FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 18 definitions, 6 theorems, 6 propositions, 8 lemmas, 3 corollaries, 7 examples, 4 constructions and 1 remark.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition / construction / notation
Let \(M\) be a finitely generated \(R\) -module. By definition its torsion submodule equals \[ \operatorname{Tors}(M) = \{m \in M \mid rm = 0 \text{ for some } r \in R \setminus \{0\}\}. \] The quotient \(M/\operatorname{Tors}(M)\) is isomorphic to a direct sum of \(\operatorname{rank}(M)\) copies of \(R\), that is, a free \(R\)-module.
Definition / construction / notation
The quotient \(M / \mathrm { T o r s } ( M )\) is isomorphic to a direct sum of \(\mathrm { r a n k } ( M )\) copies of \(R\) . Such a module is called a free \(R\) -module. Any submodule of a free module is free and a quotient of an \(R\) -module by a submodule of the same rank is torsion:
Lemma A.1.1
Lemma A.1.1. Let \(R\) be a principal ideal domain and let \(M\) be a free finitely generated \(R\) -module and \(M ^ { \prime } \subset M\) a submodule of the same rank. Then \(M / M ^ { \prime }\) is an \(R\) -torsion module and every torsion module is of this form. More precisely, \(M\) admits a basis \(\{ e _ { 1 } , \ldots , e _ { r + s } \}\) such that \(M ^ { \prime } = \oplus _ { k = 1 } ^ { r } R e _ { k } \oplus \oplus _ { k = 1 } ^ { s } R \cdot ( d _ { k } e _ { r + k } )\) with \(d _ { j }\) non-units such that \(d _ { 1 } | d _ { 2 } | \cdots | d _ { s }\) , and hence \(M / M ^ { \prime } \simeq R / d _ { 1 } R \oplus \cdot \cdot \cdot \oplus R / d _ { s } R\) .
Theorem A.1.2
Theorem A.1.2 (Elementary divisor theorem). Let \(R\) be a principal ideal domain and let \(T\) be a finitely generated torsion \(R\) -module. There is an isomorphism \(T \simeq\) \(R / d _ { 1 } R \oplus \cdot \cdot \cdot \oplus R / d _ { s } R\) , with \(d _ { j }\) non-units such that \(d _ { 1 } | d _ { 2 } \dotsm | d _ { s }\) . The ideals generated by \(d _ { j }\) , the elementary divisors, are uniquely determined by \(T\) and the resulting decomposition is the invariant factor decomposition of \(T\) .
Definition / construction / notation
There is another canonical decomposition for \(T\) . First some terminology. For \(p \in R\) irreducible, one says that a finitely generated torsion module is \(p\) -primary if every element is annihilated by some power of \(p\) . Setting \[ T _ { p } = \{ x \in T \mid p ^ { n } \cdot x = 0 \quad { \mathrm { f o r ~ s o m e ~ p o w e r ~ } } p ^ { n } { \mathrm { ~ o f ~ } } p \} , \]
Definition / construction
For an irreducible \(p\in R\), the maximal \(p\)-primary submodule is \(T_p=\{x\in T:p^n x=0\text{ for some }n\}\), and a finite torsion module has the Sylow decomposition \(T=\bigoplus_pT_p\) with each \(T_p\) decomposed into homogeneous summands \(R/p^eR\).
Definition / construction / notation
The \(s _ { p , e }\) summands of the decomposition of \(T _ { p }\) with fixed \(e\) are called the homogeneous summands of exponent \(e\) . Note that the above isomorphism for \(T _ { p }\) is not canonically determined by \(T\) as demonstrated in Example A.1.4.2 below.
Construction / comparison
The invariant-factor and primary decompositions are related by factoring each elementary divisor into prime powers and applying the Chinese remainder theorem.
Definition / construction / notation
We frequently use the concept of length: the length \(\ell ( G )\) of a finite abelian torsion \(R\) -module \(G\) is equal to the minimal number of generators (by definition). In particular, the length of \(T _ { p }\) is at most \(s\) , and for primes diving \(d _ { 1 }\) the length equals \(s\) .
