Bilinear Forms over Modules

Free Modules

To learn more about bilinear forms, it is first necessary to learn more about free modules.

Definition I.2.1

For a finitely generated free -module with basis , we call the rank of and denote this by .

Remark

Since we live over commutative rings, the rank is uniquely defined.

Definition I.2.2

Let be a bilinear module and a system of elements of . We can then construct the so-called Gram-Matrix with respect to of the system of elements as follows: . The determinant of is also called the determinant of , and denoted by .

Remark

Note that a bilinear module with a Gram matrix is symmetric, if and only if, is a symmetric matrix, that is, if . Similarly, it is screw-symmetric, if and only if, , and symplectic, if and only if, it is screw-symmetric and if in addition has as the only diagonal entry.

Remark

It is easy to see that the previously defined system of elements is linearly dependent if their determinant is not a zero divisor.

Definition I.2.3

Given an matrix with entries in , will denote the free bilinear module over with basis and bilinear form .

Definition I.2.4

Two matrices are called congruent, if there exists another matrix such, that .

Remark

Given two different bases of a finitely generated free bilinear module it is easy to see that the two different Gram matrices respective to the different bases are congruent to each other. It thus makes sense to speak of the Gram-Matrix of a bilinear module.

This last remark immediately leads to the definition of a very useful invariant of finitely generated free bilinear modules.

Definition I.2.5

The determinant of a finitely generated free bilinear module is the class modulo squares of units of the determinant of a Gram-Matrix of .

Remark

In a general setting the determinant of a finitely generated free bilinear module lives in , the so-called quotient monoid, but in the case of a free bilinear space (R is finitely generated and projective) the quotient monoid is actually a quotient group: .

Example

Over a field, the determinant of a bilinear space is either zero or an element of the quotient group.

Remark

To learn more about the groups of square classes and the quotient group, have a look at, for example, [JPS], page 14.

We now want to use the determinant as a tool to compute whether a finitely generated free module is non-degenerate or even regular. To do so, we obviously have to use , thus we need to have a closer look at , especially its basis:

Lemma

To each basis of a finitely generated free module there corresponds a unique dual basis.

Lemma

The Gram-Matrix of a finitely generated free bilinear module with respect to a given basis is equal to the transformation matrix representing with respect to the given basis and its dual basis.

With this lemmata, we can formulate the following

Theorem I.2.6

A finitely generated free bilinear module is non-degenerate if and only if its determinant is not a zero-divisor (not represented by a zero divisor). The bilinear module is regular if and only its determinant is a unit (represented by a unit), or in other words, if and only if its Gram-Matrix is invertible.

Literature

• Symmetric Bilinear Forms by J. Milnor and D. Husemoller
• Arithmetic Theory of Quadratic Forms by R. Scharlau
• [JPS] Cours d’arithmétique by J.P. Serre