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17  Bilinear and quadratic forms

Fix a commutative ring \(R\) and an \(R\)-module \(W\), the value module of the forms below. The category \(R\text{-}\mathbf{Mod}\) and extension of scalars are defined in Chapter 11.

Definition 17.1 (Bilinear and quadratic form presheaves) Let \(\operatorname{Bil}_{R,W}(M)\) be the \(R\)-module of \(R\)-bilinear maps \(M\times M\to W\), with pointwise operations. Pullback along \(f\colon M\to N\) sends \(b\) to \[ f^*b(x,y)=b(fx,fy), \] and defines a presheaf \(\operatorname{Bil}_{R,W}\colon(R\text{-}\mathbf{Mod})^{\mathrm{op}} \to R\text{-}\mathbf{Mod}\).

Let \(\operatorname{Quad}_{R,W}(M)\) be the \(R\)-module, under pointwise operations, of maps \(q\colon M\to W\) for which \(q(rx)=r^2q(x)\) and \[ b_q(x,y)=q(x+y)-q(x)-q(y) \] is \(R\)-bilinear. Pullback defines the presheaf \(\operatorname{Quad}_{R,W}\colon(R\text{-}\mathbf{Mod})^{\mathrm{op}} \to R\text{-}\mathbf{Mod}\).

Definition 17.2 (Form categories) Let \(U\colon R\text{-}\mathbf{Mod}\to\mathbf{Set}\) be the forgetful functor. Define \[ \mathcal B_{R,W}=\int_{R\text{-}\mathbf{Mod}}(U\circ\operatorname{Bil}_{R,W}), \qquad \mathcal Q_{R,W}=\int_{R\text{-}\mathbf{Mod}}(U\circ\operatorname{Quad}_{R,W}). \] An object of \(\mathcal B_{R,W}\) is a pair \((M,b)\). A morphism \((M,b_M)\to(N,b_N)\) is an \(R\)-linear map \(f\colon M\to N\) satisfying \(f^*b_N=b_M\). The projection to \(R\text{-}\mathbf{Mod}\) is the discrete fibration of Definition 3.1. The quadratic category uses the same convention.

17.1 Properties of bilinear forms

Definition 17.3 (Symmetry, alternation, and evenness) For \(b\colon M\times M\to W\):

  • \(b\) is symmetric if \(b(x,y)=b(y,x)\);

  • \(b\) is skew-symmetric if \(b(x,y)=-b(y,x)\);

  • \(b\) is alternating if \(b(x,x)=0\);

  • \(b\) is even if \(b(x,x)\in 2W\) for every \(x\).

Alternating forms are skew-symmetric. The converse holds when multiplication by \(2\) is injective on \(W\). When \(2W=W\), every bilinear form satisfies the evenness condition; quadratic refinements retain additional information in the discriminant setting.

Definition 17.4 (The adjoint maps and the radicals) The bilinear form \(b\) determines two \(R\)-linear maps to \(\operatorname{Hom}_R(M,W)\), \[ b^\sharp(x)(y)=b(x,y), \qquad {}^\sharp b(y)(x)=b(x,y). \] Their kernels are the left radical and the right radical \[ \operatorname{rad}_{\mathrm L}(M)=\{x\in M\mid b(x,M)=0\}, \qquad \operatorname{rad}_{\mathrm R}(M)=\{y\in M\mid b(M,y)=0\}. \] The form is left nondegenerate if \(b^\sharp\) is injective, right nondegenerate if \({}^\sharp b\) is injective, and nondegenerate if both hold. It is perfect if \(b^\sharp\) and \({}^\sharp b\) are isomorphisms. Perfect forms are nondegenerate; the converse requires additional hypotheses.

If \(b\) is symmetric then \({}^\sharp b=b^\sharp\), and if \(b\) is skew-symmetric then \({}^\sharp b=-b^\sharp\). In both cases the two radicals coincide, are written \(\operatorname{rad}(M)\), and the one condition \(\operatorname{rad}(M)=0\) is nondegeneracy.

