FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 6 definitions, 2 theorems, 1 proposition, 3 lemmas, 2 corollaries, 6 examples, 1 construction and 4 remarks.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition / construction / notation
The hyperplane-reflection formula is defined over \(R\) whenever \(2/b(x,x)\in R\), even when the lattice does not split as \(Rx\oplus x^\perp\).
Lemma 7.1.1
Lemma 7.1.1. Let \(( L , q )\) be a quadratic lattice. Suppose \(x \in L\) is primitive, nonisotropic such that \(b _ { q } ( x , L ^ { * } ) \subset q ( x ) \cdot R\) (in particular this is the case if \(q ( x )\) is \(a\) unit). Then the hyperplane reflection \(y \longmapsto y - ( b _ { q } ( x , y ) / q ( x ) ) \cdot x ,\) \(y \in L\) , induces the identity on the discriminant group.
Example / comparison
Let \(R = \mathbb { Z } _ { p }\) where \(p\) is any prime. Quadratic modules over this ring are the quadratic \(p\) -adic lattices. We claim that every (non-degenerate) quadratic \(p\) -adic lattice \(( L , q )\) admits reflections and so here also \(\mathsf { S O } ( L )\) is of index 2 in \(\mathrm { O } ( L )\) . We see this as follows. Since the valuation \(v _ { p }\) is non-archimedean and since for all \(z , y \in L\) , \(b _ { q } ( z , y ) = q ( z + y ) - q ( z ) - q ( y )\) , equation (A.4) implies \(v _ { p } ( b _ { q } ( z , y ) ) \geq \operatorname* { i n f } _ { u \in L , q ( u ) \neq 0 } v _ { p } ( q ( u ) )\) . That the (non-zero) infimum is attained, say at \(x\) , is clear since the valuation is discrete. It follows that \(v _ { p } \left( b _ { q } ( x , y ) / q ( x ) \right) \geq 0\) , which translates as \(b _ { q } ( x , y ) \in q ( x ) \cdot \mathbb { Z } _ { p }\) and so (6.6) holds. Hence the reflection \(\sigma _ { x }\) in \(x\) is a well-defined isometry. We set this result apart since it will be used later: \[ \operatorname* { i n f } _ { z \in L . a ( z ) \neq 0 } v _ { p } ( q ( z ) ) = v _ { p } ( q ( x ) ) \implies \sigma _ { x } { \mathrm { ~ i s ~ a ~ h y p e r p l a n e ~ r e f l e c t i o n ~ o f ~ } } L . \]
Example / comparison
In Example 6.5.5.2 we saw that the isometry group of the hyperbolic plane \(U\) over \(R\) consists of the rotations \(i _ { a } : = \left( \begin{array} { c c } { { a } } & { { 0 } } \\ { { 0 } } & { { a ^ { - 1 } } } \end{array} \right)\) 00 ??−1 , ?? ∈ ??×, and the reflections \(j _ { b } = { \binom { 0 } { b } } \quad { \binom { b ^ { - 1 } } { 0 } }\) , \(b \in R ^ { \times }\) . The condition (6.6) implies that a reflection is of the form \(\sigma _ { c }\) with \(c = u e + v f\) for some units \(u\) , \(v\) . Then \(\sigma _ { c } ( e ) = - v u ^ { - 1 } \cdot f\) and \(\sigma _ { c } ( f ) = - v ^ { - 1 } u \cdot e\) , i.e., \(\sigma _ { c } ~ = ~ j _ { b }\) with \(b \ = \ - v u ^ { - 1 }\) . It follows that products of an even number of reflections form a proper subgroup of \(\mathrm { O } ( U )\) regardless whether \(2 = 0\) in \(R\) or not.
Example / comparison
Consider the unimodular dyadic lattice \(V _ { 0 }\) , i.e., \(\mathbb { Z } _ { 2 } ^ { \oplus 2 }\) with quadratic form \(x =\) \(( x _ { 1 } , x _ { 2 } ) \mapsto q ( x ) = x _ { 1 } ^ { 2 } + x _ { 1 } x _ { 2 } + x _ { 2 } ^ { 2 }\) . It contains no isotropic vectors (such an isotropic vector would lead to a solution of the equation \(x ^ { 2 } = - 3\) in \(\mathbb { Q } _ { 2 }\) which is impossible by Theorem A.2.1-2) and all reflections are of the form \(\sigma _ { x }\) , \(q ( x ) \in \mathbb { Z } _ { 2 } ^ { * }\) . This concerns every primitive vector since then, as we readily see, \(q ( x )\) is a unit due to the special shape of the quadratic form. We end this sections with a few words about the case \(2 = 0\) . Here the situation is quite different. For example, if \(V = W \oplus k e\) is a quadratic space over a field of characteristic 2 such that \(q ( e ) = 1\) , then \(\sigma _ { e }\) is the identity. More generally, if \(( V , q )\) is quadratic \(k\) -space and \(x\) satisfies \(q ( x ) \neq 0\) and \(x \in \operatorname { r a d } ( b _ { q } )\) , then \(\sigma _ { x }\) is the identity on \(V\) . If \(q ( x ) \neq 0\) , then the corresponding map \(\sigma _ { x }\) is an example of a so-called transvection w.r.t. the bilinear form \(b _ { q }\) , see Appendix A.5.
