FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 28 definitions, 5 propositions, 6 lemmas, 16 examples, 3 constructions and 4 remarks.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition 6.1.1
Definition 6.1.1. An \(F\) -valued form \(b\) on an \(R\) module \(V\) is a bilinear form \(b : V \times V \to F\) . In other words, \(^ { b }\) has the following properties: \[ \begin{array} { r l } & { b ( r x + s y , z ) = r b ( x , z ) + s b ( y , z ) , \quad r , s \in R , x , y , z \in V \ ( \mathrm { l e f t - l i n e a r i t y } ) , } \\ & { b ( x , r y + s z ) = r b ( x , y ) + s b ( x , z ) , \quad r , s \in R , x , y , z \in V \ ( \mathrm { r i g h t - l i n e a r i t y } ) . } \end{array} \]
Definition / construction / notation
An \(F\) -valued form \(^ { b }\) is called symmetric, if \(b ( x , y ) = b ( y , x )\) for \(x , y \in V\) , skewsymmetric, if \(b ( x , y ) = - b ( y , x )\) for \(x , y \in V\) , and alternating or symplectic, if \(b ( x , x ) = 0\) for all \(x \in V\) .
Definition / construction
An \(F\)-valued symmetric \(R\)-module is a pair \((V,b)\) consisting of an \(R\)-module and an \(F\)-valued symmetric bilinear form; when \(F=R\) it is called a symmetric \(R\)-module.
Definition 6.1.2
Definition 6.1.2. An \(F\) -valued quadratic \(R\) -module consists of an \(R\) -module \(V\) equipped with an \(R\) -quadratic function \(q : V \to F\) , i.e., \(q ( r x ) = r ^ { 2 } q ( x )\) for all \(r \in R\) and \(x \in V\) , and such that the form \(b _ { q } ( x , y ) = q ( x + y ) - q ( x ) - q ( y )\) is a symmetric \(F\) -valued bilinear form on \(V\) , the polar form of \(q\) . If \(F = R\) we speak of a quadratic ??-module. It follows from the above definitions that \(b _ { q } ( x , x ) = q ( 2 x ) - 2 q ( x ) = 2 q ( x ) \in 2 F\) . In contrast to the integral situation, 2 can be a unit in \(R\) and then the bilinear form \(b = 2 ^ { - 1 } b _ { q }\) gives back \(q\) . If this is not the case, we distinguish two cases, just as in the integral situation:
Definition 6.1.3
Definition 6.1.3. Suppose \(2 \neq 0\) is not a unit in \(R\) . Then an \(F\) -valued symmetric form \(^ { b }\) on \(V\) is called even if \(b ( x , x ) \in 2 F\) for all \(x \in V\) . A form which is not even is called \(\mathbf { \Gamma } _ { o d d }\) .
Construction / comparison
If multiplication by \(2\) on the value module \(F\) is injective, then every even symmetric \(F\)-valued form is the polar form of the quadratic function \(q(x)=\frac12b(x,x)\).
Definition / construction
The general \(F\)-valued formalism includes \(R\)-lattices and \(p\)-adic lattices when \(F=R\), and symmetric or quadratic torsion forms when \(R\) is an integral domain and \(F=Q(R)/R\): these are \(Q(R)/R\)-valued forms on finitely generated torsion \(R\)-modules. For \(R=\mathbf Z\) they are forms on finite abelian groups with values in \(\mathbf Q/\mathbf Z\).
Definition 6.1.4
Definition 6.1.4. Let \(( V _ { j } , b _ { j } )\) , \(j = 1 , 2\) , be two symmetric \(F\) -valued \(R\) -bilinear modules. A homomorphism of \(R\) -modules \(\varphi : V _ { 1 } \to V _ { 2 }\) such that \(b _ { 2 } ( \varphi ( x ) , \varphi ( y ) ) =\) \(b _ { 1 } ( x , y )\) for all \(x , y \in V _ { 1 }\) is called a morphism of \(F\) -valued symmetric \(R\) - modules. If such a morphism is injective, it is called an isometric embedding and if it is bijective it is called an isometry .
Definition / construction / notation
Two symmetric \(F\) -valued \(R\) -bilinear modules \(( V _ { 1 } , b _ { 1 } )\) and \(( V _ { 2 } , b _ { 2 } )\) are called isometric, written \(V _ { 1 } \simeq V _ { 2 }\) or \(b _ { 1 } \simeq b _ { 2 }\) , if there exists an isometry \(V _ { 1 } V _ { 2 }\) . The forms \(b _ { 1 }\) and \(b _ { 2 }\) are then said to be equivalent over \(R\) . For quadratic \(F\) -valued \(R\) -modules the definitions are similar.
Definition / construction
The self-isometries of an \(F\)-valued symmetric \(R\)-module form its orthogonal group \(O(V,b)\).
Definition / construction / notation
6.1.B Examples. 1. Rank one forms. A choice of \(r \in R\) defines the symmetric form \(\langle r \rangle\) on the rank one \(R\) -module \(R \cdot e\) by setting \(e \cdot e = r\) . This is in accordance with the notation we previously used for lattices and vector spaces. Isomorphisms of \(R \cdot e\) are given by multiplying with a unit \(u\) of \(R\) . It transforms \(\langle r \rangle\) into \(\langle u ^ { 2 } r \rangle\) . So \(\langle r \rangle _ { R } \simeq \langle s \rangle _ { R }\) if and only if \(r = u ^ { 2 } s\) for a unit \(u \in R ^ { \times }\) and isometry classes of nondegenerate (see \(\mathrm { 8 ~ 6 . 1 . C }\) below) rank one symmetric \(R\) -modules are in one-to-one correspondence with the set of equivalence classes \(r \cdot ( R ^ { \times } ) ^ { 2 }\) , \(r \in R - \{ 0 \}\) .