Definition / construction
The length \(\ell(G)\) of a finite torsion module is its minimal number of generators; for a \(p\)-primary module it is the total number of homogeneous summands, and the length of a finite torsion module is the maximum of its primary lengths.
Lemma A.1.3
Lemma A.1.3. The length of a torsion \(R\) -module \(T\) of finite rank with Sylow decomposition \(T = \oplus T _ { p }\) equals max(ℓ??) = #(elementary divisors) where \(\ell _ { p }\) is the length of \(T _ { p }\) .
Example / comparison
If \(R = \mathbb { Z }\) a torsion module is the same as a finite abelian group, and a \(p\) -primary \(\mathbb { Z }\) -torsion module is the same as a \(p\) -primary group.
Example / comparison
We assume \(R = \mathbb { Z }\) . Suppose the elementary divisors of \(T\) are \(( 4 , 1 2 , 4 8 )\) . Then \(T _ { 2 } \simeq \mathbb { Z } / 4 \mathbb { Z } \oplus \mathbb { Z } / 4 \mathbb { Z } \oplus \mathbb { Z } / 1 6 \mathbb { Z }\) , \(T _ { 3 } \simeq \mathbb { Z } / 3 \mathbb { Z } \oplus \mathbb { Z } / 3 \mathbb { Z }\) and \(\ell ( T ) = 3\) . Suppose \(\{ e _ { 1 } , e _ { 2 } , e _ { 3 } \}\) is an ordered basis of \(T _ { 2 }\) , i.e., a basis reflecting the above isomorphism. Then \(\{ f _ { 1 } = e _ { 1 } + 2 e _ { 2 } + 4 e _ { 3 } , f _ { 2 } = e _ { 2 } + 4 e _ { 3 } , f _ { 3 } = 3 e _ { 3 } \}\) is also an ordered basis of \(T _ { 2 }\) and so the isomorphism \(T _ { 2 } \simeq \mathbb { Z } / 4 \mathbb { Z } \oplus \mathbb { Z } / 4 \mathbb { Z } \oplus \mathbb { Z } / 1 6 \mathbb { Z }\) is not unique. We finish this section by recalling some modular arithmetic. The multiplicative group of units \(R ^ { \times }\) of the ring \(R = \mathbb { Z } / m \mathbb { Z }\) has order \(\varphi ( m )\) , the number of elements modulo \(m\) that are coprime with \(m\) . In what follows elements \(r\) of \(R\) are viewed as additive classes of integers modulo \(m\) and we write \(r \ = \ \bar { x }\) , where \(x \in \mathbb { Z }\) . If the group \(( \mathbb { Z } / m \mathbb { Z } ) ^ { \times }\) is cyclic and \(r\) is a generator, one says that \(r\) is a primitive generator modulo \(m\) . We denote the (multiplicative) cyclic group of order \(n\) by \(C _ { n }\) (in contrast to the additive version \(\mathbb { Z } / n \mathbb { Z }\) ). The next result can be found in elementary textbooks on number theory, e.g. [113].
Definition / construction / notation
We finish this section by recalling some modular arithmetic. The multiplicative group of units \(R ^ { \times }\) of the ring \(R = \mathbb { Z } / m \mathbb { Z }\) has order \(\varphi ( m )\) , the number of elements modulo \(m\) that are coprime with \(m\) . In what follows elements \(r\) of \(R\) are viewed as additive classes of integers modulo \(m\) and we write \(r \ = \ \bar { x }\) , where \(x \in \mathbb { Z }\) . If the group \(( \mathbb { Z } / m \mathbb { Z } ) ^ { \times }\) is cyclic and \(r\) is a generator, one says that \(r\) is a primitive generator modulo \(m\) . We denote the (multiplicative) cyclic group of order \(n\) by \(C _ { n }\) (in contrast to the additive version \(\mathbb { Z } / n \mathbb { Z }\) ).