In (Milnor and Husemöller 1973) a module equipped with a perfect form is called an inner product space.

Example 17.5 (Left and right radicals can differ) Let \(b\) be the form on \(R^{2}\) with Gram matrix \(\left(\begin{smallmatrix}1&2\\3&4\end{smallmatrix}\right)\) in the basis \(e_1,e_2\). Then \(b(e_1,w)=w_1+2w_2\) while \(b(w,e_1)=w_1+3w_2\), so the two conditions \(b(e_1,w)=0\) and \(b(w,e_1)=0\) cut out different submodules of \(R^{2}\).

Definition 17.6 (Orthogonal sum) The orthogonal sum of \((M,b_M)\) and \((N,b_N)\) is \[ (M,b_M)\perp(N,b_N) = (M\oplus N,b_M\oplus b_N), \] where the mixed terms vanish. With zero module as unit and the standard associativity, symmetry, and unit isomorphisms, this defines a symmetric monoidal structure on \(\mathcal B_{R,W}\). The quadratic form category has the analogous orthogonal sum.

17.2 Orthogonality

Definition 17.7 (Orthogonal complements and isotropy) For a submodule \(N\subseteq M\) set \[ N^{\perp_{\mathrm L}}=\{x\in M\mid b(x,N)=0\}, \qquad N^{\perp_{\mathrm R}}=\{x\in M\mid b(N,x)=0\}. \] When \(b\) is symmetric or skew-symmetric these agree and are written \(N^{\perp}\).

An element \(x\in M\) is isotropic if \(b(x,x)=0\), and \(b\) is anisotropic if \(0\) is its only isotropic element. A submodule \(N\) is totally isotropic if \(b|_{N\times N}=0\), equivalently \(N\subseteq N^{\perp}\). The radical of the restricted form is \[ \operatorname{rad}(N)=N\cap N^{\perp}, \] so \(N\) is totally isotropic exactly when \(\operatorname{rad}(N)=N\) and the restriction \(b|_{N\times N}\) is nondegenerate exactly when \(N\cap N^{\perp}=0\).

Theorem 17.8 (Orthogonal decomposition) Let \(b\) be symmetric or skew-symmetric on \(M\) and let \(N\subseteq M\) be a submodule on which \(b\) restricts to a perfect form. Then \(M=N\oplus N^{\perp}\) and the sum is orthogonal, so \[ (M,b)\cong(N,b|_N)\perp(N^{\perp},b|_{N^{\perp}}) \] in the sense of Definition 17.6; see (Milnor and Husemöller 1973, I §3.1). If \(x_1,\dots,x_k\in M\) have invertible Gram matrix \(\bigl(b(x_i,x_j)\bigr)\), then they are linearly independent and this applies to the free submodule they span (Milnor and Husemöller 1973, I §3.2).

Proposition 17.9 (Forms on quotients) Let \(b\) be symmetric on \(M\) and let \(N\subseteq M\) be a submodule.

The rule \(\bar b(x+N,y+N)=b(x,y)\) defines a form on \(M/N\) if and only if \(N\subseteq\operatorname{rad}(M)\). For well-definedness one needs \(b(x+n,y+n')=b(x,y)\) for all \(n,n'\in N\); taking \(n'=0\) gives \(b(n,y)=0\) for every \(y\in M\), which is \(N\subseteq\operatorname{rad}(M)\), and that condition conversely kills all three correction terms.

The same rule defines a form on \(N^{\perp}/N\) whenever \(N\) is totally isotropic: for \(x,y\in N^{\perp}\) and \(n,n'\in N\), \[ b(x+n,y+n')=b(x,y)+b(x,n')+b(n,y)+b(n,n')=b(x,y). \]

Taking \(N=\operatorname{rad}(M)\) gives the radical quotient \(M/\operatorname{rad}(M)\), whose induced form is nondegenerate.

