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Bilinear forms
GW
# quadratic forms
A quadratic form is a map \(Q: V \rightarrow k\) such that \begin{align*} Q(\lambda v)=\lambda^{2} Q(v),\qquad \text{and } B(v, w)=\frac{1}{4}[Q(v+w)-Q(v-w)] \end{align*} defines a bilinear (necessarily symmetric) form on \(V\). A quadratic space is a pair \((V, Q)\) consisting of a vector space \(V\) and a quadratic form \(Q\) on \(V\). An isometry between two quadratic spaces \((V, Q)\) and \(\left(V^{\prime}, Q^{\prime}\right)\) is a linear transformation \(\rho: V \rightarrow V^{\prime}\) such that \begin{align*} Q(v)=Q^{\prime}(\rho(v)) \end{align*} for any \(v\) in \(V\). Of course, if \(\rho\) is invertible, then the two quadratic spaces are said to be equivalent.
Fix a quadratic space \((V, Q)\). Let \(B\) be the corresponding symmetric bilinear form. The quadratic space \(V\) is regular (non-degenerate) if \begin{align*} \{v \in V \mathrel{\Big|}B(v, u)=0 \text { for all } u \in V\}=0 . \end{align*}
The form \(B\) defines a linear map \(T: V \rightarrow V^{*}\), where \(V^{*}\) is the dual space to \(V\), by \(T(v)(u)=B(v, u)\) for all \(u \in V\).
TFAE: (i) The quadratic space \(V\) is regular. (ii) The map \(T: V \rightarrow V^{*}\) is an isomorphism. (iii) If \(A\) is a matrix of \(B\) with respect to a basis \(e_{1}, \ldots, e_{n}\) then \(\operatorname{det}(A) \neq 0\).
Sum and tensor
Signature and the hyperbolic form
Let \(V\) be a one-dimensional quadratic space. If we pick a basis vector \(e\), then the quadratic form \(Q\) is given by \(Q(x)=a x^{2}\) for some non-zero \(a\) in \(k\). We shall denote the pair \(\left(k, a x^{2}\right)\) by \(\langle a\rangle\). Thus the above proposition shows that \begin{align*} V \cong\left\langle a_{1}\right\rangle \oplus \cdots \oplus\left\langle a_{n}\right\rangle \end{align*} for some non-zero elements \(a_{1}, \ldots, a_{n}\) in \(k\). If we replace the vector \(a\) by a multiple \(b \cdot e\) then the form \(a x^{2}\) is replaced by \(\left(a b^{2}\right) x^{2}\). Thus \begin{align*} \langle a\rangle \cong\left\langle a b^{2}\right\rangle \end{align*} In particular, if \(k=\mathbb{R}\) any regular quadratic space of dimension \(n\) is isomorphic to \begin{align*} V \cong \bigoplus_{1\leq i \leq p} \left\langle{1}\right\rangle \oplus \bigoplus_{1\leq j\leq q} \left\langle{-1}\right\rangle, \qquad p+q=n \end{align*} The difference \(p-q\) is called the signature of the real quadratic space. Two real quadratic spaces are isomorphic if and only if they have the same signature. This is a consequence of Witt’s lemma.
Consider \(H\), a quadratic space of dimension 2 with a basis \(e_{1}, e_{2}\) such that the matrix of the bilinear form is \begin{align*} A=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) . \end{align*} This quadratic space is called a hyperbolic plane. Let \(U\) be a one dimensional space spanned by \(e_{1}\). Note that \(U^{\perp}=U\) in this case. However, if we consider a different basis \begin{align*} f_{1}=e_{1}+e_{2} \text { and } f_{2}=e_{1}-e_{2} \end{align*} then the lines \(U_{1}\) and \(U_{2}\) spanned by \(f_{1}\) and \(f_{2}\) respectively are perpendicular to each other. We have shown that \begin{align*} H \cong\langle 2\rangle \oplus\langle-2\rangle . \end{align*}
A quadratic space \((V, Q)\) is called isotropic if there exists a non-zero vector \(v\) in \(V\) such that \(Q(v)=0\). Otherwise, the space is called anisotropic.
An interesting property of a regular isotropic space is that for every \(a\) in \(k\) there is a vector \(u\) in \(V\) such that \(Q(u)=a\). This is seen as follows. Fix a non-zero \(v\) such that \(Q(v)=0\). Then, for every \(w\) in \(V\) consider the line \(w+t v\) through \(w\) in the direction of \(v\). Since \(V\) is regular, there exists \(w\) in \(V\) such that \(B(v, w)=1\). Then, for \(t \in k\), \begin{align*} Q(w+t v)=Q(w)+2 B(w, v) t+Q(v) t^{2}=Q(w)+2 t \end{align*} In words, \(t \mapsto Q(w+t v)\) is a linear function. It clearly takes all possible values in \(k\). (Here we definitely need that \(2 \neq 0\).)
If \(V\) is a regular isotropic space then \(V\) contains a subspace isomorphic to the hyperbolic plane.
Corollary 3.6. For any a in \(k^{\times}\), the quadratic space \(\langle a\rangle \oplus\langle-a\rangle\) is isometric to a hyperbolic plane. Proof. This is clear since \(\langle a\rangle \oplus\langle-a\rangle\) is isotropic.
Grothendieck-Witt ring
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Slogan: for a ring \(R\), the ring of virtual nondegenerate quadratic forms on \(R\).
- Has a quotient, the Witt ring (quotient by metabolic or hyperbolic forms)
- Shows up in Milnor K theory
Relation to etale cohomology and Milnor K theory:
Transfers
See transfer. 
Exercises
Witt Ring
Examples
