Example. An arithmetic family for the alternating four-cycle [fdip-000U]

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The configuration form has polynomial coefficients when the mass is fixed by \(m^2=1\). Keeping the weight \(t\) as a variable records which specializations preserve the form, its half-turn, and the operations on it. Put \(A_0=\mathbb Z[1/2,t]\) and \(s=t+2\). In the coordinate basis \(\delta _0,\ldots ,\delta _3\), set \[Q(t)=\begin {pmatrix} s&-1&0&-t\\ -1&s&-t&0\\ 0&-t&s&-1\\ -t&0&-1&s \end {pmatrix},\] \[\begin {aligned} q(x)&=x^TQ(t)x,\\ b(x,y)&=x^TQ(t)y. \end {aligned}\] Thus \(q\) is twice the earlier quadratic action. Its half-polar form is \(b\); its full polar form has matrix \(2Q\): \[q(x+y)-q(x)-q(y)=2b(x,y).\] These conventions also apply after changing the coefficient ring.

Let \(H\) have the 4 unnormalized columns \[H=\begin {pmatrix} 1&1&1&1\\ 1&-1&1&-1\\ 1&-1&-1&1\\ 1&1&-1&-1 \end {pmatrix}.\] They are neither the coordinate vectors \(\delta _i\) nor the normalized real spectral vectors \(u_a\) of the cycle spectral calculation. Direct multiplication gives \[\begin {aligned} H^TH&=4I,\\ QH&=H\operatorname {diag}(1,3,2t+1,2t+3). \end {aligned}\] Since \(2\) is a unit in \(A_0\), \(H^{-1}=H^T/4\). Similarity therefore gives \[\Delta :=\det Q=3(2t+1)(2t+3).\] The congruence formula retains a factor 4: \[H^TQH=4\operatorname {diag}(1,3,2t+1,2t+3).\] These are polynomial matrix identities, independent of any order or positivity on the coefficient ring.

The nondegenerate arithmetic base is \[R_q=A_0[\Delta ^{-1}],\qquad X_q=\operatorname {Spec}R_q,\] with quadratic module \((R_q^4,q)\). Here \(A[g^{-1}]\) denotes localization of a commutative ring \(A\) by the powers of \(g\). The spectrum \(\operatorname {Spec}A\) has prime ideals as points; its standard open \[D_{\operatorname {Spec}A}(g) =\{\mathfrak p:g\notin \mathfrak p\}\] has regular functions \(A[g^{-1}]\). The structure sheaf given by these rings and their restriction maps makes the spectrum an affine scheme, in the sense of [stacks2017stacks, Section 26.5], Tag 01HR.

Every factor of the unit product \(\Delta \) is a unit in \(R_q\): an inverse for a factor is the product of the other factors times \(\Delta ^{-1}\). The displayed similarity then makes \(Q\) invertible. Both \(Q\) and \(2Q\) identify \(R_q^4\) with its dual. This localization selects nondegenerate forms; the Clifford quotient exists over \(A_0\) even before it is made. Inverting \(2\) separately excludes characteristic two, although \(\Delta \equiv 1\pmod 2\). Inverting \(\Delta \) removes characteristic three entirely. In any residue field of characteristic other than two or three, the excluded parameters are \(-1/2\) and \(-3/2\); they are distinct because their difference is one.

For example, evaluation at \(t=1\) gives \(\det Q=45\). It is admissible over \(\mathbb Q\) and over \(\mathbb F_{11}\), but not over \(\mathbb F_5\). An arithmetic family therefore retains information that a single real matrix does not.

The same half-turn \[P(x_0,x_1,x_2,x_3)=(x_2,x_3,x_0,x_1)\] satisfies \(P^2=I\) and \(P^TQP=Q\): shifting both indices by two preserves each entry of \(Q\). By the universal property, the assignment \[\iota _q(x)\longmapsto \iota _q(Px)\] extends to an algebra endomorphism of \(\operatorname {Cl}_{R_q}(q)\). Its square fixes scalars and all vector generators, hence is the identity. It defines a \(C_2\) action by automorphisms. The coordinate basis, reference pairing \(x^Ty\), and retained partition remain additional supplied data.

Retain coordinates \((0,2)\) and eliminate \((1,3)\) as in the four-cycle Schur calculation. To distinguish its denominator from that of the trace ratio on its unit domain, put \(f=t+1\) and define \[\begin {aligned} R_{\mathrm {elim}}&=R_q[s^{-1}],\\ R_{\mathrm {obs}}&=R_{\mathrm {elim}}[f^{-1}],\\ X_{\mathrm {elim}}&=\operatorname {Spec}R_{\mathrm {elim}},\\ X_{\mathrm {obs}}&=\operatorname {Spec}R_{\mathrm {obs}},\\ U_R&=D_{X_q}(f). \end {aligned}\] The open immersions identify \[\begin {aligned} X_{\mathrm {elim}}&=D_{X_q}(s),\\ X_{\mathrm {obs}}&=D_{X_{\mathrm {elim}}}(f) =D_{X_q}(sf),\\ X_{\mathrm {obs}}&=X_{\mathrm {elim}}\cap U_R. \end {aligned}\] The independent ratio locus \(U_R\) is not identified with \(X_{\mathrm {obs}}\); at \(t=-2\), the ratio exists while the selected elimination fails.

All 4 arrows are inclusions of open subschemes. The square commutes and its upper-left open is the intersection of the other two opens inside \(X_q\). The localization homeomorphism is [stacks2017stacks, Lemma 10.17.6], Tag 00E4. The structure-sheaf construction in [stacks2017stacks, Section 26.5], Tag 01HR upgrades it to an open immersion: on a standard open inside \(D(g)\), successive localization gives the same ring and restriction map on either side. These identifications agree on intersections, so the structure sheaves agree. Mathlib's AlgebraicGeometry.Scheme.isOpenImmersion_SpecMap_localizationAway and AlgebraicGeometry.Scheme.Hom.opensRange_localizationAway give this generic map and its exact ambient image. The determinant and the particular family above are the direct calculations made here.