17.3 Gram matrices and the determinant

Proposition 17.10 (Gram matrix) Let \(M\) be free as an \(R\)-module on \(E=\{e_i\}_{i\in I}\), and let \(b\) be a bilinear form on \(M\) with values in \(R\). The Gram matrix of \(b\) with respect to \(E\) is the family \(G_{ij}=b(e_i,e_j)\) indexed by \(I\times I\). If \(v=\sum_i a_i e_i\) and \(w=\sum_j c_j e_j\), then \[ b(v,w)=\sum_{i,j\in I} a_i\, G_{ij}\, c_j, \] a finite sum by finite support of the coordinates. Every family \((G_{ij})_{i,j\in I}\) in \(R\) arises uniquely in this way.

Proposition 17.11 (Congruence) Let \(M\) be free with basis \(e_1,\dots,e_n\) and Gram matrix \(G=\bigl(b(e_i,e_j)\bigr)\) as in Proposition 17.10. Let \(e'_1,\dots,e'_n\) be a second basis and let \(P\) be the invertible matrix whose \(j\)-th column holds the coordinates of \(e'_j\) in the basis \(e\). Then the Gram matrix in the basis \(e'\) is \[ G'=P^{\mathsf T}GP, \] and two Gram matrices present isomorphic forms exactly when they are congruent in this sense (Milnor and Husemöller 1973, I §2.3).

Definition 17.12 (The Cartan matrix) Let \(b\) be a symmetric bilinear form on \(M\) with values in \(R\), and let \(\alpha_1,\dots,\alpha_\ell\) be an ordered family in \(M\) with each \(b(\alpha_j,\alpha_j)\) invertible in \(R\). The Cartan matrix of the family is \[ A_{ij}=\frac{2\,b(\alpha_i,\alpha_j)}{b(\alpha_j,\alpha_j)} , \] normalized by the second index. For the simple roots of a root system these entries are the Cartan integers (Humphreys 1972, sec. 11.1).

Write \(G\) for the Gram matrix of the family, as in Proposition 17.10, and put \(d_j=b(\alpha_j,\alpha_j)/2\). Then \[ A=G\cdot\operatorname{diag}\!\left(\tfrac{2}{G_{jj}}\right), \qquad G_{ij}=d_j\,A_{ij}. \]

Normalizing by the first index instead produces the transpose of \(A\), and the second identity then reads \(G_{ij}=d_iA_{ij}\). The two conventions give transposed matrices and different symmetrization indices; this book uses the one displayed above.

\(G\) is symmetric, while \(A\) need not be, so a Cartan matrix with \(d_i\neq d_j\) is not a Gram matrix. Take \(\ell=2\) with \[ G=\begin{pmatrix}-2&2\\2&-4\end{pmatrix}, \qquad d=(-1,-2), \qquad A=\begin{pmatrix}2&-1\\-2&2\end{pmatrix}. \] Here \(d_2A_{12}=(-2)(-1)=2=G_{12}\), whereas \(d_1A_{12}=(-1)(-1)=1\), so only the second index recovers \(G\) from \(A\).

Definition 17.13 (The determinant) Write \(R^{\bullet}\) for the group of units of \(R\) and \((R^{\bullet})^{2}\) for the subgroup of squares of units. By Proposition 17.11 the determinant of a Gram matrix changes by the square of a unit under change of basis, so a perfect form on a free module has a well-defined \[ \det(b)\in R^{\bullet}/(R^{\bullet})^{2}, \] and a general form on a free module has a well-defined class in the quotient monoid \(R/(R^{\bullet})^{2}\) (Milnor and Husemöller 1973, I §2.5). Under orthogonal sum the rank adds and the determinant multiplies (Milnor and Husemöller 1973, I §3).

