Scheme (mathematics)

In mathematics, specifically algebraic geometry, a scheme is a structure that expands the idea of an algebraic variety. Schemes help mathematicians study shapes and patterns by considering more details, like how many times equations repeat and allowing these shapes to be defined using different types of numbers, such as the integers.

Scheme theory was introduced by Alexander Grothendieck in 1960. It provides powerful tools to solve important problems in algebraic geometry and connects this area of math to number theory. This connection was crucial in proving famous theorems, like Wiles's proof of Fermat's Last Theorem.

Schemes are built using ideas from commutative algebra, topology, and homological algebra. They treat algebraic shapes by looking at the functions that can be defined on them, helping to unify many different parts of mathematics. This makes schemes a central and important concept in modern mathematics.

Development

The study of algebraic geometry started with solving polynomial equations using real numbers. Later, it was found easier to use complex numbers because they have special properties that simplify the math. In the 20th century, mathematicians began to wonder if they could study these problems using 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 developed tools from algebra to understand these geometric shapes better. They studied not just points, but also special sets of equations called prime ideals. Later, other mathematicians expanded these ideas to work with more complex shapes and settings, laying the groundwork for what we now call "schemes" in algebraic geometry.

Origin of schemes

The idea of schemes was fully developed by a mathematician named Alexander Grothendieck in his major work called Éléments de géométrie algébrique. He introduced a new way to look at mathematical spaces using something called the "spectrum" of a ring, which organizes special points called prime ideals.

Grothendieck's work let mathematicians study more complex shapes and solve important problems, even when those shapes aren't easy to picture. His ideas helped connect geometry with number theory in new and useful 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, which lets mathematicians look at things that regular geometry can't easily handle.

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 flexible tools in advanced math.

The category of schemes

Schemes are organized into a special group called a category, where 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 specific rules to decide what can be called a variety.

In this world of schemes, there are special mappings that connect the rules governing these shapes. For example, when dealing with simple building blocks of schemes, these mappings match exactly 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 are structures that expand the idea of algebraic varieties. They allow mathematicians to study problems that involve multiplicities and equations over different types of numbers. Scheme theory was developed by Alexander Grothendieck in the 1960s to solve deep problems in algebraic geometry.

One key example is the affine space, which can be thought of as all points with coordinates in a certain field. Schemes help us understand these points better by considering both the points themselves and additional information from algebraic equations. Another example is the scheme built from the ring of integers, which helps connect number theory with geometry.

Schemes also include projective spaces, which are like affine spaces but with added points at infinity, making them more complete for certain geometric studies. These examples show how schemes provide a powerful framework for exploring both algebra and geometry.

Motivation for schemes

Schemes help us understand shapes in algebra by going beyond older ideas. One way they do this is by letting us study solutions to equations over any kind of number system, not just ones where all numbers are "complete." This means we can describe shapes even when some solutions only appear when we expand our number system.

Another important idea in schemes is the "generic point." This lets us connect geometric ideas with number theory in a deeper way. Schemes also let us work with "nilpotent elements," which are like tiny changes so small that their square is zero. This lets us bring ideas from calculus, like derivatives, into the study of shapes made from equations.

Coherent sheaves

Main article: Coherent sheaf

Coherent sheaves are important ideas in scheme theory. They generalize something called vector bundles, which help describe geometric shapes. 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 can describe information 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 generalized in several ways. One way is to use the étale topology, which leads to the idea of an algebraic space. This is a structure that is built from schemes and étale equivalence relations.

Another generalization is the idea of a stack. Stacks are similar to algebraic spaces but also include an algebraic group for each point, which represents the automorphisms of that point. This helps in studying moduli spaces, which classify objects while keeping track of their symmetries.

There is also a way to generalize schemes by using ideas from homotopy theory, called derived algebraic geometry. In this approach, the usual structure of a scheme is replaced by more complex structures that can remember additional information.