Appendix 1. Quadratic Forms The theory of symmetrie bi linear forms is intimately related to the theory of quadratie forms. In fact, over a ring in whieh 2 is a unit, the two theories are indistinguishable. For this reason, it seems advisable to give abrief deseription of quadratie forms, even though our emphasis is on bilinear forms. Let X be an R-module. As usual we assurne that the ring R is eommutative with I. Definition. A quadratie form on X is a funetion q: X --> R sueh that q(:n:)=:x 2 q(x) für all 'XE R, and such that the funetion (xIY) on X x X defined by (xly)=q(x+ y)-q(x)-q(y) is bilinear over R. As an example, if ß is abilinear form (not neeessarily symmetrie), then the funetion q(x)=ß(x,x) is clearly quadratie, with assoeiated symmetrie bi linear form (xIY)= ß(x,y)+ ß(y, x). If X is projeetive, then every quadratie form on X ean be obtained in this way from a (non-symmetrie) bi linear form. Two bilinear forms ß and ß' give rise to the same quadratic form if and only if the difference ß- ß' is sympleetie. Note the identity (xlx)=2q(x). Thus the symmetrie bilinear form (xIY) assoeiated with a quadratie form must satisfy the eongruenee (xlx)=O mod 2R for all x. If 2 is not a zero-divisor in R, then any symmetrie bi linear form satisfying this eongruenee is assoeiated with a unique quadratie form q(x)=t(xlx). In partieu/ar, if 2 is a unit then every symmetrie bilinear form (xIY) comes from a 11l1ique quadratie form t(xlx). Appendix I. Quadratic Forms 111 Definition. Let (Xl' ql)' ... , (Xn, qn) be modules with quadratic forms over any commutative ring R. The orthogonal sum Xl (j) ... (j) Xn is defined to be the direct sum of the modules Xi with quadratic form q defined by the equation " q(xI(j)···(j)xn)= L.,qi(Xi) summed over 1 ~ i ~ n. The associated bi linear form (x!y) on X = Xl (j) ... (j) X n is the ortho­ gonal sum of the associated bilinear forms (X!Y)i on the Xi' Hence (x!y) is an inner product if and only if each (X!Y)i is an inner product. Definition. We will call the pair (X, q) a quadratie inner produet space, if X is finitely generated projective, and if the bilinear form (x!y) asso­ ciated with the quadratic form q is an inner product on X (Chapter I, § 1.1). We observed in Chapter I, § 4 that the Witt Cancellation Theorem (4.4) is not true for inner product spaces in characteristic 2. It is interesting to note that the analogue of (4.4) for quadratic inner product spaces is true in any characteristic. That is, if (Xl' ql)(j)(X, q)~(Xz' qz)(j) (X, q), where (XI,ql)' (Xz,qz), and (X,q) are quadratic inner product spaces over a field, then (Xl' ql)~(XZ' qz). This is proved in [Chevalley, p.16] or [Bourbaki, p. 71]. The following basic remark is due to [Fröhlich, 1969] and in­ dependently to [Sah]. If Xl is asymmetrie bilinear form module and Xz is a qUo1dratic form module, then the tensor product Xl ® X z is a quadratic form module. In fact, given a symmetric bi linear form ßI on Xl and a quadratic form qz on Xz, there is a unique quadratic form q on XI®XZ satisfying the equations, both being necessary for the definition, q (Xl ®xz) = ßI (Xl' Xl) qz (X z) and (Xl ®XZ!YI ® Yz) = ßI (Xl' Ytl (xz!Yz)· Note the isomorphism (1)®Xz ~ Xz' Ifboth Xl and Xz are quadratic form modules, then using the associated bi linear form (XtlYI) as ßI it follows that XI®XZ is also a quadratic form module. The quadratic form q on XI®XZ is determined by the equations q (Xl ® Xz) = 2ql (Xl) qz (X z) and (Xl ® XZ!YI ® Yz) = (XI!YI) (X z!Y2)· The factor of 2 is surprising but necessary. This factor was incorrect1y left out in [Bourbaki, v. 24, p. 137]. 