Proposition 17.14 (Alternating forms in a basis) Let \(b\) be skew-symmetric on a free module with basis \(e_1,\dots,e_n\). Then \(b\) is alternating if and only if \(b(e_i,e_i)=0\) for every \(i\). For \(x=\sum_ix_ie_i\), \[ b(x,x)=\sum_i x_i^{2}\,b(e_i,e_i)+\sum_{i<j}x_ix_j\bigl(b(e_i,e_j)+b(e_j,e_i)\bigr), \] and skew-symmetry kills the second sum.

Remark 17.15 The symmetric form on \(\mathbb Z^{2}\) with Gram matrix \(\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)\) has vanishing diagonal and \(b(e_1+e_2,e_1+e_2)=2\).

17.4 Tensor product and twisting

Definition 17.16 (Tensor product of forms) Let \(b_M\) have value module \(W_M\) and \(b_N\) have value module \(W_N\). There is exactly one bilinear map \[ b_M\otimes_R b_N\colon (M\otimes_RN)\times(M\otimes_RN)\longrightarrow W_M\otimes_RW_N \] with \[ (b_M\otimes_R b_N)(x\otimes_R u,\;y\otimes_R v)=b_M(x,y)\otimes_R b_N(u,v), \] obtained by factoring the four-linear map \((x,u,y,v)\mapsto b_M(x,y)\otimes_R b_N(u,v)\) through the tensor products (Milnor and Husemöller 1973, I §5.1). For \(W_M=W_N=R\) the value module is \(R\).

Call a form \(\varepsilon\)-symmetric when \(b(x,y)=\varepsilon\,b(y,x)\), so that \(1\)-symmetric means symmetric and \((-1)\)-symmetric means skew-symmetric. The tensor product of an \(\varepsilon\)-symmetric and an \(\varepsilon'\)-symmetric form is \(\varepsilon\varepsilon'\)-symmetric (Milnor and Husemöller 1973, I §5.2). If both forms are perfect and both modules are finitely generated projective, the tensor product is perfect (Milnor and Husemöller 1973, I §5.3).

Proposition 17.17 (Gram matrix and determinant of a tensor product) If \(M\) and \(N\) are free of ranks \(m\) and \(n\) with Gram matrices \(G_M\) and \(G_N\), the Gram matrix of \(b_M\otimes_R b_N\) in the product basis is the Kronecker product \(G_M\otimes_R G_N\), and \[ \det(b_M\otimes_R b_N)=\det(b_M)^{\,n}\det(b_N)^{\,m}. \]

Definition 17.18 (Twisting) For \(\lambda\in R\) the twist \(b(\lambda)\) of \((M,b)\) is the form \(\lambda b\) on the same module, written \(M(\lambda)\). Its Gram matrix in a basis is \(\lambda G\), its adjoint maps are \(\lambda\,b^{\sharp}\) and \(\lambda\,{}^{\sharp}b\), and on a free module of rank \(n\) \[ \det\bigl(b(\lambda)\bigr)=\lambda^{\,n}\det(b). \]

A morphism \(f\) of Definition 17.2 satisfies \(f^{*}b_N=b_M\) and hence \(f^{*}(\lambda b_N)=\lambda b_M\), so \((-)(\lambda)\) is an endofunctor of \(\mathcal B_{R,W}\) which is the identity on underlying maps. For \(\lambda\) a unit it is an isomorphism of categories with inverse \((-)(\lambda^{-1})\), and \((-)(-1)\) is an involution.

17.5 Signature at a real place

Definition 17.19 (Signature) Let \(F\) be an ordered field and let \(b\) be a symmetric bilinear form on a finite-dimensional \(F\)-vector space \(V\). Write \(p\) for the greatest dimension of a subspace on which \(b\) is positive definite, \(q\) for the greatest dimension of a subspace on which \(b\) is negative definite, and \(r=\dim\operatorname{rad}(V)\). The triple \((p,q,r)\) is the signature of \(b\).

For an \(R\)-module with form and a ring embedding \(\sigma\colon R\hookrightarrow\mathbb R\), the signature of \((M,b)\) at \(\sigma\) is the signature of the base change along \(\sigma\) of Definition 11.2. For \(R=\mathbb Z\) there is one such embedding and the qualifier is omitted.

