Hilbert series and Hilbert polynomial

In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related ideas that help us understand how the size of the parts of the algebra grows.

These ideas can also be used for filtered algebras, graded or filtered modules over these algebras, and even for coherent sheaves over projective schemes.

They are often applied in situations like looking at the remainder after dividing by a special set of equations in a multivariate polynomial ring, or studying how a local ring changes when we look at its maximal ideal in different ways. In the last case, the Hilbert polynomial is called the Hilbert–Samuel polynomial.

The Hilbert series and polynomial are very useful in computational algebraic geometry because they are the simplest way to find the size and other important features of shapes defined by equations. They also stay the same for families of these shapes, which helps in building the Hilbert scheme and Quot scheme.

Definitions and main properties

In algebra, we study special kinds of mathematical structures called graded commutative algebras. These algebras are built using a field (a set of numbers) and elements of positive degree. We can think of them as having layers, where each layer has a certain size.

We can measure how these layers grow using three important tools: the Hilbert function, Hilbert series, and Hilbert polynomial. The Hilbert function tells us the size of each layer. The Hilbert series is a special kind of infinite sum that captures this growth in a compact way. When the algebra is generated by elements of degree 1, the Hilbert series simplifies to a form that helps us find the Hilbert polynomial. This polynomial describes the growth of the layers for large enough layers and is a key tool in understanding the algebra’s structure.

Graded algebra and polynomial rings

Polynomial rings and their parts divided by certain rules are typical examples of graded algebras. If a graded algebra is created using special elements of degree 1, it can be matched to a part of a polynomial ring. This matching shows that graded algebras with elements of degree 1 are essentially parts of polynomial rings divided by specific rules. Because of this, the rest of this article will focus on these special parts of polynomial rings.

Properties of Hilbert series

Hilbert series and Hilbert polynomial have special rules when dealing with exact sequences. If you have a sequence of modules that fit together perfectly, the Hilbert series and polynomial of the middle one are just the sum of the Hilbert series and polynomial of the other two.

For a special kind of algebra where you remove a single element, the Hilbert series changes in a predictable way. This helps us understand how the shape of the Hilbert series relates to the dimension of the algebra.

The Hilbert series of a simple polynomial ring has a clear pattern, and this pattern helps us find the Hilbert polynomial. The form of the Hilbert series tells us about the dimension of the algebra and the degree of its Hilbert polynomial.

Degree of a projective variety and Bézout's theorem

The Hilbert series helps us find the degree of an algebraic shape. It also gives a simple way to understand Bézout's theorem.

To see how the degree of a projective algebraic set links to the Hilbert series, think of a special set of points called V. These points come from solving equations with a group of formulas called a homogeneous ideal. The Hilbert series can tell us about how these points spread out when we cut the shape with flat surfaces called hyperplanes. This helps prove Bézout's theorem, which tells us how the degrees of shapes multiply when we intersect them.

Complete intersection

A projective algebraic set is a complete intersection if it is described by a special set of equations called a regular sequence. In this case, there is a straightforward way to find the Hilbert series.

When we have k special equations of certain sizes, we can use them to understand the shape and size of the algebraic set. This helps us see that the set takes up just the right amount of space, and its overall size is linked to the sizes of these equations.

Relation with free resolutions

Every special kind of math group, called a graded module, has something called a graded free resolution. This means there is a special chain of steps connecting different parts of the group.

There are ways to use these steps to find the Hilbert series, which helps us understand how the group's parts grow. If we know certain details about these steps, we can figure out the Hilbert series using simple formulas. These methods are important in advanced math but start from the same basic tools used to find free resolutions.

Computation of Hilbert series and Hilbert polynomial

The Hilbert polynomial can be found from the Hilbert series. This section explains how to calculate the Hilbert series when dealing with a special type of math problem involving polynomial rings.

We start with a field K and a polynomial ring R. An important part of this process involves using something called a Gröbner basis, which helps simplify the problem. By using a Gröbner basis, the task becomes easier because it turns into a problem about monomials. This method is used in many computer algebra systems like Maple and Magma, where the functions are called HilbertSeries and HilbertPolynomial.

Generalization to coherent sheaves

In algebraic geometry, special kinds of math structures called graded rings can create shapes called projective schemes through a process called Proj construction. These structures also connect to things called coherent sheaves.

When we have a coherent sheaf F on a projective scheme X, we can define something called the Hilbert polynomial. This polynomial helps us understand how the sheaf grows by looking at a special number called the Euler characteristic. For larger values, this polynomial matches another important value from math theory.