Lemma A.1.5
Lemma A.1.5. 1. The units of the finite cyclic groups have the following structure: (a) For \(p\) odd the groups \(( \mathbb { Z } / p ^ { k } \mathbb { Z } ) ^ { \times }\) and \(( \mathbb { Z } / 2 p ^ { k } \mathbb { Z } ) ^ { \times }\) are cyclic of order \(p ^ { k } - p ^ { k - 1 }\) . For \(p = 2\) we have \[ ( \mathbb { Z } / 2 ^ { k } \mathbb { Z } ) ^ { \times } = { \left\{ \begin{array} { l l } { 1 } & { i f k = 1 } \\ { C _ { 2 } ~ g e n e r a t e d ~ b y } { \overline { { - 1 } } } \right.} & { i f k = 2 } \\ { C _ { 2 } \times C _ { 2 ^ { k - 2 } ~ } g e n e r a t e d ~ b y } { ( \overline { { - 1 } } } , { \overline { { \pm 3 } } } ) & { i f k \geq 3 . } \end{array} \] (b) In general, writing \(m = 2 ^ { k } \Pi _ { p } p ^ { k _ { p } }\) , where the product is over odd primes \(p\) , we have \[ ( \mathbb { Z } / m \mathbb { Z } ) ^ { \times } \simeq ( \mathbb { Z } / 2 ^ { k } \mathbb { Z } ) ^ { \times } \times \prod _ { p } ( \mathbb { Z } / p ^ { k _ { p } } \mathbb { Z } ) ^ { \times } . \] In particular, only for \(m = 2 , 4 , p ^ { k }\) and \(2 p ^ { k }\) there exist primitive generators modulo \(m\) . 2. The group \(\sf D ( \mathbb { Z } / m \mathbb { Z } )\) of units of \(\mathbb { Z } / m \mathbb { Z }\) modulo squares has the following structure: (a) In the cases \(m = p ^ { k }\) and \(2 p ^ { k }\) the group \(\sf D ( \mathbb { Z } / m \mathbb { Z } )\) is cyclic of order 2 and generated by a non-square modulo \(p ^ { k }\) : D(ℤ/????ℤ) = {1, ?? mod ????} ≃ ??2, \(p\) an odd prime and ?? a non-square modulo \(p ^ { k }\) (b) For \(p = 2\) we have: \[ { \mathsf { D } } ( \mathbb { Z } / 2 ^ { k } \mathbb { Z } ) = { \left\{ \begin{array} { l l } { \{ { \bar { 1 } } \} } & { f o r k = 1 , } \\ { \{ { \bar { 1 } } , { \bar { 3 } } \} \simeq C _ { 2 } } & { f o r k = 2 , } \\ { \{ { \bar { 1 } } , { \bar { 3 } } , - { \bar { 3 } } , - { \bar { 1 } } \} \simeq C _ { 2 } \times C _ { 2 } } & { i f k \geq 3 . } \end{array} \right. } \]
Definition / construction / notation
The \(p\) -adic valuation and \(\mathbb { Q } _ { p }\) . The field of \(p\) -adic numbers \(\mathbb { Q } _ { p }\) is the completion of \(\mathbb { Q }\) with respect to the \(p\) -adic valuation \(v _ { p }\) on \(\mathbb { Q }\) which is defined as follows: first, one sets \(v _ { p } ( 0 ) = \infty\) . Next, for a non-zero number \(\begin{array} { r } { x = p ^ { k } \frac { u } { v } } \end{array}\) with \(\operatorname* { g c d } ( u , p ) =\) \(\operatorname* { g c d } ( v , p ) = 1\) , one sets \[ v _ { p } ( x ) = k , \quad \| x \| _ { p } = p ^ { - k } . \]
Definition / construction / notation
By definition of a completion, we have a natural embedding \[ \iota _ { p } : \mathbb { Q } \hookrightarrow \mathbb { Q } _ { p } . \]
Theorem A.2.1
