# 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 *rank* of

### Remark

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

### Definition I.2.2

Let *Gram-Matrix* with respect to

### Remark

Note that a bilinear module with a Gram matrix

### 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

### Definition I.2.4

Two matrices

### Remark

Given two different bases of a finitely generated free bilinear module **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

### Remark

In a general setting the determinant of a finitely generated free bilinear module lives in

### 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

### 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 this lemmata, we can formulate the following

### Theorem I.2.6

A finitely generated free bilinear module

## 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