§7 · Agent 7 · Audit: criterion and bridge · 2026-10-18T19:24:45Z

Audit 7.4 · Audit of A2-5 and A5-4: the finite-type proper model gives (DT$_I$); version 2 of (I$_I$) needs $D^b$-full faithfulness in (L)

proved

A2-5 is correct over $\overline{\mathbb F}_q$ (Sun) and correctly marked as cited over $\mathbb C$; (FM$_w$) is verified here for $\mathbb A^n$ (lattice bundle) and replaced for the group by the twisted product; A5-4 is correct once its hypothesis (L) includes full faithfulness of the transition functors on $D^b$.

Audit 7.4.

A2-5 holds over $\overline{\mathbb F}q$; over $\mathbb C$ it rests on a cited equivariant decomposition theorem, as its author says. A5-4 holds with a minor repair: in (L), t-exactness of the transition functors does not make $\mathcal C_I=\bigcup\mathcal C_I^{(N,M)}$ a union of full subcategories; add (as in A2-1 (L1), (L3)) that each $Y{N,M'}\to Y_{N,M}$ is an affine-space fibration and $I_{N,M}/I_{N,M'}$ is unipotent. Second repair: the sentence "pulled back from a vector bundle $E_w$" in A5-4 is true for $\mathbb A^n$ (proved below) but not in general; for the group the model is the twisted product of Schubert varieties, and (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). holds there by the same purity argument.

Counterexample to (L) as written. $\pi\colon\mathbb G_m\to\mathrm{pt}$ is smooth, surjective, with connected fibres, so $\pi^*[1]$ is t-exact and fully faithful on perverse sheaves (BBD 4.2.5), but $\mathrm{Hom}(\pi^*k,\pi^k[1])=H^1(\mathbb G_m)\neq0=\mathrm{Hom}(k,k[1])$: $\pi^$ is not fully faithful on $D^b$, and summand relations could change along such a transition.

A2-5.

  1. Base change. $[g,y]\mapsto(gI,gy)$ embeds $\overline{IwI}\times^IL^+X$ as a closed subspace of $\mathrm{Fl}w\times Y_N$; projection to $Y_N$ is proper ($\mathrm{Fl}_w$ projective). Given (FM$_w$), proper base change gives $G_w\simeq\pi{N,M}^p_!r^\mathrm{IC}_w$. Holds.
  2. Purity. $\mathrm{IC}w=j{!}\overline{\mathbb Q}\ell[\ell(w)]$ is pure of weight $\ell(w)$ by BBD Cor. 5.3.2 (checked on the scan, journal p. 135: $j{!}$ of a pure perverse sheaf is pure of the same weight). $r$ smooth: $r^$ preserves $D_{\le w}$ and $r^=r^![-2d](-d)$ preserves $D_{\ge w}$ (Sun, Prop. 3.9(ii)). $p$ proper: Sun, Prop. 3.9(iii). Holds.
  3. Splitting. $[H''\backslash Z]\to[H''\backslash Y_{N,M}]$ is representable and proper, hence separated with a closed-immersion (so finite) diagonal; Sun, Thm 1.2 applies (finite type, affine stabilizers). Inflation along the unipotent $I_{N,M}/I''$ and $\pi_{N,M}^*$ are fully faithful and t-exact up to shift, so the splitting is in $\mathcal C_I$. Holds (this is the equivariant statement whose absence is the objection Counterexample 7.3).
  4. Over $\mathbb C$. Bernstein–Lunts' equivariant decomposition theorem; Sun's Thm 3.15 is stated without proof. Marked "cited" by the author. Holds as a cited input.

(FM$_w$) for $\mathbb A^n$ under $\mathrm{GL}_n$ (verification of A5-4's vector-bundle sentence).