Theorem A.2.1. 1. Suppose \(p\) is an odd prime. Then \(x \in \mathbb { Q } _ { p } ^ { \times }\) can be written uniquely as \(x = u p ^ { k }\) with \(u\) a unit in \(\mathbb { Z } _ { p }\) . It is a square if and only if \(k\) is even and \(u \equiv 1\) mod \(p\) . The group \(\mathsf { D } ( \mathbb { Q } _ { p } )\) is isomorphic to the Klein group with representatives \(1 , p\) , \(u , u p\) , where \(u\) is a unit in \(\mathbb { Z } _ { p }\) such that \(u\) mod \(p\) is a non-square. The group \(\mathsf { D } ( \mathbb { Z } _ { p } )\) is cyclic of order two, generated by a non-square in \(\mathbb { Z } _ { p } ^ { \times }\) . 2. A dyadic number \(x \in \mathbb { Q } _ { 2 }\) can be uniquely written as \(x = u \cdot 2 ^ { k }\) , \(u\) a unit in \(\mathbb { Z } _ { 2 }\) . It is a square if and only if \(k\) is even and \(u \equiv 1\) mod 8. The group \(\mathsf { D } ( \mathbb { Q } _ { 2 } )\) is isomorphic to \(C _ { 2 } \times C _ { 2 } \times C _ { 2 }\) with generators 2, 3, 5 mod 8. \(A\) unit \(u \in \mathbb { Z } _ { 2 }\) is a square if and only if \(u\) mod 8 is a square, and \(\mathsf { D } ( \mathbb { Z } _ { 2 } )\) is isomorphic to the Klein group \(( \mathbb { Z } / 8 \mathbb { Z } ) ^ { \times } = C _ { 2 } \times C _ { 2 }\) with generators \(3 , 5\) mod 8.
Definition / construction
The \(p\)-adic valuation defines the \(p\)-adic topology on \(\mathbf Q_p\); \(\mathbf Z_p\) is an open compact subring, \(\mathbf Q\) is dense in \(\mathbf Q_p\), and \(\mathbf Z\) is dense in \(\mathbf Z_p\).
Proposition A.2.2
Proposition A.2.2. The \(p\) -adic topology on \(\mathbb { Q } _ { p }\) has the following properties: 1. \(\mathbb { Q } _ { p }\) is locally compact and contains \(\mathbb { Z } _ { p }\) as an open compact subring. 2. The field \(\mathbb { Q }\) is dense in \(\mathbb { Q } _ { p }\) and the ring \(\mathbb { Z }\) is dense in \(\mathbb { Z } _ { p }\) .
Definition / construction
For a finite set of places \(S\), the local embeddings induce the diagonal embedding \(\iota_S:\mathbf Q\hookrightarrow\prod_{v\in S}\mathbf Q_v\) used in weak approximation.
Proposition A.3.1
Proposition A.3.1 (Weak approximation). If \(S\) is finite, the embedding \(\iota _ { S }\) is a dense embedding with respect to the product topology.
Corollary A.3.2
Corollary A.3.2. Suppose that for some finite set \(S \subset \mathcal { P }\) and all \(p \in S\) we are given \(c _ { p } \in \mathbb { Q } _ { p } ^ { \times }\) . Then there exists \(c \in \mathbb { Q } ^ { \times }\) such that the coset of \(c\) in \(\mathsf { D } ( \mathbb { Q } _ { p } )\) equals the coset of \(c _ { p }\) for all \(p \in S\) .
Proposition A.3.3
Proposition A.3.3 (Weak approximation for the rotation group). Let \(( V , q )\) be \(a\) (non-degenerate) quadratic \(\mathbb { Q }\) -vector space and let \(S \subset \mathcal { S }\) be a finite set of places. Then the natural diagonal embedding \(\begin{array} { r } { { \mathsf { S O } } ( V ) \hookrightarrow \prod _ { v \in S } { \mathsf { S O } } ( V _ { v } ) } \end{array}\) is dense.
Corollary A.3.4
Corollary A.3.4. Let \(( V , q )\) be a (non-degenerate) quadratic \(\mathbb { Q }\) -vector space of dimension \(\geq ~ 2\) . Let \(S ~ \subset ~ { \mathcal { P } }\) be a finite set of places. Suppose that \(t ~ \in ~ \mathbb { Q } ^ { \times }\) is represented by \(q\) . Then for given \(x _ { v } \in V _ { v }\) , \(v \in S\) with \(q ( x _ { v } ) = t\) , there exists \(x \in V\) with \(q ( x ) = t\) which is as close to each of the \(x _ { v }\) as we want.
