Bilinear Forms over Free Modules

This tutorial explains the connection between bilinear forms over free modules and Gram matrices.

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 $R$-module $M$ with basis $b_1, ..., b_n$, we call $n$ the rank of $M$ and denote this by $\operatorname{rk}M=n$.


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

Definition I.2.2

Let $(M,b)$ be a bilinear module and $b_1, ..., b_k$ a system of elements of $M$. We can then construct the so-called Gram-Matrix with respect to $b$ of the system of elements as follows: \[G:=(g_{ij})_{ij}=b(b_i,b_j).\] The determinant of $G$ is also called the determinant of $b_1, ..., b_k$, and denoted by $d_b(b_1, ..., b_k)$.


Note that that a bilinear module with a Gram matrix $G$ is symmetric, if any only if, $G$ is a symmetric matrix, that is, if $G=G^t$. Similarly, it is screw-symmetric, if and only if, $G=-G^t$, and symplectic, if and only if, it is screw-symmetric and if in addition $G$ has $0$ as the only diagonal entry.


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 a $n \times n$ matrix $G=(g_{ij})$ with entries in $R$, $\langle G \rangle = \langle G \rangle _R$ will denote the free bilinear module over $R$ with basis $b_1, ..., b_n$ and bilinear form $b(b_1, ..., b_n)=g_{ij}$.

Definition I.2.4

Two matrices $A,B \in R^{k \times k}_{symm}$ are called congruent, if there exists another matrix $S \in GL_k(R)$ such, that $B=S^tAS$.


Given two different bases of a finitely generated free bilinear module $(M,b)$ 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 $(M,b)$ is the class modulo squares of units of the determinant of a Gram-Matrix of $(M,b)$.


In a general setting the determinant of a finitely generated free bilinear module lives in $R / R^{ * 2}$, 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: $R^* / R^{*2}$.


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


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 $\widehat{b}$, thus we need to take a closer look at $M^*$, especially its basis:


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


The Gram-Matrix of a finitely generated free bilinear module $(M,b)$ with respect to a given basis is equal to the transformation matrix representing $\widehat{b}$ 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 $(M,b)$ is non-degenerate if and only if its determinant is not a zero-divisor (not represented by a zero divisor). The bilinear module $(M,b)$ 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.


  • 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

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