§7 · Agent 7 · Audit: criterion and bridge · 2026-10-18T19:23:11Z

Audit 7.1 · Audit of A2-1, A2-2: perverse hearts glue along the levels and $\mathcal D_\Sigma=\mathrm{Thick}(\Sigma)$

proved

The set-up and Lemma (PH$I$) of Claim I, version 1, are correct under (L1)–(L3), (I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).), over $\overline{\mathbb F}_q$ and over $\mathbb C$ (Betti); two citation repairs and one remark on the choice of $I{N,M}$.

Audit 7.1.

Holds with a minor repair. Under (L1)–(L3) and (I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).) of A2-1, every step of A2-1 and A2-2 is correct, over $k=\overline{\mathbb F}q$ ($\overline{\mathbb Q}\ell$) and over $k=\mathbb C$ (Betti); no weights are used. Repairs:

  1. The BBD statements used in A2-2 (Prop. 4.2.5, §4.2.6, Lemma 4.2.6.1 with the commutation of $j_{!*}$ with smooth base change stated after it, Cor. 4.2.6.2) are on journal pp. 109–111; t-exactness of a closed push-forward is BBD Prop. 1.4.16(i).
  2. (L3) must be read with $I_{N,M}$ pro-unipotent (e.g. a congruence subgroup, as in hypothesis.html, Example B), not the full kernel of the action: for the group $X=H$ the full kernel contains $\Delta Z(H)(\mathcal O)$, and when $Z(H)^\circ\neq1$ the category $D^b_c\big((I/I_{N,M})\backslash Y_{N,M}\big)$ is then not the $I$-equivariant one. The criterion is insensitive to this choice (step 6).

Audit, step by step.

  1. A2-1, idempotents. $D^b_c(H\backslash Y)$ ($H$ connected linear) carries a bounded t-structure (step 2 below), hence is idempotent complete (a triangulated category with a bounded t-structure is Karoubian: Le–Chen, J. Algebra 310 (2007); cited background). If $e$ is an idempotent of $X\in\mathcal C^{(N,M)}$ in $\mathcal C_I$, full faithfulness (L3) puts $e$ in $\mathcal C^{(N,M)}$, where it splits, $X\simeq X_1\oplus X_2$; a retract $R$ of $X$ in $\mathcal C_I$ with idempotent $e$ is isomorphic to $X_1$ (splittings of an idempotent are unique up to isomorphism). Holds.
  2. A2-2(a). For $H$ connected, BBD 4.2.5 and 4.2.6.2 (scan, pp. 109–111: $f$ smooth with geometrically connected, non-empty fibres; $f^[d]$ fully faithful on perverse sheaves with essential image stable under subquotients) applied to the action and projection $H\times Y\to Y$ make $\mathrm{Perv}H(Y)\subseteq\mathrm{Perv}(Y)$ full and closed under subquotients. Finite length: BBD Thm 4.3.1(i). A simple equivariant perverse sheaf is simple in $\mathrm{Perv}(Y)$, hence $j{!}(L[\dim V])$ (4.3.1(ii)), with $H$-stable irreducible support, i.e. an orbit closure by (I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).); equivariant irreducible local systems on $o=H/S$ are $\mathrm{Irr}\,\pi_0(S)$, trivial under (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.). Holds.
  3. A2-2(b). BBD 4.2.5 needs surjectivity and connected fibres; (L1) gives affine-space fibres. $\pi^[d]\,j_{!}=j_{!}\pi^[d]$ for smooth $\pi$ is the remark after Lemma 4.2.6.1 (p. 110), which also states that the pull-back of an irreducible local system along a smooth map with connected fibres stays irreducible. Restricting to the open orbit $o'\subseteq\pi^{-1}(o)$ keeps it irreducible because $\pi_1(o')\to\pi_1(\pi^{-1}o)$ is onto for a dense open subset of a normal variety. Closed push-forward: t-exact (BBD 1.4.16(i)), fully faithful, image closed under subquotients (a perverse subquotient of $\iota_*A$ vanishes on the open complement since $j^*$ is t-exact). Holds.
  4. A2-2(c). Per-component shifts $d(C')$ are handled; constituents are compared up to shift, so $\mathcal S_I$ and $\mathcal D_\Sigma$ are level-independent; the support bijection under (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.) uses only that the open orbit of an irreducible orbit closure is unique. Holds.
  5. A2-2(d). At one level, $\mathcal D_\Sigma\cap\mathcal C^{(N,M)}=\{F:\ h(F[n])=0\ \forall n\}$ with $h=q\circ{}^pH^0$ homological; Stacks Lemma 13.6.3 (Tag 05RD; local extract summand-stacks-kernel.txt) gives a strictly full saturated triangulated subcategory; with step 1 the union is thick. $\mathcal D_\Sigma\subseteq\mathrm{Thick}(\Sigma)$: finitely many ${}^pH^i$ (bounded), finite Jordan–Hölder filtrations, (T1). Holds.
  6. Remark on $I_{N,M}$ (group case). $(z,z)$, $z\in Z(H)(\mathcal O)$, acts trivially on $LH$. If $I_{N,M}$ is the full kernel and $Z(H)^\circ\ne1$, then $I_{N,M}/I_{N',M'}$ is still unipotent, so (L3) is internally consistent, but $D^b_c((I/I_{N,M})\backslash Y)$ differs from the $I$-equivariant category (e.g. $\mathrm{Hom}(k,k[2])$ on a fixed point differs by $H^2(BZ^\circ)$). Both categories have the same perverse heart (equivariance is a property for connected groups), bounded t-structures with finite-length hearts, and the decomposition of $G_w$ holds in both (Sun, Thm 1.2, for either quotient stack), so $\mathcal S_I$, $\mathcal M_I$, $\mathcal M_I^+$ and the truth of the criterion agree. Taking $I_{N,M}$ a congruence subgroup (hypothesis.html, Example B) removes the issue; it is also what A3-1 needs for Lang (Lemma 7.5).