Remark A.3.5
Remark A.3.5. As to lattices, since \(\mathbb { Z } _ { p }\) is open and compact in \(\mathbb { Q } _ { p }\) , any lattice \(L _ { p }\) in \(V _ { p }\) is open and compact in \(V _ { p }\) . This is not the case for the place \(\infty\) . Also, the subgroup \(\mathrm { O } ( L _ { p } )\) of \(\mathrm { O } ( V _ { p } )\) stabilizing a lattice \(L _ { p } \subset V _ { p }\) is an open subgroup.
Theorem A.3.6
Theorem A.3.6 (Strong Approximation Theorem [36, Ch. 9, Thm. 1.5]). Let \(( L , b )\) be a non-degenerate indefinite integral lattice of rank \(\geq 4\) and \(t\) a non-zero integer. Suppose that Then there exists \(x \in L\) such that \(b ( x , x ) = t\) . Further, given a finite set of places \(S \subset { \mathcal { P } }\) , we can require \(x \in L\) to be \(p\) -adically arbitrarily close to \(x _ { p }\) for all \(p \in S\) .
Corollary A.3.7
Corollary A.3.7. Let \(( L , b )\) be an indefinite non-degenerate quadratic lattice of rank \(\geq 4\) and let \(x \in L\) be a vector with \(b ( x , x ) = t \neq 0\) . Then there are infinitely many \(y \in L\) with \(b ( y , y ) = t\) .
Definition / construction / notation
Definitions and Properties. Let \(v \in \mathcal { P }\) and let \(a , b \in \mathbb { Q } _ { v } ^ { \times }\) . Define the Hilbert symbol (at \(v\) ) as follows:1 \[ \begin{array} { r } { ( a , b ) _ { v } = \left\{ \begin{array} { l l } { 1 } & { \mathrm { ~ i f ~ } z ^ { 2 } = a x ^ { 2 } + b y ^ { 2 } \mathrm { ~ h a s ~ a ~ n o n - t r i v i a l ~ s o l u t i o n ~ } ( z , x , y ) \mathrm { ~ i n ~ } \mathbb { Q } _ { v } ^ { 3 } , } \\ { - 1 } & { \mathrm { ~ o t h e r w i s e } . } \end{array} \right. } \end{array} \]
Lemma A.4.1
Lemma A.4.1 (Hensel’s lemma). Let \(f ( X _ { 1 } , \dots , X _ { n } ) \in \mathbb { Z } _ { p } [ X _ { 1 } , \dots , X _ { n } ]\) and suppose that \(\pmb { a } = ( a _ { 1 } , \ldots , a _ { n } ) \in \mathbb { F } _ { p } ^ { n }\) is a simple zero of \(f\) mod \(p\) , that is, \(f ( { \pmb a } ) \equiv 0\) mod \(p\) and the gradient \(\nabla f\) at \(\mathbf { a }\) is not identically zero mod \(p\) . Then this zero can be lifted to \(\mathbb { Z } _ { p } ^ { n }\) . In other words, there exists \(\pmb { b } \in \mathbb { Z } _ { p } ^ { n }\) with \(f ( \pmb { b } ) = 0\) and \({ \pmb a } \equiv { \pmb b }\) mod \(p\) .
Example / comparison
Example A.4.2. For odd \(p\), the equation \(ax^2+by^2=c\) over \(\mathbf F_p\), with \(a,b\ne0\), always has a solution. If \(a,b,c\) lift to \(p\)-adic units, a solution lifts by Hensel; in particular \((a,b)_p=1\) after taking \(c=1\).
Lemma A.4.3
Lemma A.4.3 (Hensel’s lemma, II). Let \(q\) be a unimodular dyadic quadratic form of rank ??, and let \(a \in \mathbb { Z } _ { 2 }\) . If a primitive dyadic solution for \(q ( x ) \equiv a\) mod 8 exists, then there is also a ”true” dyadic solution for \(q ( x ) = a\) .