Example / comparison
For odd \(p\), rank-one \(p\)-adic forms are classified up to isometry by the square class of their unit coefficient; the dyadic unit square classes are represented by \(\pm1,\pm3\) modulo \(8\).
Definition / construction / notation
- Scaling a form. In the same way as for integral lattices, if \(b\) is an \(F\) - valued symmetric form over \(R\) and \(r \in R\) , we let \(b ( r )\) be the form defined by \(( x , y ) \mapsto r \cdot b ( x , y )\) . It is called the form \(^ { b }\) scaled by \(r\) .
Definition / construction / notation
6.1.C Orthogonality and non-degeneracy. Our first task is to extend the notion of orthogonality from Section 1.3 to the present setting. Let \(( V , b )\) be a symmetric \(R\) -module. We say that \(X \subset V\) is orthogonal to \(Y \subset V\) if \(b ( x , y ) = 0\) for all \(x \in X\) and \(y \in Y\) . For a submodule \(W\) of \(V\) , the set of elements in \(V\) orthogonal to \(W\) is again a submodule of \(V\) which we write \(W ^ { \perp }\) . Similar definitions hold for quadratic modules using the corresponding polar forms.
Definition 6.1.5
Definition 6.1.5. 1. A submodule \(W \subset V\) is called isotropic if \(b ( x , y ) = 0\) for all \(x , y \in W\) . In other words \(W \subset W ^ { \bot }\) . It is called totally isotropic if \(W = W ^ { \perp }\) . A generator of a rank one isotropic submodule is called an isotropic vector . In other words \(x \neq 0\) is isotropic if \(b ( x , x ) = 0\) . A submodule \(W\) without isotropic vectors is called totally anisotropic. In other words, \(W\) is totally anisotropic if the quadratic equation \(b ( x , x ) = 0\) has only the trivial solution in \(W\) . In the quadratic module setting, one uses the polar form to define these notions. 2. The radical or null-space of a symmetric (quadratic) \(R\) -module \(( V , b )\) (respectively \(( V , q )\) ) is given by \[ \begin{array} { r l } & { \mathrm { r a d } ( b ) = V ^ { \perp } = \{ x \in V \mid b ( x , y ) = 0 \mathrm { ~ f o r ~ a l l ~ } y \in V \} } \\ & { \mathrm { r a d } ( q ) \qquad = \{ x \in \mathrm { r a d } ( b _ { q } ) \mid q ( x ) = 0 \} , \mathrm { r e s p e c t i v e l y . } } \end{array} \] 3. The form \(^ { b }\) on \(V\) is non-degenerate if \(V ^ { \perp } = 0\) , i.e. if \(b ( x , V ) = 0\) implies \(x = 0\) A quadratic form \(q\) is non-degenerate if \(\operatorname { r a d } ( b _ { q } ) = 0\) . 4. If \(V\) is a direct sum \(V = W _ { 1 } \oplus \cdots \oplus W _ { m }\) of \(R\) -submodules such that the \(W _ { j }\) are mutually orthogonal, we say that \(V\) is the (internal) orthogonal direct sum of the \(W _ { j }\) , written as \(V = W _ { 1 } \odot \cdots \odot W _ { m }\) .
Definition / construction
The external orthogonal sum of \(F\)-valued symmetric \(R\)-modules \((V_i,b_i)\) is \((\bigoplus_iV_i,\bigoplus_i b_i)\).
Lemma 6.1.6
Lemma 6.1.6. Let \(( V , q )\) be a quadratic \(R\) -module with polar form \(b _ { q }\) . Then \(\mathit { 1 }\) . rad(??) is a submodule of \(V\) . 2. \(\operatorname { r a d } ( q ) \subset \operatorname { r a d } ( b _ { q } )\) with equality if 2 is invertible in \(R\) .
Remark 6.1.7
Remark 6.1.7. Suppose that \(R\) is an integral domain. If \(( V , b )\) is non-degenerate and has values in \(R\) , then \(V\) is free of torsion. Indeed, if \(x \in V\) would be torsion, say \(m x = 0\) , then \(m b ( x , V ) = 0\) and since \(R\) has no torsion, \(x\) belongs to \(V ^ { \perp } = \{ 0 \}\) On the other hand, if \(( V , b )\) is a finitely generated non-degenerate \(Q ( R ) / R\) - valued symmetric bilinear \(R\) -module, then all elements of \(V\) are torsion. Indeed, \(b ( x , y ) \in Q ( R ) / R\) implies that there is some \(r = r ( x , y ) \in R { - } \{ 0 \}\) such that \(r b ( x , y ) =\) \(0\) . If \(\{ y _ { 1 } , \dots , y _ { m } \}\) is a set of generators of \(V\) , the non-zero product \(r ( x ) = r ( x , y _ { 1 } ) \cdot \cdot \cdot\) \(r ( x , y _ { m } )\) is such that \(r ( x ) b ( x , y ) = 0\) for all \(y \in V\) . Hence \(r ( x ) x \in V ^ { \bot } = \{ 0 \}\) and so \(x\) is torsion.