For a nondegenerate form \(r=0\), and the signature is then written as the pair \((p,q)\).

Theorem 17.20 (Sylvester's law of inertia) Let \(F\) be an ordered field and let \(b\) be a perfect symmetric bilinear form on a finite-dimensional \(F\)-vector space \(V\). Then \(V\cong V^{+}\perp V^{-}\) with \(b\) positive definite on \(V^{+}\) and negative definite on \(V^{-}\), and the two dimensions depend only on the isomorphism class of \((V,b)\): \(\dim V^{+}\) is the greatest dimension of a positive definite subspace of \(V\) (Milnor and Husemöller 1973, III §2.5). So the signature of Definition 17.19 is \((\dim V^{+},\dim V^{-},0)\) and \(p+q=\dim V\).

For a degenerate \(b\), choose a complement \(N\) to \(\operatorname{rad}(V)\) in \(V\). Every pairing involving \(\operatorname{rad}(V)\) vanishes, so \(V\cong\operatorname{rad}(V)\perp N\) and \(b|_N\) is nondegenerate, hence perfect because \(N\) is finite dimensional. Applying the above to \(N\) gives \(V\cong V^{+}\perp V^{-}\perp\operatorname{rad}(V)\), so \(p+q+r=\dim V\) in every case.

Definition 17.21 (Definiteness)

Let \(b\) be a symmetric bilinear form of signature \((p,q,r)\) and set \(n=p+q+r\). Then \(b\) is

  • degenerate if \(r>0\);

  • positive definite if \((p,q,r)=(n,0,0)\), and negative definite if \((p,q,r)=(0,n,0)\);

  • definite if it is positive definite or negative definite;

  • indefinite if \(p\geq1\) and \(q\geq1\);

  • hyperbolic, or of Lorentzian signature, if \(n\geq2\) and \((p,q,r)=(1,n-1,0)\);

  • parabolic if \(n\geq2\) and \((p,q,r)=(0,n-1,1)\).

Convention 17.22 Root lattices are taken negative definite here, so the forms on \(A_n\), \(D_n\) and \(E_n\) are negative definite and the hyperbolic signature is \((1,n-1,0)\).

17.6 Isotropy and Witt decomposition

Definition 17.23 (Rank one forms, the hyperbolic plane, and split forms) For a unit \(u\in R^{\bullet}\) write \(\langle u\rangle\) for the free module of rank \(1\) on a generator \(e\) with \(b(e,e)=u\). Then \(\langle u\rangle\cong\langle u'\rangle\) if and only if \(u'=\alpha^{2}u\) for some \(\alpha\in R^{\bullet}\) (Milnor and Husemöller 1973, I §2.4).

The hyperbolic plane \(U\) over \(R\) is the free module on \(e,f\) with \[ b(e,e)=b(f,f)=0,\qquad b(e,f)=b(f,e)=1, \] so its Gram matrix is \(\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)\), its form is perfect and even, and its signature over an ordered field is \((1,1,0)\).

A perfect symmetric form on \(M\) is split if \(M\) has a direct summand \(N\) with \(N=N^{\perp}\) (Milnor and Husemöller 1973, I §6.1). Over a ring whose finitely generated projective modules are free and in which \(2\) is a unit, a split form is an orthogonal sum of hyperbolic planes (Milnor and Husemöller 1973, I §6.3); this applies to a field of characteristic other than \(2\).

Definition 17.24 (Witt index) The Witt index \(i(V)\) of a perfect symmetric form on a finite-dimensional vector space \(V\) over a field is the greatest dimension of a totally isotropic subspace. It satisfies \[ 0\leq i(V)\leq\tfrac12\dim V, \] with \(i(V)=0\) exactly when the form is anisotropic and \(i(V)=\tfrac12\dim V\) exactly when the form is split (Milnor and Husemöller 1973, III §1.2).