Builds on

  • Setup 2.1 · Set-up: level hypotheses, simple objects, $\mathcal M_I\subseteq\mathcal M_I^+$
  • Lemma 2.2 · Lemma (PH$_I$): perverse hearts glue; $\mathcal C_I=\mathrm{Thick}(\mathcal S_I)$

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
Original record, as the agent wrote it at 19:23 UTC

Audit of A2-1, A2-2: perverse hearts glue along the levels and $\mathcal D_\Sigma=\mathrm{Thick}(\Sigma)$

Audit, step by step.

  1. A2-1, idempotents. $D^b_c(H\backslash Y)$ ($H$ connected linear) carries a bounded t-structure (step 2 below), hence is idempotent complete (a triangulated category with a bounded t-structure is Karoubian: Le–Chen, J. Algebra 310 (2007); cited background). If $e$ is an idempotent of $X\in\mathcal C^{(N,M)}$ in $\mathcal C_I$, full faithfulness (L3) puts $e$ in $\mathcal C^{(N,M)}$, where it splits, $X\simeq X_1\oplus X_2$; a retract $R$ of $X$ in $\mathcal C_I$ with idempotent $e$ is isomorphic to $X_1$ (splittings of an idempotent are unique up to isomorphism). Holds.
  2. A2-2(a). For $H$ connected, BBD 4.2.5 and 4.2.6.2 (scan, pp. 109–111: $f$ smooth with geometrically connected, non-empty fibres; $f^[d]$ fully faithful on perverse sheaves with essential image stable under subquotients) applied to the action and projection $H\times Y\to Y$ make $\mathrm{Perv}H(Y)\subseteq\mathrm{Perv}(Y)$ full and closed under subquotients. Finite length: BBD Thm 4.3.1(i). A simple equivariant perverse sheaf is simple in $\mathrm{Perv}(Y)$, hence $j{!}(L[\dim V])$ (4.3.1(ii)), with $H$-stable irreducible support, i.e. an orbit closure by (I1); equivariant irreducible local systems on $o=H/S$ are $\mathrm{Irr}\,\pi_0(S)$, trivial under (I2). Holds.
  3. A2-2(b). BBD 4.2.5 needs surjectivity and connected fibres; (L1) gives affine-space fibres. $\pi^[d]\,j_{!}=j_{!}\pi^[d]$ for smooth $\pi$ is the remark after Lemma 4.2.6.1 (p. 110), which also states that the pull-back of an irreducible local system along a smooth map with connected fibres stays irreducible. Restricting to the open orbit $o'\subseteq\pi^{-1}(o)$ keeps it irreducible because $\pi_1(o')\to\pi_1(\pi^{-1}o)$ is onto for a dense open subset of a normal variety. Closed push-forward: t-exact (BBD 1.4.16(i)), fully faithful, image closed under subquotients (a perverse subquotient of $\iota_*A$ vanishes on the open complement since $j^*$ is t-exact). Holds.
  4. A2-2(c). Per-component shifts $d(C')$ are handled; constituents are compared up to shift, so $\mathcal S_I$ and $\mathcal D_\Sigma$ are level-independent; the support bijection under (I2) uses only that the open orbit of an irreducible orbit closure is unique. Holds.
  5. A2-2(d). At one level, $\mathcal D_\Sigma\cap\mathcal C^{(N,M)}=\{F:\ h(F[n])=0\ \forall n\}$ with $h=q\circ{}^pH^0$ homological; Stacks Lemma 13.6.3 (Tag 05RD; local extract summand-stacks-kernel.txt) gives a strictly full saturated triangulated subcategory; with step 1 the union is thick. $\mathcal D_\Sigma\subseteq\mathrm{Thick}(\Sigma)$: finitely many ${}^pH^i$ (bounded), finite Jordan–Hölder filtrations, (T1). Holds.
  6. Remark on $I_{N,M}$ (group case). $(z,z)$, $z\in Z(H)(\mathcal O)$, acts trivially on $LH$. If $I_{N,M}$ is the full kernel and $Z(H)^\circ\ne1$, then $I_{N,M}/I_{N',M'}$ is still unipotent, so (L3) is internally consistent, but $D^b_c((I/I_{N,M})\backslash Y)$ differs from the $I$-equivariant category (e.g. $\mathrm{Hom}(k,k[2])$ on a fixed point differs by $H^2(BZ^\circ)$). Both categories have the same perverse heart (equivariance is a property for connected groups), bounded t-structures with finite-length hearts, and the decomposition of $G_w$ holds in both (Sun, Thm 1.2, for either quotient stack), so $\mathcal S_I$, $\mathcal M_I$, $\mathcal M_I^+$ and the truth of the criterion agree. Taking $I_{N,M}$ a congruence subgroup (hypothesis.html, Example B) removes the issue; it is also what A3-1 needs for Lang (Lemma 7.5).

Sources