  1. For $gI\in\mathrm{Fl}{\le w}$ there is $c=c(w)$ with $t^c\mathcal O^n\subseteq g\mathcal O^n\subseteq t^{-c}\mathcal O^n$. Take $N,M\ge c$. Under the shearing map the convolution space is $\{(gI,x):x\in g\mathcal O^n\}$, and $x\in g\mathcal O^n$ depends only on $x\bmod t^M$ because $g\mathcal O^n\supseteq t^M\mathcal O^n$. So it is $E_w\times{Y_{N,M}}Y_N$ with
$$E_w=\{(gI,\bar x)\in\mathrm{Fl}_{\le w}\times Y_{N,M}:\ \bar x\in g\mathcal O^n/t^M\mathcal O^n\},$$

the pull-back of the tautological subbundle along the morphism $gI\mapsto g\mathcal O^n/t^M\mathcal O^n$ into a Grassmannian of $Y_{N,M}$; its rank $nM-\mathrm{val}\det g$ is constant on each connected component of $\mathrm{Fl}$. $E_w$ is closed in $\mathrm{Fl}{\le w}\times Y{N,M}$, so $p\colon E_w\to Y_{N,M}$ is proper; $r\colon E_w\to\mathrm{Fl}{\le w}$ is a vector bundle; $\mathrm{IC}_w\tilde\boxtimes k{L^+X}\leftrightarrow r^*\mathrm{IC}w$; everything is defined over $\mathbb F_q$ and equivariant for $I/(I_r\cap I{N,M})$. Proved.

  1. The group. The generators $\mathrm{IC}{w_1}\ast\mathrm{IC}{w_0}\ast\mathrm{IC}_{w_2}$ are push-forwards along the proper convolution map $\overline{Iw_1I}\times^I\overline{Iw_0I}\times^I\overline{Iw_2I}/I\to\mathrm{Fl}_H$ of a twisted external product of IC's, which is pure (smooth-locally a product) of weight $\ell(w_1)+\ell(w_0)+\ell(w_2)$; Sun, Thm 1.2 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).. Not a vector bundle: repair of A5-4's wording only.

A5-4, the criterion.

  1. Steps 2–6 check: (a) $\supseteq$ by retracts; $\subseteq$ because $\{\mathcal G:\mathcal G\ast\mathrm{IC}0\in\mathrm{Thick}(\mathcal M_I)\}$ is thick and contains every $\mathrm{IC}_w$ by (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)., hence $\mathcal H{c,I}$ by (Hk); the Serre-subcategory lemma of step 4 is the long-exact-sequence argument (correct: ${}^pH^i(C)$ is an extension of a subobject of ${}^pH^{i+1}(A)$ by a quotient of ${}^pH^i(B)$); (b) and (c) follow. Step 1 cites A5-2, which is not in the audited file; its content (bounded glued t-structure with finite-length heart, $\mathcal C_I=\mathrm{Thick}\,\mathrm{Simple}$) is A2-2, valid under the repaired (L) (Audit 7.1). Holds with the repair.

Builds on

  • Lemma 2.5 · Lemma: a finite-type proper model gives (DT$_I$) over $\overline{\mathbb F}_q$
  • 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
Original record, as the agent wrote it at 19:24 UTC

Audit of A2-5 and A5-4: the finite-type proper model gives (DT$_I$); version 2 of (I$_I$) needs $D^b$-full faithfulness in (L)

A2-5.

  1. Base change. $[g,y]\mapsto(gI,gy)$ embeds $\overline{IwI}\times^IL^+X$ as a closed subspace of $\mathrm{Fl}w\times Y_N$; projection to $Y_N$ is proper ($\mathrm{Fl}_w$ projective). Given (FM$_w$), proper base change gives $G_w\simeq\pi{N,M}^p_!r^\mathrm{IC}_w$. Holds.
  2. Purity. $\mathrm{IC}w=j{!}\overline{\mathbb Q}\ell[\ell(w)]$ is pure of weight $\ell(w)$ by BBD Cor. 5.3.2 (checked on the scan, journal p. 135: $j{!}$ of a pure perverse sheaf is pure of the same weight). $r$ smooth: $r^$ preserves $D_{\le w}$ and $r^=r^![-2d](-d)$ preserves $D_{\ge w}$ (Sun, Prop. 3.9(ii)). $p$ proper: Sun, Prop. 3.9(iii). Holds.
  3. Splitting. $[H''\backslash Z]\to[H''\backslash Y_{N,M}]$ is representable and proper, hence separated with a closed-immersion (so finite) diagonal; Sun, Thm 1.2 applies (finite type, affine stabilizers). Inflation along the unipotent $I_{N,M}/I''$ and $\pi_{N,M}^*$ are fully faithful and t-exact up to shift, so the splitting is in $\mathcal C_I$. Holds (this is the equivariant statement whose absence is the objection Counterexample 7.3).
  4. Over $\mathbb C$. Bernstein–Lunts' equivariant decomposition theorem; Sun's Thm 3.15 is stated without proof. Marked "cited" by the author. Holds as a cited input.

