§1 · Agent 1 · Trace formalism · 2026-10-17T21:10:33Z

Audit 1.9 · Check: Grothendieck groups modulo roots of unity cannot decide \((\mathrm{SG})_1\)

proved

Chen–Yi define their affine Hecke modules with Weil structures for sufficiently divisible \(n\) and eigenvalues in \(\overline{\mathbb Q}_\ell^\times/\mu\). The specialization \(\mathrm{ev}_1\) does not factor through \(\mathbb Z[\overline{\mathbb Q}_\ell^\times/\mu]\), so statements in their modules control \((\mathrm{SG})_n\) only for \(n\) divisible enough; the \(\pm\) cancellations that decide \((\mathrm{SG})_1\) in Counterexample 1.5 are invisible there.

Audit 1.9.

Let \(\pi:R\to\mathbb Z[\mathsf A]\), \(\mathsf A=\overline{\mathbb Q}_\ell^\times/\mu\) (\(\mu\) the roots of unity), be the quotient used by Chen–Yi, §8.2. For \(r\in R\) supported on eigenvalues \(\zeta q^{k/2}\) with \(\zeta^N=1\) (the eigenvalues allowed in Chen–Yi's categories), \(\mathrm{ev}_n(r)\) is determined by \(\pi(r)\) when \(N\mid n\), but not in general: \(\pi([\gamma]+[-\gamma])=\pi(2[\gamma])\) while \(\mathrm{ev}_1([\gamma]+[-\gamma])=0\neq2\gamma=\mathrm{ev}_1(2[\gamma])\). Hence no identity in Chen–Yi's modules \(M,M'\) (or their spherical analogues) decides \((\mathrm{SG})_1\). Under (R), (C), if all eigenvalues of the \(A_{\nu,i}(\lambda)\) lie in \(q^{\mathbb Z/2}\), then \(m^{(n)}=M(q^{n/2})\) for an integer Laurent matrix \(M(v)\), and the coarse module loses nothing.

  1. What Chen–Yi define. §8.2.1: \(\mathrm{Shv}(I_0\backslash LX)^{\mathrm{Weil}}=\operatorname{colim}_n\mathrm{Shv}(I_0\backslash LX)^{F^n}\), objects with \(\Phi_n:(F^n)^*\mathcal F\simeq\mathcal F\) for sufficiently divisible \(n\), Frobenius eigenvalues defined as \(n\)-th roots of eigenvalues of \(\Phi_n\), taken in \(\mathsf A=\overline{\mathbb Q}_\ell^\times/\mu\); \(M=K(C_{LX})\) is free over \(\mathbb Z[q^{\pm1/2}]\subset\mathbb Z[\mathsf A]\). §8.2.3: the IC classes are triangular with respect to the standard classes in this Grothendieck group.
  1. Factorization. For \(\alpha=\zeta q^{k/2}\) with \(\zeta^N=1\) and \(N\mid n\), \(\alpha^n=q^{nk/2}\) depends only on \(\pi(\alpha)=q^{k/2}\bmod\mu\); so on such \(r\), \(\mathrm{ev}_n=\mathrm{ev}_n\circ\pi\) is well defined. For \(n=1\) (or \(N\nmid n\)) it is not: the displayed pair has the same image under \(\pi\) and different values under \(\mathrm{ev}_1\).
  1. In the torus example (Counterexample 1.5 step 6) the cokernel at odd labels is \(R/I\), \(I=\langle[\gamma]+[-\gamma]\rangle\); after \(\pi\) it becomes \(\mathbb Z[\mathsf A]/2\), i.e. invisible after inverting 2, although \((\mathrm{SG})_n\) fails for all odd \(n\). (That example is outside (R), but the same loss occurs for any multiplicity space whose Frobenius has eigenvalues \(\pm\alpha\).)
  1. Under (R), (C). The canonical Weil structures of \(\mathrm{IC}_\nu\) and \(V_\lambda\) have eigenvalue 1 on the lowest stalk of the open stratum, so \(M\) is determined by the \((E_{\nu,i}(\lambda),A_{\nu,i}(\lambda))\). If all their eigenvalues lie in \(q^{\mathbb Z/2}\) (no root-of-unity factor), then \(M\in\mathrm{Mat}(\mathbb Z[q^{\pm1/2}])\) and \(m^{(n)}=M|_{q^{1/2}\mapsto(q^{1/2})^n}\): \((\mathrm{SG})_n\) is \((\mathrm{SG})\) for the residue field \(\mathbb F_{q^n}\), and Chen–Yi's coarse module loses nothing. Root-of-unity factors arise, for instance, when Frobenius permutes irreducible components of a convolution fibre \(m^{-1}(x_\mu)\); a convolution fibre whose top-dimensional components are permuted without fixed points contributes \(0\) to \(m^{(1)}\) and a positive amount to \(m^{(n)}\) for suitable \(n\). This is the mechanism left open in Proposition 1.8(g). \(\square\)

Builds on

  • Setup 1.1 · Setup: Weil Grothendieck groups over \(R=\mathbb Z[\overline{\mathbb Q}_\ell^\times]\)
  • Counterexample 1.5 · Counterexample outside (R): the norm torus quotient, and the torus criterion

Built on by

  • Dead end 1.11 · Dead end: forcing \((\mathrm{SG})_1\) to fail inside (R)+(C) by cancellation or gluing

Sources