Proposition 6.1.8
Proposition 6.1.8. 1. Any non-degenerate \(\mathbb { Q } / \mathbb { Z }\) -valued cyclic symmetric torsion form on \(C _ { m }\) is given by \(( x , y ) \mapsto a \cdot m ^ { - 1 } \cdot x y \in \mathbb { Q } / \mathbb { Z }\) for some integer ?? with \(( a , m ) = 1\) . In other words, it is isometric to \(\langle a \cdot m ^ { - 1 } \rangle\) . 2. Isometry classes of non-degenerate bilinear cyclic torsion forms on \(C _ { m }\) are classified by \(\sf D ( \mathbb { Z } / m \mathbb { Z } )\) . 3. Any quadratic torsion form on \(C _ { m }\) is of the form \(\textstyle \left[ { \frac { 1 } { 2 } } a m ^ { - 1 } \right]\) , i.e., \(x \mapsto a ( 2 m ) ^ { - 1 } x ^ { 2 }\) , with ???? even. Its polar form is \(\langle a m ^ { - 1 } \rangle\) . The quadratic form is non-degenerate if and only if \(( a , m ) = 1\) . 4. The non-degenerate quadratic forms on \(C _ { m }\) are classified by elements in \(\mathsf { D } ( \mathbb { Z } / 2 m \mathbb { Z } )\) .
Example / comparison
\(C _ { 9 }\) . Units in this group are \(\{ \overline { { 1 } } , \overline { { 2 } } , \overline { { 4 } } , \overline { { 5 } } , \overline { { 7 } } , \overline { { 8 } } \}\) and these form a cyclic group generated by \(\overline { { 2 } }\) . The non-zero squares are \(\overline { { 1 } } , \overline { { 4 } } , \overline { { 7 } }\) and \(\overline { { 2 } } , \overline { { 8 } } , \overline { { 5 } }\) are the non-squares. Therefore, there are two non-degenerate non-isometric symmetric forms: \(\left. { \frac { 1 } { 9 } } \right.\) and \(\big < \frac { 2 } { 9 } \big >\) . Next we turn to quadratic forms. The group \(\mathsf { D } ( \mathbb { Z } / 1 8 \mathbb { Z } )\) is the cyclic group generated by \(\bar { 5 }\) , the class of 5 mod 18. The squares are \(\{ \overline { { 1 } } , \overline { { 7 } } , \overline { { 1 3 } } \}\) and the non-squares \(\{ \overline { { 5 } } , \overline { { 1 1 } } , \overline { { 1 7 } } \}\) . We find the two non-isometric quadratic forms \(\left[ { \frac { 1 } { 9 } } \right]\) and \(\textstyle { \left[ { \frac { 2 } { 9 } } \right] }\) .
Example / comparison
\(C _ { 6 }\) . Consider the set \(\{ 1 , 5 \}\) of numbers coprime to 6, one a square and one a nonsquare. So there are two symmetric forms \(\langle { \scriptstyle { \frac { 1 } { 6 } } } \rangle\) and \(\big < \frac { 5 } { 6 } \big >\) . For the quadratic forms we consider the set \(\{ 1 , 5 , 7 , 1 1 \}\) of numbers coprime to 12. Since \(5 ^ { 2 } = 7 ^ { 2 } = 1 1 ^ { 2 } = 1\) and \(5 \cdot 7 = 1 1\) the group \(( \mathbb { Z } / 1 2 \mathbb { Z } ) ^ { \times }\) modulo squares is isomorphic to the product of two cyclic groups of order 2 and hence there are four non-degenerate non-isometric quadratic forms \(\textstyle { \left[ { \frac { a } { 1 2 } } \right] }\) , \(a \in \{ 1 , 5 , 7 , 1 1 \}\) , while the polar forms for \(a = 1 , 7\) as well as for \(a = 5 , 1 1\) are the same and give the two non-isometric symmetric torsion forms on \(C _ { 6 }\) . We now pass to non-degenerate torsion forms on the groups \(C _ { m }\) , \(m = p ^ { k }\) a prime power. These have been enumerated in Example 1.9.5. Here we discuss the isometry problem. We have just seen that the symmetric forms on \(C _ { p ^ { k } }\) are classified by \(\mathsf { D } ( \mathbb { Z } / p ^ { k } \mathbb { Z } )\) and the quadratic forms by \(\mathsf { D } ( \mathbb { Z } / 2 p ^ { k } \mathbb { Z } )\) . Recall (Lemma A.1.5) that for odd primes \(p\) the group \(\mathsf { D } ( \mathbb { Z } / 2 p ^ { k } \mathbb { Z } ) \simeq \mathsf { D } ( \mathbb { Z } _ { p ^ { k } } ) \simeq \mathsf { D } ( \mathbb { Z } _ { p } )\) is cyclic, generated by a non-square modulo \(p\) . The groups \(\mathsf { D } ( \mathbb { Z } / 2 ^ { k } \mathbb { Z } )\) which enter the classification for \(p = 2\) can be viewed as subgroups of the multiplicative group \(( \mathbb { Z } / 2 ^ { 3 } \mathbb { Z } ) ^ { \times }\) : for \(k = 1\) the group consists of the class of 1, for \(k = 2\) one has the cyclic group \(\{ 1 , 3 \}\) , and for \(k \geq 2\) one has the full multiplicative group of \(\mathbb { Z } / 2 ^ { 3 } \mathbb { Z }\) . Since \(\mathsf { D } ( \mathbb { Z } / 2 ^ { k } \mathbb { Z } )\) and \(\mathsf { D } ( \mathbb { Z } / 2 ^ { k + 1 } \mathbb { Z } )\) differ for \(k = 1 , 2\) , some non-isometric quadratic forms have isometric polar forms as indicated in Table 6.1.1 by putting them in the same row. Table 6.1.1: Cyclic torsion forms of prime power order 
Definition / construction
The adjoint/correlation map associated to a bilinear form sends a vector to its pairing functional.
