Affine variety

In algebraic geometry, an affine variety or affine algebraic variety is a special type of shape that lives inside a flat space called an affine space. These shapes are defined using equations with many variables, called polynomials. When we solve these polynomial equations together, we find points that make up the affine variety.

An affine algebraic set is simply the collection of points where certain polynomials equal zero. But an affine variety is a bit more special—it cannot be broken down into smaller pieces that are also algebraic sets. This means the equations that define it are tightly linked together in a very important way.

One-dimensional affine varieties are known as affine algebraic curves, and two-dimensional ones are affine algebraic surfaces. These ideas help mathematicians understand the solutions to polynomial equations and their properties. For example, Fermat's Last Theorem is about a special affine variety that has no simple solutions for certain powers.

Introduction

An affine algebraic set is a group of points that solve a system of equations made from polynomials. These equations have their answers in a special kind of number system. When we find all the points that make all these equations true at the same time, we get an affine algebraic set.

An affine variety is a special type of affine algebraic set. It cannot be split into smaller, simpler groups of these points. We call this kind of set "irreducible." The dimension of a variety is a number that tells us about the variety, and it can be defined in many different ways.

Examples

The space around a special shape called a hypersurface in an affine variety remains affine. This means we can describe it using certain equations.

For example, if we take a line and remove just one point, it stays affine and looks like a special curve in a two-dimensional space. However, if we take a flat plane and remove just one point, it is no longer an affine variety.

Special shapes called hypersurfaces in an affine space are defined by just one polynomial equation. Also, making a simpler version of an irreducible affine variety stays affine.

Rational points

Main article: rational point

For a special kind of shape called an affine variety, we can look at points that have certain types of numbers as their positions. If the numbers are rational numbers (like fractions), we call these points "rational points." If the numbers are real numbers (like decimals that don’t end), we call them "real points."

For example, the point (1, 0) is both a rational point and a real point for a shape called a circle. But a point like (√2/2, √2/2) is only a real point, not a rational point, because its numbers are not simple fractions. Some shapes, like a certain circle, don’t have any rational points at all.

Singular points and tangent space

Let V be a special shape in math made from equations. We can look at a point a on this shape and study how the shape behaves near that point.

We use something called the Jacobian matrix to understand this. If the Jacobian matrix has a certain size at point a, the point is called "regular." If not, it is "singular." For regular points, we can describe a flat space called the tangent space that touches the shape just at that point.

The Zariski topology

Main article: Zariski topology

In algebraic geometry, we talk about special ways to organize points in space. One of these ways is called the Zariski topology. It helps us understand how points are grouped together when we look at solutions to certain equations.

When we use the Zariski topology, the points that solve a group of equations form closed sets. These sets follow special rules, like how they can overlap or combine. We can also describe these sets using "open" areas, which are areas where a specific equation does not equal zero. This gives us a complete way to study the shape and structure of solutions in space.

Geometry–algebra correspondence

The shape and structure of an affine variety are closely connected to the algebra of its coordinate ring. For an affine variety ( V ), the set of all polynomials that vanish on ( V ) forms an ideal in the polynomial ring. There is a special relationship between these ideals and the geometric subsets of ( V ).

Special ideals called radical ideals correspond to algebraic subsets of ( V ). When two such ideals are compared, their corresponding geometric sets are also compared in a matching way. Prime ideals in the coordinate ring correspond to affine subvarieties, which are special subsets that cannot be broken down further into smaller algebraic sets.

Maximal ideals correspond to individual points in the variety. This means each point can be linked directly to a specific ideal in the coordinate ring, creating a clear one-to-one connection between points and certain algebraic structures.

Products of affine varieties

When we combine two affine varieties, we use a special rule. Think of it like putting two puzzle pieces together to make a new, bigger puzzle. If we have one shape in a space with n directions and another in a space with m directions, we can combine them into a new space with n+m directions.

We describe these shapes using math rules called polynomials. When we combine the shapes, we use all the rules from both shapes to define the new combined shape. This new shape keeps the special properties of the original shapes.

The way we study these combined shapes using open sets is also unique and different from simply combining the open sets of each individual shape. Some math rules that work in the combined space cannot be made by just multiplying rules from each separate space.

Morphisms of affine varieties

Main article: Morphism of algebraic varieties

A morphism, or regular map, of affine varieties is a special kind of function between these spaces. It uses polynomials to connect points from one space to another. For example, if we have two spaces, V and W, a morphism from V to W is a map that takes each point in V and turns it into a point in W using polynomial formulas.

There is a special connection between these morphisms and the coordinate rings of the spaces. Each morphism between affine varieties corresponds to a certain kind of matching between their coordinate rings, going in the opposite direction. This matching helps show how the shapes and their algebraic descriptions are linked.

Structure sheaf

With a special structure called a sheaf, an affine variety becomes a locally ringed space.

For an affine variety X and its coordinate ring A, the sheaf O X is made by taking the regular functions on open parts of X. These open parts are sets where a certain function does not equal zero.

The important idea is that the functions on these open parts are linked to the coordinate ring in a clear way. This helps show that X is a locally ringed space and that O X behaves nicely as a sheaf.

Serre's theorem on affineness

Main article: Serre's theorem on affineness

A theorem by Serre explains when a special kind of space called an affine variety occurs. It tells us that a space is affine if certain mathematical measurements, called cohomology groups, are zero for all measurements greater than zero. This idea helps mathematicians understand affine varieties better, unlike other spaces where these measurements are very important.

Affine algebraic groups

Main article: linear algebraic group

An affine variety over a special kind of math field is called an affine algebraic group if it has special rules for combining points. These rules include a way to multiply points that follows certain patterns, a special point that acts like a "do nothing" button, and a way to undo the multiplication.

A common example of an affine algebraic group is a set of special transformations of space. These groups help organize and understand certain types of mathematical structures.

Generalizations

Affine varieties can also be studied over fields that are not algebraically closed, like the real numbers. A part of an affine variety that is open is called a quasi-affine variety, and all affine varieties are examples of these. Quasi-affine varieties are also a type of quasi-projective variety.

Affine varieties help describe larger shapes called algebraic varieties, such as projective varieties, by fitting together many affine varieties. Simple structures connected to varieties, like tangent spaces or parts of algebraic vector bundles, are also affine varieties.

In modern mathematics, affine varieties are a special case of objects called affine schemes, which are studied in scheme theory. Each affine variety matches up with an affine scheme, which includes points that represent both the variety itself and its smaller parts. This gives a clearer way to think about special points on the variety. An affine scheme is an affine variety when it meets certain conditions, like being simple and of a specific type over an algebraically closed field.