Definition / construction
The Hilbert symbol at a place \(v\) is the symmetric bilinear pairing \((\ ,\ )_v:D(\mathbf Q_v)\times D(\mathbf Q_v)\to\{\pm1\}\); by bilinearity it is determined by its values on generators.
Theorem A.4.4
Let \(a, b \in \mathbb{Q}_{p}\) with \(p\) prime, and write \(a = p^{\alpha}u\), \(b = p^{\beta}v\) with \(u, v \in \mathbb{Z}_{p}^{\times}\). Recall \[ \varepsilon(u) = \begin{cases} 0 & \text{if } u \equiv 1 \bmod 4, \\ 1 & \text{if } u \equiv -1 \bmod 4, \end{cases} \qquad \omega(u) = \begin{cases} 0 & \text{if } u \equiv \pm 1 \bmod 8, \\ 1 & \text{if } u \equiv \pm 3 \bmod 8. \end{cases} \] Then: 1. If \(p\) is odd, \[ (a,b)_{p} = (-1)^{\alpha\beta\varepsilon(p)} \left(\frac{u}{p}\right)^{\beta} \left(\frac{v}{p}\right)^{\alpha}; \] in particular \((a,b)_{p} = 1\) when \(a\) and \(b\) are units. 2. For \(p = 2\), writing \(a = 2^{\alpha}u\), \(b = 2^{\beta}v\) with \(u, v \in \mathbb{Z}_{2}^{\times}\), \[ (a,b)_{2} = (-1)^{\varepsilon(u)\varepsilon(v) + \alpha\omega(v) + \beta\omega(u)}. \]
Construction / comparison
The Hilbert symbol satisfies symmetry, triviality on squares, \((a,-a)_v=(a,1-a)_v=1\), multiplicativity in each argument, and the displayed elementary identities used to reduce calculations.
Construction / comparison
Viewed as an \(\mathbf F_2\)-vector space, \(D(\mathbf Q_p)\) has dimension \(2\) for odd \(p\) and dimension \(3\) for \(p=2\); Hilbert-symbol equations therefore define linear or affine hyperplanes in this square-class space.
Lemma A.4.5
Lemma A.4.5. 1. Let \(r = \dim _ { \mathbb { F } _ { 2 } } \mathsf { D } ( \mathbb { Q } _ { p } )\) and let \(a \in \mathsf { D } ( \mathbb { Q } _ { p } )\) . For \(a \ne 1\) and \(\varepsilon \in \{ 1 , - 1 \}\) , the equation \(( a , x ) _ { p } = \varepsilon\) has \(2 ^ { r - 1 }\) solutions. For \(a = 1\) one has \(( a , x ) _ { p } = 1\) for all \(x \in \mathsf { D } ( \mathbb { Q } _ { p } )\) and \(( 1 , x ) _ { p } = - 1\) has no solutions. 2. Let \(a , a ^ { \prime } \in \mathsf { D } ( \mathbb { Q } _ { p } )\) . Assume that for some \(\varepsilon , \varepsilon ^ { \prime } \in \{ 1 , - 1 \}\) each of the two equations \(( a , x ) _ { p } = \varepsilon\) and \(( a ^ { \prime } , x ) _ { p } = \varepsilon ^ { \prime }\) has a solution in \(\mathsf { D } ( \mathbb { Q } _ { p } )\) . Then no common solution exists if and only if one has simultaneously \(\boldsymbol { a } \ = \ \boldsymbol { a } ^ { \prime }\) and \(\varepsilon = - \varepsilon ^ { \prime }\) .
Theorem A.4.6
Theorem A.4.6 (Hilbert’s product formula). Given \(a , b \in \mathbb { Q } ^ { \times }\) , then \(( a , b ) _ { v } = 1\) for all but a finite number ?? of places and for those \(\begin{array} { r } { \prod _ { v \in S } ( a , b ) _ { v } = 1 } \end{array}\) .