Definition / construction / notation
The kernel of the adjoint map equals \(V ^ { \perp }\) and so, by definition, \(^ { b }\) is non-degenerate if and only if \(b _ { V }\) is injective. If, moreover, \(b _ { V }\) is an isomorphism, we say that \(( V , b )\) is unimodular. For a quadratic \(R\) -module \(( V , q )\) the correlation map \(b _ { V } : V \to V _ { F } ^ { * }\) is the one associated to the polar form of \(q\) and if it is injective, respectively bijective we speak of a non-degenerate, respectively unimodular quadratic \(R\) -module. 2
Remark: Remark
Remark. 1. Observe also that unimodularity imposes a rather severe restriction on the possible \(R\) -modules \(F\) in which \(b\) takes values. 2. Starting from an \(F\) -valued symmetric \(R\) -module \(( V , b )\) , there is a canonical way to transport the structure to rings \(S\) that are \(R\) -algebras. Take for example \(R = \mathbb { Z }\) and \(S = \mathbb { R }\) . The correlation morphism for \(( V , b ) _ { S } = ( V \otimes _ { R } S , b \otimes 1 )\) is the composition \[ V \otimes _ { R } S \xrightarrow { b _ { V } \otimes \mathrm { i d } } V _ { F } ^ { * } \otimes S = \mathrm { H o m } _ { R } ( V , F ) \otimes _ { R } S \xrightarrow { \alpha } \mathrm { H o m } _ { S } ( V _ { S } , F _ { S } ) , \] where the rightmost map is the natural map. In many cases, if \(b _ { V }\) is an isomorphism, then so is \(b _ { V _ { S } } = \alpha \circ ( b _ { V } \otimes \mathrm { i d } )\) , e.g. if \(F = R\) and \(V\) is a finitely generated free \(R\) -module. In particular, if \(V\) is a finitely generated free \(R\) -module and \(( V , b )\) is unimodular, then so is \(( V , b ) _ { S }\) .
Definition / construction
For a submodule \(W\subset V\), the relative correlation map is \(\beta_W:V\to W_F^*\), \(x\mapsto b(x,-)|_W\); the ordinary correlation \(b_W:W\to W_F^*\) is its restriction to \(W\).
Lemma 6.2.1
Lemma 6.2.1. Let \(V\) be an \(F\) -valued symmetric \(R\) -module of finite rank and \(W \subset\) \(V\) a submodule. Then 1. \(\beta _ { W }\) is injective on \(W\) if and only if \(^ { b }\) restricts to a non-degenerate form on \(W\) . 2. If \(\beta _ { W } | _ { W }\) is an isomorphism, that is, \(( W , b | _ { W } )\) is a unimodular submodule, then \(V = W \oplus W ^ { \perp }\) . In other words, unimodular submodules split off.
Definition / construction / notation
An \(R\)-lattice is a finite-rank free \(R\)-module equipped with an \(R\)-valued symmetric bilinear form.
Definition 6.3.1
Definition 6.3.1. The discriminant \(\operatorname { d i s c } ( b )\) of \(b\) is the value of \(\operatorname* { d e t } ( B _ { E } )\) up to squares of units.
Proposition 6.3.2
Proposition 6.3.2. Let \(( L , b )\) be an \(R\) -lattice. Then: 1. ?? is non-degenerate if \(\operatorname { d i s c } ( b )\) is not a zero-divisor. It is unimodular if \(\operatorname { d i s c } ( b ) \in \mathsf { D } ( R ) = R ^ { \times } / ( R ^ { \times } ) ^ { 2 }\) . 2. For \(R = k\) a field, \(^ { b }\) is non-degenerate if and only if \(^ { b }\) is unimodular.
Construction / comparison
If \(2\) is not a unit, the Gram determinant of the polar form of an odd-rank quadratic \(R\)-module is \(2P\) for a polynomial invariant \(P\). Hence such a quadratic module is never unimodular; the class of \(P\) modulo squares is the semi-discriminant used in Definition 6.3.3 and may be invertible.
Definition 6.3.3
Definition 6.3.3. For a quadratic \(R\) -module \(( V , q )\) of odd rank over a ring \(R\) the semi-discriminant is the invariant \(\mathrm { s d i s c } ( q ) \in R / ( R ^ { \times } ) ^ { 2 }\) . The module \(( V , q )\) (or the form \(q\) ) is called semi-unimodular if the semi-discriminant is invertible, that is \(\operatorname { s d i s c } ( q ) \in \mathsf { D } ( R )\) . If 2 is invertible, then \(q\) is semi-unimodular if and only if it is unimodular.
Example / comparison
\(n = 1\) . Here the bilinear form is never unimodular, but since \(\mathrm { s d i s c } ( q ) = a _ { 1 1 } =\) \(q ( e _ { 1 } , e _ { 1 } )\) , the form is semi-unimodular if and only if \(a _ { 1 1 }\) is a unit.
Example / comparison
\(n = 3\) . Expanding the determinant we find \(\mathrm { s d i s c } ( q ) = 4 a _ { 1 1 } a _ { 2 2 } a _ { 2 3 } + a _ { 1 2 } a _ { 2 3 } a _ { 1 3 } -\) \(a _ { 1 3 } ^ { 2 } a _ { 1 2 } - a _ { 2 3 } ^ { 2 } a _ { 1 1 } - a _ { 1 2 } ^ { 2 } a _ { 2 3 }\) . In particular, if \(B\) is diagonal \(\operatorname { s d i s c } ( q )\) is not invertible. This is evidently true for all odd \(n \geq 3\) , and so diagonal forms of odd rank \(\geq 3\) are not semi-unimodular.
Remark 6.3.5
Remark 6.3.5. An \(R\) -valued symmetric \(R\) -module \(( L , b )\) such that \(^ { b }\) is unimodular is also called an inner product space over \(R\) . Similarly, one speaks of a quadratic inner product space over \(R\) . In Chapter 1 we used this terminology for a field \(R\) , but in the literature it is often used when working over integral domains. As before, we can speak of primitive sublattices. Observe however that for free modules over general commutative rings \(R\) the characterizations of Definition 1.2.4 valid for \(R = \mathbb { Z }\) do not all make sense, since \(R\) and hence \(L\) can have non-trivial zero-divisors. However, they remain valid for rings without zero-divisors. The following definition makes sense in general.
