Full-rank Gram torsor descent and extension across lower-rank strata

Put $Y=\mathrm{Spec}R$, where $R$ is the rank ring in [F1], and let $Y^{\circ }$ denote its rank-$m$ locus. Write $H=Z_{I,I}$ for a principal $m\times m$ block, and work over $\det H\mathrm{\ne }0$ (over a Pfaffian open in the alternating case). These opens cover $Y^{\circ }$: a rank-$m$ bilinear form on $k^t$ induces a nondegenerate form on its quotient by the radical, and some $m$ coordinate vectors form a basis of that quotient. In the block decomposition $Z=\left(\begin{array}{cc}H& C\\ ϵC^{\mathsf{T}}& D\end{array}\right)$, with $ϵ=1$ or $-1$, Gaussian elimination gives $D=ϵC^{\mathsf{T}}{H}^{-1}C$. This identity holds in the localized ring, since it holds at every closed point and [F1] says that ring is reduced. Thus this chart is the product of the space of invertible symmetric or alternating $H$ and the affine space of $C$. A tuple over this chart has an invertible column matrix $T$ on the labels $I$, satisfying $T^{\mathsf{T}}BT=H$; its remaining columns are $T{H}^{-1}C$. These formulas are identities over any chart algebra, not only at closed points.

We verify the normal-extension interface precisely. The coordinate rank ring and its principal-open localizations are Noetherian by [F3], and normal domains by [F1]. If $A$ is such a normal domain and $f\in \mathrm{Frac}A$ is regular at every height-one prime, then [F4] gives $f\in A$. Consequently a regular function on an open subset with complement of codimension at least two extends: the open contains every height-one point, and its function defines a rational function at the generic point; apply [F4]. Its pullback agrees with the original function on the open because both are the same rational function in the domain. Uniqueness follows from that domain property as well. All hypotheses of the published supplier are satisfied, including AC.

Here is a cover on which such a $T$ exists regularly. A nondegenerate symmetric $H$ has a vector of nonzero norm among $e_i$ and $e_i+e_j$: if all these norms vanish, polarization makes every entry zero. Cover by the corresponding norm-unit opens, split off that vector by subtracting its $H$-orthogonal projections, and repeat on the complement. This diagonalizes $H$ over a finite Zariski cover as $Q^{\mathsf{T}}HQ=\mathrm{diag}(d_1,\dots ,d_m)$ with each $d_i$ a unit. Adjoin $u_i$ with $u_i^2=d_i$. Each extension is finite free of rank two and étale because $2u_i$ is a unit; their composite is faithfully flat. Fixing a constant isometry of $B$ with the identity form, the matrix $\mathrm{diag}(u_i)Q^{-1}$, transported by that isometry, is the required $T$. In the alternating case choose a unit entry $h_{ij}$, rescale one of the corresponding vectors so their pairing is one, subtract projections onto this symplectic plane, and induct. This gives a regular symplectic basis on a Zariski cover, and hence a $T$ without adjoining square roots. There are finitely many pivot choices, and their opens cover because every residue form is nondegenerate.

On any of these covers let $S$ be the section given by $T$ and $T{H}^{-1}C$. Every spanning tuple $X$ with Gram matrix $Z$ has unique $g=X_I{T}^{-1}$ and $g^{\mathsf{T}}Bg=B$. Conversely $gS$ is a tuple with Gram matrix $Z$. These formulas and their inverses are regular over every base algebra, so $X^{\circ }{\times }_{Y^{\circ }}U\simeq G\times U$ over each chosen étale cover $U$. They also identify $G\times X^{\circ }$ with $X^{\circ }{\times }_{Y^{\circ }}{X}^{\circ }$ via $(g,X)\mapsto (gX,X)$; the inverse uses the same invertible block formula. This establishes the scheme-theoretic torsor assertion. To apply flat descent, fix an affine principal open $V=\mathrm{Spec}A\subseteq Y^{\circ }$. Its preimage $X_V$ is affine. The finitely many pivot opens from step 2.1 restrict to principal opens $D(f_j)$ covering $V$. Let $C_j$ be the chart algebra over $A_{f_j}$: it is finite free of positive rank in the symmetric case (the successive square-root extension) and $A_{f_j}$ in the alternating case. Put $C={\prod }_j{C}_j$. Then $A\to C$ is faithfully flat: if an $A$-module $M$ satisfies $M{\otimes }_{A}C=0$, the faithful flatness of each chart algebra gives $M_{f_j}=0$; since the $f_j$ generate the unit ideal, this forces $M=0$. Over $C$, the torsor map is $G\times \mathrm{Spec}C\to \mathrm{Spec}C$, whose coordinate algebra $C{\otimes }_{k}k[G]$ is faithfully flat over $C$ because $k[G]$ is a nonzero $k$-vector space. The base change of $X_V\to V$ is therefore flat, so [F9] gives flatness before base change. If $M{\otimes }_A\mathcal{O}(X_V)=0$, faithful flatness after base change gives $M{\otimes }_{A}C=0$, hence $M=0$; thus $X_V\to V$ is faithfully flat. These affine opens cover $Y^{\circ }$, proving that $X^{\circ }\to Y^{\circ }$ is faithfully flat. The étale cover and its base changes are legitimate by [F8].

We spell out function descent. For a faithfully flat ring map $A\to C$, the sequence $0\to A\to C\rightrightarrows C{\otimes }_{A}C$ is an equalizer. Indeed tensor its difference complex with $C$; multiplication of the initial two $C$ factors gives a contracting map, so the tensored complex is exact. Flatness preserves kernels and faithful flatness detects a zero homology module, proving the original equalizer assertion. On a trivial torsor $G\times U$, a left-translation-invariant function equals its value along the identity section: apply the counit on the translation factor to the coaction identity. Hence an invariant regular function on $X^{\circ }$ restricts on each $U$ to the pullback of a unique regular function on $U$. On overlaps the two functions pull back to the same function on the torsor, so faithful flatness and the equalizer just proved identify them and descend them to $Y^{\circ }$. Equivalently, the torsor isomorphism in step 3.1 identifies its invariance equation with equality of its two pullbacks to $X^{\circ }{\times }_{Y^{\circ }}{X}^{\circ }$. Applying the affine equalizer on the stated covers and gluing proves both existence and uniqueness of descent.

The dimension formulas in [F2] give, for symmetric matrices, $d_m-d_{m-1}=t-m+1$ and, for alternating matrices, $d_m-d_{m-2}=2(t-m)+1$. All lower strata have smaller dimensions, so these are the codimensions of the respective omitted closed loci. Since $t\ge m+1$, the codimensions are at least two and three, respectively. Apply step 1.2 to the normal domain in [F1] and the descended function from step 4.1. This proves the asserted unique extension.