112 Appendix I. Quadratic Forms The Witt algebra WQ (R) of quadratie inner produet spaees over R ean now be defined as folIows. (Compare [Bass], [Sah].) A quadratie inner produet spaee (X, q) is said to be splir if the module X eontains a direet summand N with N=Nl. and q(N)=O. Two quadratic inner produet spaees (Xl' ql) and (X2, q2) belong to the same Witt dass if (Xl' ql)ffi(SI' q~)~(X2' q2)ffi(S2' q~) where the (Si' q;) are split. The Witt dasses of quadratie inner produet spaees over R now form the required algebra WQ(R). Clearly WQ(R) is a eommutative assoeiative algebra over the Witt ring W(R). This algebra does not possess a 1 element in general. There is a eanonieal augmen­ tation a: WQ(R)- W(R), and the produet of two arbitrary elements Wl and w2 in WQ(R) is deter­ mined by the identity wl w2 = a(wl ) w2 • Of course if 2 is a unit in R, then a is an isomorphism. In the ease of a field F, it is easily verified that every quadratie inner produet spaee over F is the orthogonal sum of a split quadratie inner produet spaee and an anisotropie quadratie inner produet spaee (i. e., one with Q (x)::j= 0 for x::j= 0). Furthermore, any two split spaees of the same rank are isomorphie. Using the Witt eaneellation theorem, it follows that every Witt dass in WQ(F) possesses one and up to isomorphism only one anisotropie representative. Suppose in particular that F is a field of eharaeteristie 2. Then the bi linear form (xIY) assoeiated with a quadratie form is neeessarily sym­ pleetie (xlx)=2q(x)=0. Thtrefore the augmentation homomorphism WQ(F)- W(F) is identically zero. For if (X, q) is a quadratie inner produet spaee, then X possesses a sympleetie basis (Chapter I, § 3.5), and henee represents the zero element of the Witt ring W(F). An important invariant of a quadratic inner produet spaee in ehar­ aeteristie 2 has been defined by C. Arf. Let p: F-F denote the additive homomorphism fJ (~) = ~2 +~. Choosing a sym­ pleetic basis Xl"'" x n ' Yl"'" Yn for X, with inner produet matrix G ~), Arf showed that the residue dass of q(xl ) q(Yl)+ ... +q(xn) q(}'n) Appendix 1. Quadratic Forms 113 modulo fJ (F) is an invariant of (X, q). This Arf invariant depends only on the Witt class of (X, q). and hence gives rise to an additi\e surjection LI: WQ(F)->F/pF. The kernel of LI has been computed by H. Sah as fallows. Let 1 c W(F) be the fundamental ideal. Then kerncl(LI)= I· WQ(F). The resulting additive isomorphism WQ(F)/I· WQ(F)~F/.pF should perhaps be regarded as an analogue of pfister's isomorphism I/I 2 ~ F· / F· 1 . (Compare Chapter III, § 5.2.) Here is a simple example to illustrate the Arf invariant. Let (X, q) be a quadratic inner product space of rank 2. We continue to assume that the field F has characteristic 2. Ta any basis x, y with (xIY)= 1 we associate the residue class q(x) q(y) (mod gafF)). This is an invariant of (X, q). for under an elementary change of basis .x = x + er:y we have q (.x) = q (x) + er: + er: l q (y), and therefore q(x) q(y)=q(x) q(y)+ f,J(etq(y)). From this formula we see that the quadratic form q represellts 0 if ami only if q(x) q(y)=O mod p(F). For if q(x) q(y)=O and q(Y)=FO, then we can choose 'l. so that q (x) q (y) = O. In fact we can actually choose a symplectic basis x, y with q(x)=q(y)=O, simply by setting y= ßx+ y and choosing ß appropriately. IJ the field F is perfect, then the residue class q(x) q(y) mod gJ(F) is always a complete invariant Jor the quadratic inner product space. Proo! We may assume that q(x) =1=0. Choosing an arbitrary re­ presentative Ja =q(x) q(y)+ fJ(er:) for the Arf invariant. the symplectic basis ~ x=x/Vq(x) y=:xx+.n/q('() will satisfy q(x)= 1, qCY)=Ll a. Thus Ll a determines the isomorphism class of (X, q). 0 In particular, if F is a finite fielel oJ characteristic 2 it follows that there are precisely two isomorphism classes of qlladratic inner product spaces oJ rank t\\'o over F. For by inspecting the additi\'e exact se­ quence 0 -> F2 -> F ~ F we see that the cokernel Fi p (Fl has order two. Appendix 2. Hermitian Forms Let R be an associative ring with 1, not necessarily commutative. Byan involution of R (or more precisely an "involutory anti-automorphism") is meant an additive homomorphism a H aJ from R to itself satisfying (aß)J = ßJ aJ and (aJ)J = a for all CI. and ß. Note that 1J = 1. Examples. If R is commutative, then the identity map of R is an involution. For any multiplicative group n, the integral group ring zn possesses a canonical involution which maps each group element (J to (J-l.