in modern math.

Development

The study of algebraic geometry began with solving polynomial equations using real numbers. Later, using complex numbers made the math easier because of their special properties. In the 20th century, mathematicians wondered if they could use other types of numbers, like those with positive characteristic or number rings such as the integers.

Over time, mathematicians like Emmy Noether and Wolfgang Krull used algebra to better understand these shapes. They studied points and special sets of equations called prime ideals. Other mathematicians then expanded these ideas to more complex shapes, leading to what we now call "schemes" in algebraic geometry.

Origin of schemes

The idea of schemes was created by a mathematician named Alexander Grothendieck. He wrote a big book called Éléments de géométrie algébrique. In this book, he showed a new way to think about math spaces. He used something called the "spectrum" of a ring. This helps organize special points called prime ideals.

Grothendieck's work helped mathematicians study more complex shapes. It also helped solve important problems. His ideas connected geometry with number theory in new ways.

Definition

A scheme is a special kind of space used in a part of math called algebraic geometry. It helps mathematicians study shapes and patterns in a more detailed way. Think of it like putting together puzzle pieces where each piece is called an "affine scheme." These pieces are glued together using a special method.

One simple example of an affine scheme is something called "affine n-space." This is like the space you’re used to — only with n directions instead of just the three we experience everyday. Schemes can be built over different kinds of number systems, making them very useful tools in advanced math.

The category of schemes

Schemes are organized into a special group called a category. Connections between schemes are called morphisms. When we talk about one scheme over another, we mean there is a special kind of connection between them.

For schemes over a field, there are rules to decide what can be called a variety.

In this world of schemes, there are special mappings that connect the rules for these shapes. For example, when dealing with simple building blocks of schemes, these mappings match with mappings between certain number systems. Schemes also have special points that match with solutions to equations, extending ideas from older math.

One important feature is that combining two schemes over a common one always works and gives another scheme. For example, combining two simple space shapes over a field results in a larger space shape.

Examples

In algebraic geometry, schemes expand the idea of algebraic varieties. They help mathematicians study problems with equations and different types of numbers. Scheme theory was developed by Alexander Grothendieck in the 1960s.

One key example is the affine space, which is all points with coordinates in a certain field. Schemes help us understand these points better by looking at the points and algebraic equations. Another example is the scheme from the ring of integers, which connects number theory with geometry.

Schemes also include projective spaces, which are like affine spaces but with added points at infinity. These examples show how schemes help explore algebra and geometry.

Motivation for schemes

Schemes help us understand shapes in algebra by using newer ideas. They let us study solutions to equations with any kind of number system, not just simple ones. This helps us describe shapes even when some answers only show up when we use more complex numbers.

Schemes also introduce the "generic point," which connects geometry with number theory in a deeper way. They let us work with "nilpotent elements," which are very small changes that disappear when multiplied by themselves. This lets us use ideas from calculus, like derivatives, when studying shapes made from equations.

Coherent sheaves

Main article: Coherent sheaf

Coherent sheaves are important ideas in scheme theory. They help us understand shapes by describing how things are arranged on them. For a scheme (X), we look at special structures called (\mathcal{O}_X)-modules, which are like bundles of functions on (X).

Coherent sheaves include vector bundles and more. They help us learn about smaller shapes inside a bigger shape. A key tool that uses these sheaves is called coherent sheaf cohomology, which helps solve problems in algebraic geometry.

Generalizations

A scheme can be made more general in a few ways. One way is to use something called the étale topology. This leads to the idea of an algebraic space. This is a structure made from schemes and étale equivalence relations.

Another way is the idea of a stack. Stacks are like algebraic spaces but they also include an algebraic group for each point. This helps when studying moduli spaces, which classify objects and their symmetries.

There is also a way to generalize schemes using ideas from homotopy theory, called derived algebraic geometry. In this method, the usual structure of a scheme is changed to more complex structures that can hold extra information.