Definition 6.3.6
Definition 6.3.6. A sublattice \(S\) of \(L\) is called primitive if it is a direct summand of \(L\) . Equivalently: any basis of \(S\) can be extended to a basis of \(L\) . A vector \(x \in L\) is called primitive if \(R x\) is a primitive sublattice of \(L\) .
Lemma 6.3.7
Lemma 6.3.7. Let \(( L , b )\) be a unimodular \(R\) -lattice. 1. For any basis \(\{ e _ { 1 } , \ldots , e _ { n } \}\) of \(L\) there exists \(a\) (unique) \(^ { b }\) -dual basis \(\{ e _ { 1 } ^ { \# } , . . . , e _ { n } ^ { \# } \}\) for \(L\) , that is, a basis for which \(b ( e _ { i } , e _ { j } ^ { \# } ) = \delta _ { i j }\) . 2. If fir \(S\) s a primit?? vectors f m \(L\) and a basi \(\{ e _ { 1 } , \ldots , e _ { n } \}\) an \(L\) ch that theis a basis \(\{ e _ { 1 } , \ldots , e _ { m } \}\) \(S\) \(\{ e _ { m + 1 } ^ { \# } , \hdots , e _ { n } ^ { \# } \}\) \(S ^ { \bot }\) \(( S ^ { \bot } ) ^ { \bot } = S\) 3. Let \(R x \subset L\) be primitive. Then there exists a vector \(y \in L\) for which \(b ( x , y ) =\) 1.
Lemma 6.3.8
Lemma 6.3.8. Let \(R\) be a ring in which 2 is invertible. The hyperbolic plane \(U _ { R }\) over \(R\) represents all elements of \(R\) . A unimodular quadratic \(R\) -module \(( L , q )\) represents 0 if and only if \(L\) splits off a hyperbolic plane.
Example / comparison
For symmetric \(\mathbf Z\)-lattices, the general ring-theoretic notions of non-degeneracy and unimodularity agree with the integral-lattice notions from Chapter 1.
Example / comparison
A rank-one integral or dyadic quadratic lattice is never unimodular because its polar form is even.
Example / comparison
The hyperbolic plane is the split rank-two quadratic space.
Example / comparison
For the dyadic lattices \(V_k\), \(V_0\) is unimodular whereas \(V_k\) for \(k\ge1\) is non-degenerate but not unimodular.
Definition / construction
For a quadratic field extension \(K/k\), the field norm gives a binary quadratic norm form; its polarization is the associated trace form.
Lemma 6.3.9
Lemma 6.3.9. A quadratic binary form \(q ( x , y ) = a x ^ { 2 } + c y ^ { 2 }\) over a field of characteristic different from 2 is a norm form if and only if \(- c / a\) is not \(a\) square. If \(- c / a\) is a square, then \(q\) is isometric to the hyperbolic plane.
Construction / comparison
Over a perfect field of characteristic \(2\), a non-degenerate binary quadratic form is either hyperbolic or, up to scale, the norm form of a quadratic field extension.
Example / comparison
For a local ring \((R,\mathfrak m)\), reduction of an \(R\)-lattice modulo \(\mathfrak m\) induces a symmetric form over the residue field; non-degeneracy of the reduction implies non-degeneracy upstairs, while the converse can fail.
Proposition 6.3.10
Proposition 6.3.10 (Splitting off units). Let \(( L , b )\) be an \(R\) -valued symmetric form, \(v \in L\) , such that \(b ( v , v ) = u\) , a unit in \(R\) . Then \(L = R v \oplus ( R v ) ^ { \perp } \simeq \langle u \rangle \oplus ( R v ) ^ { \perp }\) . Consequently, there exists an orthogonal splitting \[ \boldsymbol { L } \simeq \left. \boldsymbol { u } _ { 1 } \right. \mathbb { \Phi } \cdots \mathbb { O } \left. \boldsymbol { u } _ { s } \right. \mathbb { O } \boldsymbol { N } , \] where \(u _ { 1 } , \ldots , u _ { s }\) are units in \(R\) , and \(b ( x , x )\) is a non-unit for every \(x \in N\) .
Proposition 6.3.11
Proposition 6.3.11 (Splitting over local rings). Let \(R\) be a local ring in which 2 is a unit and let \(( L , b )\) be a unimodular form on a free \(R\) -module of finite rank. Then ?? is diagonalizable.
Proposition 6.3.12
Proposition 6.3.12 (Embedding in hyperbolic modules). Let \(L\) be \(a\) free \(R\) -module of finite rank ?? endowed with a non-degenerate \(R\) -valued quadratic form \(q\) . 1. The embedding \(j : L \to L \oplus L ^ { * }\) , \(j ( x ) = ( x , 0 )\) induces an isometric embedding from \(( L , q )\) into \(U _ { L , q }\) such that \(j ( L ) ^ { \perp } \simeq L ( - 1 )\) . 2. If \(L\) is unimodular, then \(U _ { L , q } = j ( L ) \oplus j ( L ) ^ { \perp } \simeq L \oplus L ( - 1 )\) .
Definition / construction
The discriminant torsion module of a non-degenerate \(R\)-lattice is the quotient of the dual lattice by the lattice.
