§7 · Agent 7 · Audit: criterion and bridge · 2026-10-18T19:24:07Z

Counterexample 7.3 · Objection to A5-7 step 3 and A5-4 (DT over $\mathbb C$): BBD 5.4.5 splits only non-equivariantly

proved ★ key step

$\mathcal M_I$ is defined by direct summands in the equivariant category $\mathcal C_I$; a splitting in $D^b_c(Y)$ (BBD 5.4.5, 6.2.5) does not give one, as $\mathbb G_m$ acting on a point shows; the stack versions in Sun's paper repair A5-7, and Bernstein–Lunts covers $\mathbb C$.

Counterexample 7.3.

Holds with a minor repair (A5-7, step 3; A5-4, "Where the hypotheses hold", the clause over $\mathbb C$).

Lemma (counterexample). Let $H=\mathbb G_m$ act trivially on $Y=\mathrm{pt}$, let $c\in\mathrm{Hom}{D_H}(k,k[2])=H^2(B\mathbb G_m)$ be a generator and $K=\mathrm{Cone}(c)$. Then the underlying complex of $K$ is $k[2]\oplus k[1]$, each ${}^pH^i(K)$ is simple, but the simple object $k$ does not lie in $\mathrm{Thick}{D_H(\mathrm{pt})}\{K\}$; in particular $k[s]$ is not a retract of $K$ in $D_H(\mathrm{pt})$.

Consequence. "$P_i$ semisimple and $G_w\simeq\bigoplus_iP_i[-i]$ after base change to $\overline{\mathbb F}q$" in $D^b_c(Y{N,M})$ (BBD 5.4.5, 5.3.8) does not show that a constituent $S$ of ${}^pH^i(G_w)$ lies in $\mathcal M_I$ or in $\mathcal C_I^{\mathrm{Iwa}}$. Repair: apply Sun, Thm 3.12 and 3.11 (stack versions of BBD 5.4.5 and 5.3.8) or Thm 1.2 to the finite-type quotient stack $[Y_{N,M}/(I/I_{N,M})]$, which has affine stabilizers; this uses exactly hypothesis (P) of A5-7, so A5-7 survives. Over $\mathbb C$, replace BBD 6.2.5 by the equivariant decomposition theorem (Bernstein–Lunts, LNM 1578, §5; Sun, Thm 3.15, states the stack version without proof). A2-5 (step 4) and A3-1 (D3) already cite Sun and need no repair.

Proof of the lemma.

  1. $\mathrm{Hom}_{D_H(\mathrm{pt})}(k,k[n])=H^n(B\mathbb G_m;k)=k[c]$ in degree $n$ ($\deg c=2$). The forgetful functor sends $c$ to $0\in H^2(\mathrm{pt})$, so the underlying complex of $K$ is $\mathrm{Cone}(0\colon k\to k[2])\simeq k[2]\oplus k[1]$; its perverse (= ordinary) cohomology is $k$ in degrees $-2$ and $-1$, simple.
  2. Let $\mathcal T=\{A\in D_H(\mathrm{pt}):\ \sum_n\dim\mathrm{Hom}(k,A[n])<\infty\}$. It is closed under shifts, under cones (long exact $\mathrm{Hom}(k,-)$ sequence) and under retracts (a summand of a finite-dimensional graded space): $\mathcal T$ is thick.
  3. In the long exact sequence for $k\xrightarrow{c}k[2]\to K\to k[1]$, multiplication by $c\colon H^n(B\mathbb G_m)\to H^{n+2}(B\mathbb G_m)$ is injective for all $n$ and surjective except onto $H^0$. Hence $\sum_n\dim\mathrm{Hom}(k,K[n])=1$ and $K\in\mathcal T$, while $\sum_n\dim H^n(B\mathbb G_m)=\infty$, so $k\notin\mathcal T\supseteq\mathrm{Thick}\{K\}$. $\square$

Why it bears on the claims.

  1. $\mathcal M_I$ (A5-4, A5-7) is "summand of $\mathrm{IC}w\ast_I\mathrm{IC}_0$ in $\mathcal C_I$", and $\mathcal C_I^{(N,M)}$ is the equivariant category. The groups $I/I{N,M}$ contain the torus $T$ and have fixed points or non-reductive stabilizers containing tori (e.g. $0\in Y_{N,M}$ for $\mathbb A^n$ is fixed by all of $I/I_{N,M}$), so equivariant extension classes of the kind in step 1 are present in $\mathcal C_I$ and the lemma's phenomenon is not excluded a priori.
  2. A5-7 step 3 writes "By (P) and BBD 5.4.5, over $k$ $\mathrm{IC}w\ast\mathrm{IC}_0\simeq\bigoplus_iP_i[-i]$" and concludes "$m{S,w}\neq0$ only if $S$ is a shifted summand, i.e. $S\in\mathcal M_I$". The trace identity itself needs only the perverse truncation triangles (Laumon (1.1.1.1)) and semisimplicity of $P_i$, which is fine in $\mathrm{Perv}_H\subseteq\mathrm{Perv}$ (full, closed under subobjects); the passage to $\mathcal M_I$, and the final appeal to (DT)(DT) · Decomposition: every generator IC_w ∗_I IC_0 is, equivariantly, a finite direct sum of shifted simple objects of C_I. Holds over F̄_q from a finite-type proper model of the convolution (purity and the stack decomposition theorem). in A5-4(c), need the equivariant splitting.
  3. Sun, Thm 3.12 (local extract summand-sun-main.txt / summand-sun-stacks.txt): for an $\mathbb F_q$-algebraic stack of finite type with affine stabilizers, an $\iota$-pure $K_0$ satisfies $K\simeq\bigoplus_i({}^pH^iK)[-i]$ on the stack over $\overline{\mathbb F}q$; Thm 3.11: $\iota$-pure perverse sheaves are geometrically semisimple. For $[Y{N,M}/(I/I_{N,M})]$ the stabilizers are closed subgroups of a linear group, hence affine. With (P) this gives (DT)(DT) · Decomposition: every generator IC_w ∗_I IC_0 is, equivariantly, a finite direct sum of shifted simple objects of C_I. Holds over F̄_q from a finite-type proper model of the convolution (purity and the stack decomposition theorem). in $\mathcal C_I^{(N,M)}$.
  4. Shared cited input (both versions, wherever Sun is applied): the Laszlo–Olsson derived category of the quotient stack $[Y/H]$ agrees with the Bernstein–Lunts equivariant category $D^b_{H,c}(Y)$ used to define $\mathcal C_I^{(N,M)}$. Standard; not re-proved in the claims or here.

