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