Lemma 7.2.1
Lemma 7.2.1. Let \(( V , b )\) be an inner product space over \(R\) . If \(x , y \in V\) , \(x \neq y\) , are such that \(b ( x , x ) = b ( y , y ) \in R ^ { \times }\) , there exists an isometry \(\sigma\) sending \(x\) to \(y\) which is a product of at most two hyperplane reflections.
Remark 7.2.2
Remark 7.2.2. If \(( V , b )\) has rank 1, say \(V \ = \ R x\) , the above proof can easily be adapted to show that the only reflection is \(- \mathrm { i d } = \sigma _ { x }\) .
Definition / construction / notation
There is a more specific version of Lemma 7.2.1 in the \(p\) -adic setting which we are going to use later. Its statement involves the \(p\) -adic invariant \(m _ { q }\) associated to the non-degenerate quadratic form \(q\) defined by \[ m _ { q } = \operatorname* { i n f } _ { z \in V } v _ { p } ( q ( z ) ) , \]
Lemma 7.2.3
Lemma 7.2.3. Let \(R = \mathbb { Z } _ { p }\) ( \(p \neq 2\) ) and let \(( V , q )\) be a non-degenerate quadratic \(\mathbb { Z } _ { p }\) -module. If \(x , y \in V\) are such that \(q ( x ) = q ( y )\) have \(p\) -adic valuation \(m _ { q }\) , then there is an isometry which is the product of at most two hyperplane reflections \(\sigma _ { z }\) and carries \(x\) to \(y\) . Moreover, we can choose \(z\) such that \(v _ { p } ( z ) = m _ { q }\) .
Theorem 7.2.4
Theorem 7.2.4 (Cartan–Dieudonn´e). Every isometry of an inner product space \(( V , b )\) of rank ?? over a local ring \(R\) in which 2 is a unit, is a product of at most \(2 n\) hyperplane reflections.
Corollary 7.2.5
Corollary 7.2.5. If \(R = \mathbb { Z } _ { p }\) , \(p\) odd, we can write an isometry of a quadratic inner product space \(( V , q )\) over \(R\) as a product of hyperplane reflections in vectors ?? for which \(v _ { p } ( z ) = m _ { q }\) (cf. Eqn. (7.3)).
Remark 7.2.6
Remark 7.2.6. Cartan–Dieudonné extends to quadratic inner-product spaces over local rings except for the stated rank-\(4\), residue-characteristic-\(2\) exception; over fields of characteristic not \(2\), at most \(\dim V\) reflections suffice.
Theorem 7.2.7
Theorem 7.2.7 (Witt’s cancellation theorem). Let \(V , W _ { 1 } , W _ { 2 }\) be inner product spaces over \(R\) . If \(V \oplus W _ { 1 } \simeq V \oplus W _ { 2 }\) , then \(W _ { 1 } \simeq W _ { 2 }\) . 2
Definition / construction / notation
Next, we investigate extendability of isometries between subspaces \(W , W ^ { \prime }\) of \(V\) . We say that two embeddings \(i : W \hookrightarrow V\) and \(i ^ { \prime } : W ^ { \prime } \hookrightarrow V\) are equivalent if there exists an isometry \(\varphi : V \to V\) making the following diagram commutative: \[ \begin{array} { r l r } { W \langle \underbrace { \imath } _ { i } \rangle } & { { } \underset { \rho } { \uparrow } } & { { } V } \\ { \varphi | _ { W } \sqrt { \underset { \imath } { \simeq } } } & { { } \quad \varphi \underset { \rho } { \downarrow } } \\ { W ^ { \prime } \langle \underbrace { \imath } _ { i ^ { \prime } } \rangle } & { { } V . } \end{array} \]
Corollary 7.2.8
Corollary 7.2.8 (Witt’s extension theorem). Let \(( V , b )\) be an inner product space over \(R\) and \(W _ { 1 } , W _ { 2 } \subset V\) two \(R\) -submodules such that \(b | _ { W _ { 1 } } , \ b | _ { W _ { 2 } }\) are unimodular. Any isometry \(W _ { 1 } \stackrel { \simeq } { \to } W _ { 2 }\) can be extended to an isometry of \(V\) . In particular, any two primitive embeddings \(W \hookrightarrow V\) , \(W\) unimodular, are equivalent. In other words, \(\operatorname { O } ( V )\) acts transitively on such primitive embeddings.
Remark 7.2.9
Remark 7.2.9. Witt extension and cancellation are equivalent; over local rings they remain valid for primitive free submodules with surjective correlation map. For merely symmetric inner-product spaces when \(2\) is not a unit they can fail: over \(\mathbf Z_2\), \(\langle-1\rangle^{\oplus2}\oplus\langle1\rangle\simeq\langle-1\rangle\oplus U\), while \(U\not\simeq\langle-1\rangle\oplus\langle1\rangle\).