Builds on

  • Proposition 5.7 · (II$_I$) (IG) over $\mathbb F_q((t))$ puts every simple object in $\mathcal M_I$
  • Proposition 5.4 · (I$_I$) The Iwahori summand criterion: generation iff $\mathcal M_I$ is all

Built on by

  • Summary 7.11 · Verdict of the audit: (I$_I$) over $\overline{\mathbb F}_q$ and $\mathbb C$, (II$_I$) over $\overline{\mathbb F}_q$ only, converses as stated
  • Summary 10.2 · Verdict: the criterion and the bridge hold; the decided rows of Table 1 agree; GL₂/T fails at both levels
Original record, as the agent wrote it at 19:24 UTC

Objection to A5-7 step 3 and A5-4 (DT over $\mathbb C$): BBD 5.4.5 splits only non-equivariantly

Proof of the lemma.

  1. $\mathrm{Hom}_{D_H(\mathrm{pt})}(k,k[n])=H^n(B\mathbb G_m;k)=k[c]$ in degree $n$ ($\deg c=2$). The forgetful functor sends $c$ to $0\in H^2(\mathrm{pt})$, so the underlying complex of $K$ is $\mathrm{Cone}(0\colon k\to k[2])\simeq k[2]\oplus k[1]$; its perverse (= ordinary) cohomology is $k$ in degrees $-2$ and $-1$, simple.
  2. Let $\mathcal T=\{A\in D_H(\mathrm{pt}):\ \sum_n\dim\mathrm{Hom}(k,A[n])<\infty\}$. It is closed under shifts, under cones (long exact $\mathrm{Hom}(k,-)$ sequence) and under retracts (a summand of a finite-dimensional graded space): $\mathcal T$ is thick.
  3. In the long exact sequence for $k\xrightarrow{c}k[2]\to K\to k[1]$, multiplication by $c\colon H^n(B\mathbb G_m)\to H^{n+2}(B\mathbb G_m)$ is injective for all $n$ and surjective except onto $H^0$. Hence $\sum_n\dim\mathrm{Hom}(k,K[n])=1$ and $K\in\mathcal T$, while $\sum_n\dim H^n(B\mathbb G_m)=\infty$, so $k\notin\mathcal T\supseteq\mathrm{Thick}\{K\}$. $\square$

Why it bears on the claims.

  1. $\mathcal M_I$ (A5-4, A5-7) is "summand of $\mathrm{IC}w\ast_I\mathrm{IC}_0$ in $\mathcal C_I$", and $\mathcal C_I^{(N,M)}$ is the equivariant category. The groups $I/I{N,M}$ contain the torus $T$ and have fixed points or non-reductive stabilizers containing tori (e.g. $0\in Y_{N,M}$ for $\mathbb A^n$ is fixed by all of $I/I_{N,M}$), so equivariant extension classes of the kind in step 1 are present in $\mathcal C_I$ and the lemma's phenomenon is not excluded a priori.
  2. A5-7 step 3 writes "By (P) and BBD 5.4.5, over $k$ $\mathrm{IC}w\ast\mathrm{IC}_0\simeq\bigoplus_iP_i[-i]$" and concludes "$m{S,w}\neq0$ only if $S$ is a shifted summand, i.e. $S\in\mathcal M_I$". The trace identity itself needs only the perverse truncation triangles (Laumon (1.1.1.1)) and semisimplicity of $P_i$, which is fine in $\mathrm{Perv}_H\subseteq\mathrm{Perv}$ (full, closed under subobjects); the passage to $\mathcal M_I$, and the final appeal to (DT) in A5-4(c), need the equivariant splitting.
  3. Sun, Thm 3.12 (local extract summand-sun-main.txt / summand-sun-stacks.txt): for an $\mathbb F_q$-algebraic stack of finite type with affine stabilizers, an $\iota$-pure $K_0$ satisfies $K\simeq\bigoplus_i({}^pH^iK)[-i]$ on the stack over $\overline{\mathbb F}q$; Thm 3.11: $\iota$-pure perverse sheaves are geometrically semisimple. For $[Y{N,M}/(I/I_{N,M})]$ the stabilizers are closed subgroups of a linear group, hence affine. With (P) this gives (DT) in $\mathcal C_I^{(N,M)}$.
  4. Shared cited input (both versions, wherever Sun is applied): the Laszlo–Olsson derived category of the quotient stack $[Y/H]$ agrees with the Bernstein–Lunts equivariant category $D^b_{H,c}(Y)$ used to define $\mathcal C_I^{(N,M)}$. Standard; not re-proved in the claims or here.

Sources