Lemma 6.4.1
Lemma 6.4.1. 1. Under the above identification of \(L _ { Q ( R ) }\) with \(L _ { Q ( R ) } ^ { * }\) we have \[ L ^ { * } = \{ y \in L _ { Q ( R ) } \mid b _ { Q ( R ) } ( y , x ) \in R f o r \ a l l \ x \in L \} . \] 2. If \(A = B _ { E }\) is the Gram matrix of \(b\) with respect to the basis \(\boldsymbol { E } = \{ e _ { 1 } , \ldots , e _ { n } \}\) , then \(A ^ { - 1 }\) is the Gram matrix of \(b _ { Q ( R ) }\) with respect to the dual basis \(\pmb { E } ^ { * }\) . This is also the matrix expressing the basis \(E ^ { * }\) of \({ L } _ { Q ( R ) } ^ { * } = { L } _ { Q ( R ) }\) in the basis \(\pmb { { \cal E } }\) . 3. There is a canonical identification \(L = ( L ^ { * } ) ^ { * }\) induced by \(b _ { Q ( R ) } ^ { * } { \circ } b _ { Q ( R ) }\)
Definition / construction
For a non-degenerate symmetric \(R\)-lattice, the bilinear extension to \(L_{Q(R)}\) induces the discriminant bilinear form \(b_L^\#: \operatorname{dg}_L\times\operatorname{dg}_L\to Q(R)/R\), \((\bar x,\bar y)\mapsto b_{Q(R)}(x,y)\bmod R\).
Definition / construction
For a non-degenerate quadratic \(R\)-lattice, the quadratic extension induces the discriminant quadratic form \(q_L^\#: \operatorname{dg}_L\to Q(R)/R\), \(\bar x\mapsto q_{Q(R)}(x)\bmod R\); the associated discriminant bilinear form is non-degenerate.
Example / comparison
Example 6.5.1. Consider a non-degenerate quadratic form \(q\) on a vector space \(V\) over a field of characteristic 2 such that \(q\) has an isotropic vector, e.g. if \(q\) is an orthogonal direct sum of hyperbolic planes. The so-called symplectic transvection \(t ( x ) = x + b _ { q } ( u , x ) u\) , \(u \in V\) isotropic, preserves \(b _ { q }\) as one directly checks. However, since we are in characteristic two, \(q ( t ( x ) ) = q ( x ) + b _ { q } ( u , x ) ^ { 2 }\) and so \(t \not \in { \cal O } ( q )\) . The simplest such example is the hyperbolic plane \(U _ { \mathbb { F } _ { 2 } }\) over \(\mathbb { F } _ { 2 }\) . Then \(\mathrm { O } ( U _ { \mathbb { F } _ { 2 } } ) .\) ) consists of the six transformations \(\mathrm { i d } , t : = \left( \begin{array} { l l } { 0 } & { 1 } \\ { 1 } & { 0 } \end{array} \right) , s : = \left( \begin{array} { l l } { 1 } & { 1 } \\ { 1 } & { 0 } \end{array} \right) , s ^ { 2 } , s t = t s ^ { 2 }\) and \(s t ^ { 2 } = t s\) , while \({ \sf O } ( q ) = \{ { \mathrm { i d } } , t \}\) .
Definition / construction / notation
An isometry of determinant \(1\) is called a rotation; an isometry of determinant \(-1\) is called a reflection.
Definition 6.5.2
Definition 6.5.2. Suppose \(2 \neq 0\) in \(R\) . The special orthogonal group (or the group of rotations of \(L\) ) is the group \(\operatorname { S O } ( L ) = \{ g \in \operatorname { O } ( L ) \mid \operatorname* { d e t } ( \varphi ) = 1 \}\) , a normal subgroup of \(\mathrm { O } ( L )\) of index at most 2. An isometry \(\varphi\) with \(\operatorname* { d e t } \varphi = - 1\) will be called a reflection.
Example / comparison
Example 6.5.3. If a non-isotropic \(x\) in an \(R\)-lattice satisfies \(2b(x,L)\subset b(x,x)R\), then \(\sigma_x(y)=y-2b(x,y)b(x,x)^{-1}x\) is the hyperplane reflection in \(x\) and has determinant \(-1\). For a quadratic lattice this is \(\sigma_x(y)=y-b_q(x,y)q(x)^{-1}x\).
Remark 6.5.4
Remark 6.5.4. 1. One can define the determinant for an isometry with values in any commutative ring \(A\) . However, if \(A\) has zero-divisors, this is a less useful concept, especially if all elements are zero-divisors since then \(1 = 0\) . This applies for example to the ring \(Q ( R ) / R\) and hence to \(R\) -torsion modules. 2. There are lattices \(( L , b )\) that do not admit reflections, i.e., for which \({ \sf O } ( L ) =\) \(\mathsf { S O } ( L )\) . See Example 6.5.5.4 below.
Definition / construction
Every lattice isometry acts on \(L^*/L\) and yields reduction homomorphisms \(r_L^b:O(L,b)\to O(b_L^\#)\) and, for a quadratic lattice, \(r_L^q:O(L,b_q)\to O(q_L^\#)\hookrightarrow O(b_L^\#)\). These maps need not be injective or surjective; scaling a lattice by a non-zero scalar does not change its isometry group.