Theorem A.4.7
Theorem A.4.7 (Existence of rational numbers with given Hilbert symbol). Given finitely many \(a_j\in\mathbf Q^\times\) and signs \(\varepsilon_{j,v}\in\{\pm1\}\) satisfying finite support, the product formula in every \(j\), and local simultaneous solvability at every place \(v\), there exists \(x\in\mathbf Q^\times\) with \((a_j,x)_v=\varepsilon_{j,v}\) for all \(j,v\); conversely, existence of such an \(x\) implies these three conditions.
Definition / construction / notation
The pair \(( V , b )\) is called a symplectic \(k\) -space. As for symmetric bilinear forms, one can define orthogonality and orthogonal direct sums. A symplectic form \(^ { b }\) is non-degenerate if any \(x\) for which \(b ( x , V ) = 0\) necessarily vanishes.
Example / comparison
The standard example of a non-degenerate symplectic space is the symplectic plane \(J\) with basis \(\{ e , f \}\) and symplectic form \(\left( \begin{array} { l l } { 0 } & { 1 } \\ { - 1 } & { 0 } \end{array} \right)\) The basis \(\{ e , f \}\) is called a symplectic pair .
Example / comparison
We get examples in all even dimensions by taking orthogonal direct sums \(J ^ { \boldsymbol { \Theta } n } = J \boldsymbol { \Theta } \cdot \cdot \cdot \boldsymbol { \Theta } J\) ( \(n\) copies). Reordering basis vectors, we find that \(J ^ { \ ( \perp ) n }\) is isometric to the standard symplectic form \[ J _ { n } = \left( \begin{array} { l l } { \mathbf { 0 } _ { n } } & { \mathbf { 1 } _ { n } } \\ { - \mathbf { 1 } _ { n } } & { \mathbf { 0 } _ { n } } \end{array} \right) . \] In other words, in a suitable basis \(\pmb { { \cal E } } = \{ e _ { 1 } , \dots , e _ { 2 n } \}\) , the form becomes \[ b _ { E } ( x , y ) = \sum _ { i = 1 } ^ { n } \left( x _ { i } y _ { n + i } - y _ { i } x _ { n + i } \right) , \quad x = \sum _ { j = 1 } ^ { 2 n } x _ { j } e _ { j } , y = \sum _ { j = 1 } ^ { 2 n } y _ { j } e _ { j } . \]
Example / comparison
If \(\operatorname { c h a r } ( k ) = 2\) , the polar form \(b _ { q }\) of a non-degenerate quadratic form \(q\) is a non-degenerate symplectic form. This explains why symplectic forms come up naturally in the study of quadratic forms in characteristic 2. A \(k\) -linear map between symplectic \(k\) -spaces preserving the symplectic form is called a symplectic map and two symplectic spaces are isometric if there exists a symplectic isomorphism between them. Isometries of \(( V , b )\) are called symplectic automorphisms and these form a subgroup Sp(??) of \(\operatorname { G L } ( V )\) , the symplectic group of \(V\) . The classification of non-degenerate symplectic forms up to symplectic isomorphism is very straighforward since any non-degenerate symplectic form is isometric to the standard form:
Definition / construction / notation
A \(k\) -linear map between symplectic \(k\) -spaces preserving the symplectic form is called a symplectic map and two symplectic spaces are isometric if there exists a symplectic isomorphism between them. Isometries of \(( V , b )\) are called symplectic automorphisms and these form a subgroup Sp(??) of \(\operatorname { G L } ( V )\) , the symplectic group of \(V\) . The classification of non-degenerate symplectic forms up to symplectic isomorphism is very straighforward since any non-degenerate symplectic form is isometric to the standard form:
Proposition A.5.2
Proposition A.5.2. A non-degenerate \(k\) -symplectic space \(( V , b )\) is isometric to \(( k ^ { 2 n } , J ^ { \oplus n } )\) , and in particular must be even dimensional. In a suitable basis the Gram matrix of \(^ { b }\) is \(J _ { n }\) , the standard symplectic form.
Example / comparison
For a symplectic vector space, a standard transvection has the displayed form determined by a vector \(u\) and non-zero scalar \(a\).
Lemma A.5.3
Lemma A.5.3. In a non-degenerate symplectic \(k\) -space, a symplectic transvection ?? is of the form ?? = ????,?? .