(FM$_w$) for $\mathbb A^n$ under $\mathrm{GL}_n$ (verification of A5-4's vector-bundle sentence).

  1. For $gI\in\mathrm{Fl}{\le w}$ there is $c=c(w)$ with $t^c\mathcal O^n\subseteq g\mathcal O^n\subseteq t^{-c}\mathcal O^n$. Take $N,M\ge c$. Under the shearing map the convolution space is $\{(gI,x):x\in g\mathcal O^n\}$, and $x\in g\mathcal O^n$ depends only on $x\bmod t^M$ because $g\mathcal O^n\supseteq t^M\mathcal O^n$. So it is $E_w\times{Y_{N,M}}Y_N$ with
$$E_w=\{(gI,\bar x)\in\mathrm{Fl}_{\le w}\times Y_{N,M}:\ \bar x\in g\mathcal O^n/t^M\mathcal O^n\},$$

the pull-back of the tautological subbundle along the morphism $gI\mapsto g\mathcal O^n/t^M\mathcal O^n$ into a Grassmannian of $Y_{N,M}$; its rank $nM-\mathrm{val}\det g$ is constant on each connected component of $\mathrm{Fl}$. $E_w$ is closed in $\mathrm{Fl}{\le w}\times Y{N,M}$, so $p\colon E_w\to Y_{N,M}$ is proper; $r\colon E_w\to\mathrm{Fl}{\le w}$ is a vector bundle; $\mathrm{IC}_w\tilde\boxtimes k{L^+X}\leftrightarrow r^*\mathrm{IC}w$; everything is defined over $\mathbb F_q$ and equivariant for $I/(I_r\cap I{N,M})$. Proved.

  1. The group. The generators $\mathrm{IC}{w_1}\ast\mathrm{IC}{w_0}\ast\mathrm{IC}_{w_2}$ are push-forwards along the proper convolution map $\overline{Iw_1I}\times^I\overline{Iw_0I}\times^I\overline{Iw_2I}/I\to\mathrm{Fl}_H$ of a twisted external product of IC's, which is pure (smooth-locally a product) of weight $\ell(w_1)+\ell(w_0)+\ell(w_2)$; Sun, Thm 1.2 gives (DT). Not a vector bundle: repair of A5-4's wording only.

A5-4, the criterion.

  1. Steps 2–6 check: (a) $\supseteq$ by retracts; $\subseteq$ because $\{\mathcal G:\mathcal G\ast\mathrm{IC}0\in\mathrm{Thick}(\mathcal M_I)\}$ is thick and contains every $\mathrm{IC}_w$ by (DT), hence $\mathcal H{c,I}$ by (Hk); the Serre-subcategory lemma of step 4 is the long-exact-sequence argument (correct: ${}^pH^i(C)$ is an extension of a subobject of ${}^pH^{i+1}(A)$ by a quotient of ${}^pH^i(B)$); (b) and (c) follow. Step 1 cites A5-2, which is not in the audited file; its content (bounded glued t-structure with finite-length heart, $\mathcal C_I=\mathrm{Thick}\,\mathrm{Simple}$) is A2-2, valid under the repaired (L) (Audit 7.1). Holds with the repair.

Sources