Quadratic forms were probably first studied by Babylonian mathematicians and from its birth until the beginning of the twentieth century, quadratic forms were only studied over the field of real numbers, the complex field or the ring of integers. It was only in 1937 that Ernst Witt published a groundbreaking paper that would become the founding stone to incorporate the study of quadratic forms into the theory of pure algebra.
We want to embark on a journey to study quadratic (and hermitian) forms on modules over general rings, but first, we will give an introduction into the theory of bilinear forms over rings (our rings will always be associative with identity element 1).
Bilinear Forms
Definition I.1.1
Let be a left -module. A bilinear form on is a function , which is a linear function for both arguments. A bilinear form is called symmetric, if , for all . The form is called screw-symmetric, if , for all , and symplectic or alternating, if , for all .
An object , where is a -module and a symmetric bilinear form on , is called a symmetric bilinear module over . Since we are mostly only interested in the theory of symmetric bilinear forms, we will omit the word symmetric in this notion for the rest of this blog entry.
Remark
A symplectic form is always screw-symmetric, for if are arbitrarily chosen, it is easy to see that .
Remark
If , then a form is symplectic if, and only if, it is screw-symmetric, which follows immediately from for all .
Definition and Remark I.1.2
Now let be two bilinear modules. An injective module homomorphism which preserves the bilinear forms is called an isometry and we say that the two bilinear modules are isometric if there exists a bijective homomorphism between them. The relation of being isometric is an equivalence relation, and thus it makes sense to say that a bijective isometry is an isomorphism in the category of bilinear modules. (We will talk more about that category in a later entry.)
Definition I.1.3
Two elements of a bilinear module are called orthogonal, if . For a submodule , we define the orthogonal submodule as follows
Remark
It is easy to see that , and it is interesting to note that .
Definition I.1.4
A sum of submodules of is called an internal orthogonal sum, or orthogonal sum, if it is a direct sum, that is, if and for all .
The external orthogonal sum of two submodules is defined as with .
Remark
Those two types of orthogonal sums are obviously canonically isomorphic.
Definition and Remark I.1.5
For a given bilinear module we have a natural -module-homomorphism , and conversely, a linear form defines a bilinear form by , with .
Remark
We thus have a module isomorphism between the set of all bilinear forms on and all homomorphisms from to .
The next definition is essential to further specify the bilinear forms we are actually interested in:
Definition I.1.6
A bilinear module is called non-degenerate, if is injective. It is called regular, if is bijective.
Remark
For finite vector spaces over fields those two definitions are the same, as then .
Remark
It is easy to see that is non-degenerate if, and only if, , as clearly .
Remark
In other words: Regularity means that for a given , , there exists one, and only one, such, that . This is the famous representation theorem: Each linear function is defined by a unique vector.
We will now specify the modules we are truly interested in:
Definition I.1.7
A -module is projective, if there exists an -module such, that is free, that is, isomorphic to a direct sum of copies of .
Definition I.1.8
A bilinear module is called a regular bilinear module if is regular. It is called a bilinear space if additionally is finitely generated and projective over .
Remark
If is a regular bilinear module, then the only element orthogonal to every is . Over a field, the converse statement is true: If is a bilinear module over a field such that only the zero element is orthogonal to every other element, that is, if , then is a regular bilinear module.
Theorem I.1.9
An orthogonal sum of bilinear modules is non-degenerate / regular if and only if each summand is a non-degenerate / regular bilinear module.
As a last result for this introductory chapter, we want to state an easy to prove theorem over fields:
Theorem I.1.10
Let be a subspace of a -vector space, being a field and consider the non-degenerate bilinear space , then the following statements are equivalent:
,
,
and
is non-degenerate.
Literature
Symmetric Bilinear Forms by J. Milnor and D. Husemoller
Arithmetic Theory of Quadratic Forms by R. Scharlau