Proposition A.5.4
Proposition A.5.4. The symplectic group \(\mathsf { S p } ( V )\) is generated by symplectic transvections.
Definition / construction / notation
A \(G\) -module is an abelian group \(M\) equipped with a \(G\) -action, or, equivalently, a \(\mathbb { Z } [ G ]\) -module. Given a \(G\) -module \(M\) , we set \[ \begin{array} { l } { { M ^ { G } = \{ x \in M \mid g x = x , \forall g \in G \} , } } \\ { { M _ { G } = M / I M , \quad I = \langle g x - x \rangle , x \in M , g \in G . } } \end{array} \]
Definition / construction / notation
Group (co)homology of \(G\) can be defined topologically as the (co)homology of a topological space \(X\) which has fundamental group \(G\) and has no higher homotopy groups. These groups turn out to be independent of the choice of such \(X\) .
Lemma A.6.1
Lemma A.6.1. 1. Let \(M\) be a \(G\) -module. Then \(H ^ { 0 } ( G , M ) = M ^ { G }\) and \(H _ { 0 } ( G , M ) =\) \(M _ { G }\) . In particular, \(H ^ { \cup } ( G ) = H _ { 0 } ( G ) = \mathbb { Z }\) . 2. If \(G\) is a finite group of order \(n\) , then the groups \(H ^ { q } ( G , M )\) , \(H _ { q } ( G , M )\) , \(q \geq 1\) , are \(n\) -torsion. If \(G\) is a finite abelian group and \(G\) acts trivially on \(M\) , then \(H ^ { 1 } ( G , M ) \simeq \operatorname { H o m } ( G , M )\) , \(H _ { 1 } ( G , M ) \simeq G \otimes M\) and, in particular, \(H ^ { 1 } ( G ) = 0\) 1 and \(H _ { 1 } ( G ) \simeq G\) . 3. If \(G\) is a finite abelian group \(H ^ { 2 } ( G ) = \operatorname { E x t } ^ { 1 } ( G , \mathbb { Z } ) \simeq G\) . 4. If \(G\) is cyclic of order \(n\) , then \(H ^ { j } ( G ) = 0\) for \(j\) odd and \(H ^ { j } ( G ) \simeq G\) for \(j > 0\) even, while \(H _ { j } ( G ) \simeq G\) for \(j\) odd and \(H _ { j } ( G ) = 0\) for \(j > 0\) even.
Construction / comparison
For a free action of a finite group \(G\) on a CW-complex \(X\) with quotient \(Y\), the Cartan–Leray spectral sequence has \(E_2^{p,q}=H^p(G,H^q(X))\) and converges to \(H^{p+q}(Y)\).
Proposition A.6.2
Proposition A.6.2. Suppose \(X\) is a connected CW-complex with finite rank cohomology groups and with \(H _ { 1 } ( X ) = 0\) . If \(G\) is a finite abelian group acting freely on \(X\) with quotient map \(\pi : X \to Y = X / G\) , then 1. \(H ^ { 2 } ( X )\) has no torsion and Tors \(H ^ { 2 } ( Y ) = \operatorname { E x t } ^ { 1 } G\) . 2. There is a natural map \(\delta : H ^ { 2 } ( X ) ^ { G } H ^ { 3 } ( G )\) and an exact sequence \[ 0 \to \operatorname { E x t } ^ { 1 } G \to H ^ { 2 } ( Y ) \stackrel { \pi ^ { * } } { \longrightarrow } \ker ( H ^ { 2 } ( X ) ^ { G } \stackrel { \delta } { \to } H ^ { 3 } ( G ) ) \to 0 . \] Moreover, there is an induced isomorphism \[ H ^ { 2 } ( Y ) / \mathrm { T o r s } ( H ^ { 2 } ( Y ) ) \stackrel { \sim } { } \ker ( \delta ) . \] In case \(G\) is cyclic, \(\delta = 0\) and hence \(\pi ^ { * }\) surjects onto \(H ^ { 2 } ( X ) ^ { G }\) .