§4 · Agent 4 · Audit: counterexample and mechanism · 2026-10-17T22:58:41Z

Audit 4.14 · Audit: Theorem A, rows versus columns of the \(R\)-matrix \(M\)

proved

All parts of Theorem 1.2 are re-derived. The weight argument for (i)(⇒) is correct: eigenvalues from different perverse degrees have different weights, so they cannot cancel in \(R\). Thomason's classification applies to \(\operatorname{Tri}\{V_\lambda\ast\mathrm{IC}_0\}\) because its summand closure is thick. The Laumon citation is accurate.

Audit 4.14.

Under (R), (C), D0–D3, with \(Q=\operatorname{coker}(M)\): (i) \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\iff\) no row of \(M\) vanishes in \(R\); (ii) \((\mathrm{SG})_n\iff Q\otimes_{R,\mathrm{ev}_n}\overline{\mathbb Q}_\ell=0\); (iii) if \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\), then \(\operatorname{Tri}\{V_\lambda\ast\mathrm{IC}_0\}=\mathcal C_k\iff Q\otimes_{R,\varepsilon}\mathbb Z=0\); (iv) \(Q=0\Rightarrow(\mathrm{SG})_n\) for all \(n\).

Outcome: holds. Thomason, Compositio 105 (1997), Thm 2.1 is not in the local sources but is a published, checkable statement, used correctly.

  1. (i), the weight argument. \(V_\lambda\ast\mathrm{IC}_0\) is pure of some weight \(w\) (D2; \(m\) is proper on \(\operatorname{Gr}_{\le\lambda}\mathbin{\tilde\times}L^+X\)). So \({}^p\mathcal H^i\) is pure of weight \(w+i\): BBD Cor. 5.4.4, per the local reviewed excerpt bbd-split.txt ("the purity criterion for perverse cohomology"). \(\mathrm{IC}_\mu\) with its canonical structure is pure of weight \(w_\mu\). Hence every eigenvalue of \(A_{\mu,i}(\lambda)\) is pure of weight \(w+i-w_\mu\) for every embedding \(\overline{\mathbb Q}_\ell\hookrightarrow\mathbb C\). Two eigenvalues from different \(i\) are therefore different elements of \(\overline{\mathbb Q}_\ell^\times\), while within one \(i\) the coefficients of \([E,A]=\sum d_\alpha[\alpha]\) are positive. So \(M_{\mu\lambda}=\sum_i(-1)^i[E_{\mu,i},A_{\mu,i}]\) has no cancellation, and \(M_{\mu\lambda}\ne0\iff\exists i:E_{\mu,i}(\lambda)\ne0\). With Proposition I this is (i) in both directions.
  2. (ii), (iv). Right exactness of \(-\otimes_{R,\mathrm{ev}_n}\overline{\mathbb Q}_\ell\) gives \(\operatorname{coker}\mathrm{ev}_n(M)=Q\otimes_{\mathrm{ev}_n}\overline{\mathbb Q}_\ell\); then IIb(iii) and Audit 4.13(c).
  3. (iii), \(K_0\). The perverse t-structure on \(\mathcal C_k\) is bounded with finite-length heart whose simples are the \(\mathrm{IC}_\nu\) (by (C)), so \(K_0(\mathcal C_k)=\bigoplus_\nu\mathbb Z[\mathrm{IC}_\nu]\) and \([V_\lambda\ast\mathrm{IC}_0]=\sum_\nu\varepsilon(M_{\nu\lambda})[\mathrm{IC}_\nu]\).
  4. (iii), density. Let \(\mathcal D=\operatorname{Tri}\{V_\lambda\ast\mathrm{IC}_0\}\) and \(\mathcal D'\) the class of direct summands of objects of \(\mathcal D\). \(\mathcal D'\) is closed under shifts and summands, and under cones: if \(x\oplus x',y\oplus y'\in\mathcal D\), the cone of \(f\oplus0\) is \(\operatorname{cone}(f)\oplus x'[1]\oplus y'\in\mathcal D\). So \(\mathcal D'\) is thick and contains the generators, hence \(\mathcal D'=\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\): \(\mathcal D\) is dense. Thomason's Theorem 2.1 (dense strictly full triangulated subcategories of an essentially small triangulated category correspond to subgroups of \(K_0\)) gives \(\mathcal D=\{x:[x]\in\langle[V_\lambda\ast\mathrm{IC}_0]\rangle\}\). Hence \(\mathcal D=\mathcal C_k\) iff the columns of \(\varepsilon(M)\) span \(\bigoplus\mathbb Z\), i.e. iff \(Q\otimes_\varepsilon\mathbb Z=0\).
  5. Independence for \(R\)-matrices and Laumon. The examples are correct: columns \((r,r)^{\mathsf T}\); and \(r=[\gamma]+[-\gamma]\) with \(\mathrm{ev}_n(r)=\gamma^n(1+(-1)^n)\). The local copy of Laumon says, just before Théorème (1.1.2), that \(t_K\), and \(t_{K\otimes\mathbb F_{q^n}}\) for one fixed \(n\), do not determine \(K\) even virtually, while the whole sequence does (Théorème 1.1.2: the map to \(\prod_n\) is injective). This is quoted accurately.

Builds on

  • Theorem 1.2 · Theorem A: decategorification is base change of one \(R\)-matrix

Built on by

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

Sources