Theorem 17.25 (Witt decomposition) Every perfect symmetric form on a finite-dimensional vector space \(V\) over a field \(F\) decomposes as \[ V\cong S\perp A \] with \(S\) split and \(A\) anisotropic (Milnor and Husemöller 1973, III §1.1), and \(A\) is determined up to isomorphism by the Witt class of \(V\) (Milnor and Husemöller 1973, III §1.7). Here \(\dim S=2\,i(V)\) and \(\dim A=\dim V-2\,i(V)\).

Example 17.26 (Witt index and signature) Over a real closed field every positive element is a square, so by Theorem 17.20 a perfect symmetric form of signature \((p,q,0)\) is \(p\langle1\rangle\perp q\langle-1\rangle\). Each summand \(\langle1\rangle\perp\langle-1\rangle\) is split (Milnor and Husemöller 1973, I §6.1), so the Witt index is \(\min(p,q)\).

Over \(\mathbb Q\) the Witt index can be strictly smaller. The form \(x^{2}+y^{2}-3z^{2}\) has signature \((2,1,0)\), so \(\min(p,q)=1\), and it is anisotropic over \(\mathbb Q\), so its Witt index is \(0\). For a primitive integral solution of \(x^{2}+y^{2}=3z^{2}\), reduction modulo \(3\) forces \(3\mid x\) and \(3\mid y\) because \(-1\) is not a square modulo \(3\); then \(9\mid3z^{2}\) gives \(3\mid z\), contradicting primitivity.

Remark 17.27 (Signature, signature difference, and Witt index) Three integers are attached to a nondegenerate symmetric form over an ordered field, and they are distinct invariants. The pair \((p,q)\) is the signature of Definition 17.19. The integer \(p-q\) is called the signature of the form in (Milnor and Husemöller 1973, III §2.5), where it is the value of a ring homomorphism from the Witt ring to \(\mathbb Z\). The Witt index of Definition 17.24 is a third quantity, equal to \(\min(p,q)\) over a real closed field and smaller over \(\mathbb Q\) in the case above.

17.7 Diagonal and polarization

Definition 17.28 (Diagonal and polarization functors) Diagonal and polarization define natural transformations \[ \operatorname{diag}\colon\operatorname{SymBil}_{R,W} \Longrightarrow\operatorname{Quad}_{R,W}, \qquad \operatorname{polar}\colon\operatorname{Quad}_{R,W} \Longrightarrow\operatorname{SymBil}_{R,W}, \] with \[ \operatorname{diag}(b)(x)=b(x,x), \qquad \operatorname{polar}(q)(x,y)=q(x+y)-q(x)-q(y). \] Here \(\operatorname{SymBil}_{R,W}\) is the \(R\)-submodule-valued presheaf of symmetric bilinear forms. The composites satisfy \[ \operatorname{polar}(\operatorname{diag}(b))=2b, \qquad \operatorname{diag}(\operatorname{polar}(q))=2q. \] If \(2\) is invertible on \(W\), division by \(2\) gives inverse equivalences between symmetric bilinear forms and quadratic forms. For a general value module, they define parallel functors whose composites are multiplication by \(2\).

For discriminant forms, the isomorphism \[ \mathbb Q/2\mathbb Z\longrightarrow\mathbb Q/\mathbb Z, \qquad w\longmapsto w/2, \] followed by polarization gives the bilinearization functor from \(\mathbb Q/2\mathbb Z\)-valued quadratic forms to \(\mathbb Q/\mathbb Z\)-valued symmetric bilinear forms.

Bilinear forms on the underlying module of an associative unital \(R\)-algebra are treated in Chapter 18: module bilinear forms are \(\operatorname{Hom}_{R\text{-}\mathbf{Mod}}(M\otimes_R M,R)\), and algebra bilinear forms are \(\operatorname{Hom}_{R\text{-}\mathbf{Alg}}(A\otimes_R A,R)\) with the tensor product taken in \(R\text{-}\mathbf{Alg}\).