Example / comparison
Rank one lattices. The isometry group of \(\langle a \rangle\) , \(a \in R - \{ 0 \}\) is the kernel \(R _ { [ 2 ] } ^ { \times }\) of the squaring map on \(R ^ { \times }\) . If \(( R , \mathfrak { m } )\) is a local ring with 2 a unit, this group only consists of , as we shall see in Remark 7.2.2. The isometry group of the discriminant bilinear form \(\langle a ^ { - 1 } \rangle\) is the kernel \(( R / a R ) _ { [ 2 ] } ^ { \times }\) of the squaring map on \(( R / a R ) ^ { \times }\) . Indeed, in the latter case an isometry is given by an invertible element \(\xi \in a ^ { - 1 } R / R\) such that \(a ^ { - 1 } \xi ^ { 2 } x y - a ^ { - 1 } x y = 0\) in \(a ^ { - 1 } R / R\) for all \(x , y \in a ^ { - 1 } R\) , which is equivalent to \(\xi ^ { 2 } - 1 = 0\) in \(R / a R\) . The reduction homomorphism is the quotient morphism \(R _ { [ 2 ] } ^ { \times } ( R / a R ) _ { [ 2 ] } ^ { \times }\) . by ow to theand has adratic form as its polar f \([ \mathinner { \frac { 1 } { 2 } } a ]\) , \(a \in 2 R\) ,s n. \(x \mapsto { \textstyle { \frac { 1 } { 2 } } } a x ^ { 2 }\) \(\langle a \rangle\) \({ \cal O } ( [ \textstyle \frac { 1 } { 2 } a ] ) = { \cal O } ( \langle a \rangle ) = R _ { [ 2 ] } ^ { \times }\) The isometry group of its discriminant quadratic form \([ \textstyle { \frac { 1 } { 2 } } a ^ { - 1 } ]\) is the same as for its polar form. Indeed, \({ \scriptstyle { \frac { 1 } { 2 } } } a ^ { - 1 } \xi ^ { 2 } x ^ { 2 } - { \scriptstyle { \frac { 1 } { 2 } } } a ^ { - 1 } x ^ { 2 } = 0\) in \(a ^ { - 1 } R / R\) for all \(x \in a ^ { - 1 } R / R\) is equivalent to \(\xi ^ { 2 } - 1 = 0\) in \(2 a ^ { - 1 } R / 2 R \simeq R / a R\) . Consider for example \(R = \mathbb { Z }\) and \(a = 2 ^ { k }\) , \(k \geq 1\) . Then \(\operatorname { O } ( \langle 2 ^ { - k } \rangle )\) is the identity for \(k = 1\) , the cyclic group generated by \(-\) id for \(k = 2\) , and the Klein 4-group \(C _ { 2 } { \times } C _ { 2 }\) generated by \(-\) id and multiplication by \(- 1 + 2 ^ { k - 1 }\) mod \(2 ^ { k }\) for all \(k \geq 3\) (since its square is \(1\) in \(\mathbb { Z } / 2 ^ { k } \mathbb { Z }\) ). Since the isometry group of rank one integral lattices consists of \(\pm\) id, we see that the reduction homomorphism (for the symmetric as well as the quadratic form) is not surjective if \(k \geq 3\) . If instead \(R = \mathbb { Z } _ { 2 }\) , the group \(R _ { [ 2 ] } ^ { \times }\) is isomorphic to \(C _ { 2 } \times C _ { 2 }\) , since the units up to squares are represented by \(\pm 1 , \pm 3\) , each with square 1. The preceding calculation for the isometry groups of the discriminant forms for \(R = \mathbb { Z }\) is also valid for \(R = \mathbb { Z } _ { 2 }\) . So in this case the reduction morphisms are surjective.
Example / comparison
2. Binary forms. (a) We first consider the hyperbolic plane \(U _ { R } = R e \oplus R f\) , \(e \cdot e = f \cdot f = 0\) , \(e \cdot f = 1\) Assume \(2 \neq 0\) . Up to units \(e\) and \(f\) are the only primitive isotropic vectors. Since these are preserved under isometries, the only isometries are the rotations \(i _ { a } : = \left( \begin{array} { c c } { { a } } & { { 0 } } \\ { { 0 } } & { { a ^ { - 1 } } } \end{array} \right)\) , \(a \in R ^ { \times }\) , and the reflections \(j _ { b } : = \binom { 0 } { b } \begin{array} { r } { b ^ { - 1 } } \\ { 0 } \end{array}\) , \(b \in R ^ { \times }\) . Observe that \(j _ { b }\) is the hyperplane reflection in \(e - b f\) . Since \(j _ { 1 } \circ j _ { a } = i _ { a }\) , all isometries of \(U _ { R }\) are products of at most two hyperplane reflections. Note that these isometries are also the isometries for the corresponding quadratic form. In case \(2 = 0\) , also the vector \(e + f\) is a primitive isotropic vector and as we saw in Example 6.5.1, this gives an isometry of the bilinear form but not of the quadratic form. Indeed, if \(\left( \begin{array} { c c } { { a _ { 1 1 } } } & { { a _ { 1 2 } } } \\ { { a _ { 2 1 } } } & { { a _ { 2 2 } } } \end{array} \right)\) represents an isometry with respect to the quadratic form, then \(( a _ { 1 1 } x + a _ { 1 2 } y ) ( a _ { 2 1 } x + a _ { 2 2 } y ) = x y\) . This implies \(a _ { \mathrm { 1 1 } } a _ { \mathrm { 2 1 } } = a _ { \mathrm { 1 2 } } a _ { \mathrm { 2 2 } } = 0\) and \(a _ { 1 1 } a _ { 2 2 } + a _ { 1 2 } a _ { 2 1 } = 1\) so that the only isometries preserving the quadratic form are just the \(j _ { a }\) and the \(i _ { b }\) for suitable units \(a\) , \(b\) . For a non-zero \(s\) which is not a unit, the lattice \(U _ { R } ( s )\) is not unimodular and its discriminant torsion module \([ s ^ { - 1 } R / R ] ^ { \oplus 2 }\) inherits the hyperbolic form. Its isometry group consists