Proposition 7.2.10
Proposition 7.2.10. Let \(( V , q )\) be a quadratic inner product space over \(R\) . Up to isometry \(V\) has a unique orthogonal decomposition \[ V = \widehat { U } \oplus V ^ { \prime } , \quad \widehat { U } \simeq \underbrace { U \oplus \cdots \oplus U } _ { m \ c o p i e s } , \quad V ^ { \prime } \ t o t a l l y \ a n i s o t r o p i c , \] the Witt decomposition. The subspace \(\hat { U }\) contains a maximal totally isotropic sublattice of rank \(m\) . Moreover, all maximal isotropic sublattices of \(V\) have rank \(m\) .
Definition 7.2.11
Definition 7.2.11. Let \(( V , q )\) be an inner-product space over a local ring. The Witt index \({ \sf W } _ { \tau } ( V , q )\) of \(( V , q )\) is the dimension of a maximal isotropic submodule of \(V\) .
Construction / comparison
Over a local ring where \(2\) need not be invertible, the Witt decomposition persists after replacing a sum of hyperbolic planes by a suitable split inner-product space.
Remark 7.2.12
Remark 7.2.12. The Witt index is not stable under ring extension since a form which does not represents zero over \(R\) may very well do so over a larger ring. However, if \(R\) is an integral domain with quotient field \(Q ( R )\) , the Witt indices over \(R\) and \(Q ( R )\) are the same. This applies to integral lattices \(L\) and shows that for those the Witt index is an invariant of \(L _ { \mathbb { Q } }\) and \(\mathsf { W } _ { \tau } ( L _ { \mathbb { Q } } ) \leq \mathsf { W } _ { \tau } ( L _ { \mathbb { R } } )\) . The equality may be strict. We have seen this already in Example 1.10.6 where we showed that \(x ^ { 2 } + y ^ { 2 } - 3 z ^ { 2 }\) has no isotropic vector over \(\mathbb { Q }\) and so has Witt index 0 while over the reals it has Witt index 1. Note that the latter is directly related to the signature. Indeed, if \(L\) is non-degenerate with signature \(( s , t )\) one has \(\mathsf { W } _ { \tau } ( L _ { \mathbb { R } } ) = \operatorname* { m i n } ( s , t )\) .
Definition 7.3.1
Definition 7.3.1. Two quadratic spaces are Witt equivalent if they become isometric after adding split inner-product spaces. The equivalence classes form the Witt group \(W(R)\) under orthogonal sum; the tensor-product form \(q\otimes q\prime\) induces multiplication, making \(W(R)\) the Witt ring.
Example / comparison
Since the rank of a split inner product space is even (cf. (6.5)), the parity of the rank is a well-defined ring homomorphism $W ( R ) $ \(\mathbb { Z } / 2 \mathbb { Z }\) . For any algebraically closed field \(R\) of characteristic not equal to 2 this gives an isomorphism. This follows since according to Proposition 6.3.11 all forms can be diagonalized so that the Witt ring is generated by the class of a form of odd rank.
Example / comparison
The index of a split inner product space over \(\mathbb { R }\) is zero and induces an isomorphism \(W ( \mathbb { R } ) \xrightarrow { \sim } \mathbb { Z }\) . This is a direct consequence of Sylvester’s theorem (cf. Corollary 8.1.3) and the uniqueness of the Witt splitting.
Example / comparison
The Witt groups for the basic finite fields \(\mathbb { F } _ { p }\) , \(p\) prime, are as follows. \[ W ( \mathbb { F } _ { p } ) \simeq \left\{ \begin{array} { l l } { \mathbb { Z } / 2 \mathbb { Z } } & { \mathrm { f o r ~ } p = 2 } \\ { \mathbb { Z } / 4 \mathbb { Z } } & { p \mathrm { ~ o d d ~ } , \quad p \equiv 3 \mathrm { ~ m o d ~ } 4 } \\ { \mathbb { Z } / 2 \mathbb { Z } \times \mathbb { Z } / 2 \mathbb { Z } } & { p \mathrm { ~ o d d ~ } , \quad p \equiv 1 \mathrm { ~ m o d ~ } 4 . } \end{array} \right. \] The Witt ring of \(\mathbb { Q }\) can be described by a split exact sequence \[ 0 \to \mathbb { Z } { \overset { i } { \to } } W ( \mathbb { Q } ) { \overset { \partial } { \to } } \bigoplus _ { p { \mathrm { ~ p r i m e } } } W ( \mathbb { F } _ { p } ) \to 0 . \] Here \(i\) is defined by sending 1 to [⟨1⟩] and \(\partial\) sends a class of a form \(\langle a \rangle\) to the class of its localizations (obviously, Witt equivalence is compatible with localization). See [151, IV.2].
Definition / construction / notation
Here \(i\) is defined by sending 1 to [⟨1⟩] and \(\partial\) sends a class of a form \(\langle a \rangle\) to the class of its localizations (obviously, Witt equivalence is compatible with localization). See [151, IV.2].