Orthogonal Decompositions

To further investigate bilinear forms, we want to take a closer look at orthogonal sums of bilinear modules. For the definition of an orthogonal sum, see I.1.4. Theorem I.1.9 explained when an orthogonal sum is non-degenerate or regular. Following I.2, we add that for finitely generated free bilinear modules , the rank and .

In theorem I.1.10 we saw an example of an orthogonal decomposition over fields. A similar result is true (and important) for bilinear modules:

Lemma I.3.1

Let be a symmetric bilinear module and . If is regular, then M is isomorphic to the orthogonal sum .

Theorem I.3.2

Let be a symmetric bilinear module and a free module spanned by linear independent elements (see remark after definition I.2.2), then .

As an application of this theorem, we immediately get the following

Corollary I.3.3

For a finitely generated symmetric bilinear module there exists an orthogonal decomposition of the following form: where the are units and is not a unit for every .

Investigating this module a bit more precisely, we notice that over a field must be symplectic, thus over a field of characteristic not equal to two, the bilinear form restricted to must be zero (symmetric and symplectic). In a general setting, we get the following:

Corollary I.3.4

Over a local ring of characteristic not two, every symmetric bilinear space possesses an orthogonal basis.

Remark

The proof is relatively easy if we remember that every finitely generated projective module over a local ring is free.

As a last example or theorem in this rather short chapter, we give an orthogonal decomposition for vector spaces over fields (basically the previous corollary with different words):

Corollary I.3.5

Over an arbitrary field, every finite-dimensional bilinear space has an orthogonal decomposition into subspaces of dimension at most two. Over a field of characteristic not two, every finite-dimensional bilinear space possesses an orthogonal basis.

Literature

• Symmetric Bilinear Forms by J. Milnor and D. Husemoller
• Arithmetic Theory of Quadratic Forms by R. Scharlau