of the isometries \(i _ { \bar { a } }\) and \(j _ { \bar { b } }\) where now \(\bar { a } , \bar { b } \in ( R / s R ) ^ { \times }\) . Hence the reduction homomorphism ?????? is surjective. (b) Next, we return to the situation (see Section 6.3.B) where \(k = Q ( R )\) , \(K = Q ( S )\) \(S = R [ \xi ]\) , a quadratic extension with \(\xi ^ { 2 } + u \xi + v = 0\) , and associated quadratic form \(\mathsf { N } _ { K / k } ( x + y \xi ) = x ^ { 2 } - u x y + v y ^ { 2 }\) on \(K\) (i.e., \(x ^ { 2 } - u x y + v y ^ { 2 }\) on \(\Phi ^ { 2 } k\) , where \(( x , y )\) is identified with \(x + y \xi\) ). (b1) Determining the rotations in case \(2 \neq 0\) . Multiplication with any \(U \in S\) with \(N _ { K / k } ( U ) = 1\) induces a \(k\) -linear map from \(K\) to \(K\) that preserves the quadratic form since \(N _ { K / k }\) is multiplicative. The matrix of multiplication by \(U = a + b \xi\) , \(a , b , \in R\) , in the basis \(\{ 1 , \xi \}\) is given by \[ M _ { a , b } = \left( \begin{array} { c c } { { a } } & { { - b v } } \\ { { b } } & { { a - u b } } \end{array} \right) . \] Since \(\operatorname* { d e t } \bigl ( M _ { a , b } \bigr ) = a ^ { 2 } - u a b + v b ^ { 2 } = N _ { K / k } ( U ) = 1\) , this is a rotation. We claim that all rotations occur this way. Let \(\rho\) be any rotation and let \(\rho ( 1 ) = a \cdot 1 + b \xi\) . Then \(N _ { K / k } \big ( a + b \xi \big ) = 1\) and \(\rho ^ { - 1 } \cdot M _ { a , b } ( 1 ) = 1\) . A small matrix computation shows that the only rotation which fixes 1 is the identity. It follows that \(\rho = M _ { a , b }\) . (b2) On reflections in case \(2 \neq 0\) . The reflection in \(a - 1 + b \xi\) with \(N _ { K / k } ( a + b \xi ) =\) \(a ^ { 2 } - u a b + v b ^ { 2 } = 1\) has matrix \[ M _ { a , b } ^ { \prime } = \left( \begin{array} { c c } { { a } } & { { - u a + v b } } \\ { { b } } & { { - a } } \end{array} \right) . \] As in the case of rotations, any orthogonal transformation with determinant \(^ { - 1 }\) is of this shape. Observe also that \(M _ { a , b } ^ { \prime } \circ M _ { 1 , 0 } ^ { \prime } = M _ { a , b }\) , and so all isometries are products of at most two reflections. (b3) The case \(2 = 0\) . We still have the preceding wo types of orthogonal transformations despite both having determinant 1. The first type \(M _ { a , b }\) comes from multiplication with norm 1 elements. However, a matrix for a reflection cannot come from multiplication with norm 1 vectors, since if this were the case, we would have \(u b = 0 = u a\) and so \(u = 0\) but then the extension \(R ( \xi ) / R\) would not be separable. We shall see later (cf. Example 16.2.5.3) that in this case the two types of orthogonal transformations are distinguished by their Dickson invariants.
Example / comparison
The form \(V\) over \(\mathbb { Z } _ { 2 }\) , that is, the norm form for the extension \(\mathbb { Z } _ { 2 } [ \xi ]\) with \(\xi ^ { 2 } + \xi + 1 = 0\) . Hence we can apply the calculation in \(\ S\) 6.3.B, example 4: The rotation group of \(V\) (and so of \(V _ { k }\) ) consists of matrices of the form \(\left( { a \atop b } \quad { - b \atop a - b } \right)\) and those with determinant equal to \(^ { - 1 }\) are given by \(\left( { a \atop b } \right. \left. { - a + b \atop - a } \right)\) Since \(a ^ { 2 } - a b + b ^ { 2 } = 1\) either \(a\) or \(^ { b }\) must be a dyadic unit. As in the previous example \(0 ( q _ { V _ { k } } ^ { \# } )\) is of similar shape where one replaces by residue classes of integers modulo with \(a ^ { 2 } - a b + b ^ { 2 } \equiv 1 { \bmod { 2 } } ^ { k }\) . In particular, the reduction homomorphism \(r _ { V _ { k } } ^ { q }\) is surjective.
Example / comparison
Consider the quadratic form \(q ( x , y ) = a x ^ { 2 } + 2 x y + c y ^ { 2 }\) on \(\mathbb { Z } ^ { 2 }\) with \(a , c \in \mathbb { Z }\) and \(a \ge 2\) and \(c > a\) . The only integral solution for \(q ( x , y ) = a\) is \(( x , y ) = ( \pm 1 , 0 )\) . Indeed, writing \[ q ( x , y ) = a \left( x + a ^ { - 1 } y \right) ^ { 2 } + \left( c - a ^ { - 1 } \right) y ^ { 2 } , \] and observing that \(c - a ^ { - 1 } > a\) , one sees that if \(| y | \ge 1\) , there is no solution, and if \(y = 0\) , then \(x = \pm 1\) . Hence, an isometry of \(q ( x , y )\) must preserve the first basis vector up to sign. Then the matrix representing the isometry is of the shape \(\left( { \begin{array} { c c } { \pm 1 } & { * } \\ { 0 } & { \pm 1 } \end{array} } \right)\) and a small calculation shows that if \(a > 2\) such an isometry can only be ±id and \(q\) does not admit reflections. However, for \(a = 2\) there are two reflections, \(\pm \left( { \begin{array} { r r } { 1 } & { 1 } \\ { 0 } & { - 1 } \end{array} } \right)\) .