§4 · Agent 4 · Audit: counterexample and mechanism · 2026-10-17T22:58:17Z

Audit 4.13 · Audit: Weil Grothendieck groups over \(R\) and the multiplicity matrix \(M\)

proved

The setup of Setup 1.1 is re-derived. Simple Weil perverse sheaves are the \(\mathrm{IC}_\nu(\alpha)\), so \(K_0\) is free over \(R\) on the \([\mathrm{IC}_\nu]\). Convolution with \(\mathrm{IC}_0\) induces an \(R\)-linear map with matrix \(M\), and \(\mathrm{tr}_n\) is base change along \(\mathrm{ev}_n\), giving \(m^{(n)}=\mathrm{ev}_n(M)\).

Audit 4.13.

Under (R), (C), D0–D3: (a) the simple objects of \(\mathcal C^W\) (Weil, \(L^+G\)-equivariant perverse sheaves of finite support) are the \(\mathrm{IC}_\nu(\alpha)=(\mathrm{IC}_\nu,\alpha\cdot\mathrm{can})\), every object has finite length, and \(K_0(\mathcal C^W)=\bigoplus_\nu R[\mathrm{IC}_\nu]\); likewise \(K_0(\mathcal H^W)=\bigoplus_\lambda R[V_\lambda]\); (b) \([F]\mapsto\sum_i(-1)^i[{}^p\mathcal H^i(F\ast\mathrm{IC}_0)]\) is \(R\)-linear with \(\mathrm{act}[V_\lambda]=\sum_\nu M_{\nu\lambda}[\mathrm{IC}_\nu]\), \(M_{\nu\lambda}=\sum_i(-1)^i[E_{\nu,i}(\lambda),A_{\nu,i}(\lambda)]\); (c) \(\mathrm{tr}_n\) factors through \(-\otimes_{R,\mathrm{ev}_n}\overline{\mathbb Q}_\ell\), giving isomorphisms onto \(C_{K_n}\otimes\overline{\mathbb Q}_\ell\) and \(\mathcal H_{K_n}\otimes\overline{\mathbb Q}_\ell\) that intertwine \(\mathrm{act}\) with \(-\ast\varphi_0\); hence \(m^{(n)}=\mathrm{ev}_n(M)\).

Outcome: holds.

  1. Simples. For a simple Weil object \((P,\Phi)\), \(\mathrm{Fr}^*\) is an autoequivalence of perverse sheaves over \(k\), so \(\Phi\) carries \(\mathrm{soc}(\mathrm{Fr}^*P)=\mathrm{Fr}^*\mathrm{soc}(P)\) onto \(\mathrm{soc}(P)\). Thus the socle is a nonzero Weil subobject and equals \(P\), and \(P\) is semisimple over \(k\). Under (R) each \(\mathcal O_\mu\) has a rational point, so \(\mathrm{Fr}^*\mathrm{IC}_\nu\cong\mathrm{IC}_\nu\) canonically, and under (C) the simple equivariant perverse sheaves are the \(\mathrm{IC}_\nu\). Schur's lemma gives \(\Phi=\bigoplus\mathrm{can}_\nu\otimes A_\nu\), and simplicity forces one \(\nu\) and a simple \(\overline{\mathbb Q}_\ell[A^{\pm1}]\)-module, which is one-dimensional because \(\overline{\mathbb Q}_\ell\) is algebraically closed.
  2. \(K_0\). Weil subobjects are subobjects over \(k\), so lengths are finite; Jordan–Hölder gives the \(\mathbb Z\)-basis \([\mathrm{IC}_\nu(\alpha)]\), i.e. the \(R\)-basis \([\mathrm{IC}_\nu]\), and an \(A\)-stable flag gives \([\mathrm{IC}_\nu\otimes(E,A)]=[E,A][\mathrm{IC}_\nu]\). Schubert cells of split \(G\) have connected stabilizers, so the same holds on \(\operatorname{Gr}\).
  3. \(\mathrm{act}\). Convolution with \(\mathrm{IC}_0\) is exact and commutes with \(\mathrm{Fr}^*\) (proper base change), so a short exact sequence of Weil objects gives a Frobenius-compatible long exact sequence of perverse cohomology, whose alternating sum vanishes in \(K_0\). \(R\)-linearity is the projection formula for \(-\otimes L_\alpha\).
  4. Traces. Laumon (1.1.1.1)–(1.1.1.2) (checked in the local copy) give additivity and \(\mathrm{tr}_n(\mathrm{IC}_\nu(\alpha))=\alpha^n\mathrm{tr}_n\mathrm{IC}_\nu\). The bases go to the bases of IIb(i) and IIa step 5, and the square commutes on the basis \([V_\lambda]\) by IIa.
  5. The remark on Chen–Yi. Their §8.2.1 indeed uses \(\mathbb Z[A]\), \(A=\overline{\mathbb Q}_\ell^\times/\mu\), with Weil structures given for sufficiently divisible \(n\) (checked in chen-yi-v4.pdf). The finer ring \(R\) is what keeps \(n=1\) visible.

Builds on

  • Setup 1.1 · Setup: Weil Grothendieck groups over \(R=\mathbb Z[\overline{\mathbb Q}_\ell^\times]\)

Built on by

  • Summary 4.20 · Verdict of the second audit: the mechanism claims hold, with minor repairs

Sources