--- date: 2022-02-23 18:45 modification date: Wednesday 23rd February 2022 18:45:08 title: quadratic form aliases: [quadratic forms, bilinear form, Grothendieck-Witt ring, Grothendieck-Witt, GW] --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags - #motivic/homotopy - Refs: - #todo/add-references - Links: - [rank](rank) - [signature](signature.md) - [trace](trace) - [Milnor K theory](Milnor%20K%20theory) --- # Bilinear forms ![](attachments/Pasted%20image%2020220323134526.png) ![](attachments/Pasted%20image%2020220526142006.png) ![](attachments/Pasted%20image%2020220526142107.png) ![](attachments/Pasted%20image%2020220526142158.png) # GW ![](attachments/Pasted%20image%2020220410211605.png) ![](attachments/Pasted%20image%2020220410215111.png) ![](attachments/Pasted%20image%2020220410215136.png) # quadratic forms ![](attachments/Pasted%20image%2020220410212542.png) ![](attachments/Pasted%20image%2020220410212610.png) ![](attachments/Pasted%20image%2020220410212624.png) ![](attachments/Pasted%20image%2020220410212645.png) ![](attachments/Pasted%20image%2020220323134620.png) ![](attachments/Pasted%20image%2020220410212755.png) A **quadratic form** is a map $Q: V \rightarrow k$ such that \[ Q(\lambda v)=\lambda^{2} Q(v),\qquad \text{and } B(v, w)=\frac{1}{4}[Q(v+w)-Q(v-w)] \] 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 \[ Q(v)=Q^{\prime}(\rho(v)) \] 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 \[ \{v \in V \mid B(v, u)=0 \text { for all } u \in V\}=0 . \] 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 ![](attachments/Pasted%20image%2020220410214932.png) ## 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 \[ V \cong\left\langle a_{1}\right\rangle \oplus \cdots \oplus\left\langle a_{n}\right\rangle \] 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 \[ \langle a\rangle \cong\left\langle a b^{2}\right\rangle \] In particular, if $k=\mathbb{R}$ any regular quadratic space of dimension $n$ is isomorphic to \[ V \cong \bigoplus_{1\leq i \leq p} \gens{1} \oplus \bigoplus_{1\leq j\leq q} \gens{-1}, \qquad p+q=n \] 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](Witt's%20lemma.md). Consider $H$, a quadratic space of dimension 2 with a basis $e_{1}, e_{2}$ such that the matrix of the bilinear form is \[ A=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) . \] 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 \[ f_{1}=e_{1}+e_{2} \text { and } f_{2}=e_{1}-e_{2} \] 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 \[ H \cong\langle 2\rangle \oplus\langle-2\rangle . \] 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 $t \in k$, \[ Q(w+t v)=Q(w)+2 B(w, v) t+Q(v) t^{2}=Q(w)+2 t \] 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. ![](attachments/Pasted%20image%2020220323135355.png) ![](attachments/Pasted%20image%2020220323135403.png) # Grothendieck-Witt ring ![](attachments/Pasted%20image%2020220425094315.png) - 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](Milnor%20K%20theory.md) ![](attachments/Pasted%20image%2020220323135522.png) ![](attachments/Pasted%20image%2020220323135638.png) ![](attachments/Pasted%20image%2020220323135704.png) ![](attachments/Pasted%20image%2020220323135934.png) Relation to [etale cohomology](Unsorted/l-adic%20cohomology.md) and [Milnor K theory](Milnor%20K%20theory): ![](attachments/Pasted%20image%2020220410215423.png) ![](attachments/Pasted%20image%2020220410215457.png) ![](attachments/Pasted%20image%2020220410215526.png) # Transfers See [transfer](Unsorted/transfers.md). ![](attachments/Pasted%20image%2020220410215601.png) ![](attachments/Pasted%20image%2020220410215651.png) # Exercises ![](attachments/Pasted%20image%2020220323135947.png) # Witt Ring ![](attachments/Pasted%20image%2020220410215356.png) # Examples ![](attachments/Pasted%20image%2020220410212707.png) ![](attachments/Pasted%20image%2020220410215306.png) ![](attachments/Pasted%20image%2020220410215316.png) ![](attachments/Pasted%20image%2020220410215332.png)![](attachments/Pasted%20image%2020220410215406.png)