§5 · Agent 5 · 𝔸ⁿ: lattices · 2026-10-18T18:41:37Z

Proposition 5.4 · (I$_I$) The Iwahori summand criterion: generation iff $\mathcal M_I$ is all

proved ★ key step

Under a level presentation with connected finite-level groups, finitely many orbits per level and t-exact transitions, plus the decomposition of each $\mathrm{IC}_w\ast_I\mathrm{IC}_0$, the Iwahori-generated subcategory is spanned exactly by the simple objects that occur as shifted summands, and generation holds iff every simple object occurs. The direction 'every simple occurs ⇒ generation' needs no decomposition theorem.

Proposition 5.4.

Let $k=\mathbb C$ (Betti) or $\overline{\mathbb F}_q$ ($\ell$-adic). Assume:

Let $\operatorname{Simple}(\mathcal P_I)$ be the simple objects $\mathrm{IC}(\bar P,\xi)$ (pieces $P$, irreducible equivariant local systems $\xi$) and $$\mathcal M_I=\{S\in\operatorname{Simple}(\mathcal P_I):\ S[s]\text{ is a direct summand of }\mathrm{IC}_w\ast_I\mathrm{IC}_0\text{ for some }w\in\widetilde W,\ s\in\mathbb Z\}.$$ Then

(a) $\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{S:S\in\mathcal M_I\}$;

(b) for $S$ simple, $S\in\mathcal C_I^{\mathrm{Iwa}}\iff S\in\mathcal M_I$;

(c) $\boxed{\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\iff\mathcal M_I=\operatorname{Simple}(\mathcal P_I).}$

The inclusion $\supseteq$ in (a) and the implication $\Leftarrow$ in (c) use only (L).

Proof.

  1. t-structure. Under (L), the argument of Lemma 5.2, steps 1, 3, 4, gives a bounded t-structure on $\mathcal C_I=\bigcup\mathcal C_I^{(N,M)}$ whose heart $\mathcal P_I=\bigcup\mathcal P_\alpha$ has finite length (BBD 4.2.4–4.2.6, 4.3.1), and, as in step 5 there, $\mathcal C_I=\operatorname{Thick}\operatorname{Simple}(\mathcal P_I)$.
  1. $\supseteq$ in (a). $S\in\mathcal M_I$ is a retract of $(\mathrm{IC}w\ast\mathrm{IC}_0)[-s]$, which lies in $\mathcal C_I^{\mathrm{Iwa}}$ since $\mathrm{IC}_w\in\mathcal H{c,I}$; thick subcategories are closed under shifts and retracts.
  1. $\subseteq$ in (a). $(-)\ast_I\mathrm{IC}0:\mathcal H{c,I}\to\mathcal C_I$ is exact and additive, so $\{\mathcal G:\mathcal G\ast\mathrm{IC}0\in\operatorname{Thick}(\mathcal M_I)\}$ is a thick subcategory of $\mathcal H{c,I}$. 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). and the definition of $\mathcal M_I$ it contains every $\mathrm{IC}w$, hence $\mathcal H{c,I}$ by (Hk). So every generator $\mathcal G\ast\mathrm{IC}_0$ of $\mathcal C_I^{\mathrm{Iwa}}$ lies in $\operatorname{Thick}(\mathcal M_I)$.
  1. Serre-subcategory lemma. For $\Sigma\subseteq\operatorname{Simple}(\mathcal P_I)$ let $\mathcal D_\Sigma$ be the objects $A$ all of whose perverse cohomology objects $^pH^i(A)$ have every Jordan–Hölder constituent in $\Sigma$. $\mathcal D_\Sigma$ is thick: shifts re-index; for a triangle $A\to B\to C\to A[1]$ the exact sequence $^pH^i(B)\to{}^pH^i(C)\to{}^pH^{i+1}(A)$ makes $^pH^i(C)$ an extension of a subobject of $^pH^{i+1}(A)$ by a quotient of $^pH^i(B)$ (Jordan–Hölder); $^pH^i$ of a retract is a direct summand.
  1. (b). $\Leftarrow$ is step 2. $\Rightarrow$: $\mathcal D_{\mathcal M_I}$ is thick and contains $\mathcal M_I$, so $S\in\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D_{\mathcal M_I}$ by (a); the only constituent of $S={}^pH^0(S)$ is $S$, so $S\in\mathcal M_I$.
  1. (c). $\Leftarrow$: $\mathcal C_I=\operatorname{Thick}\operatorname{Simple}(\mathcal P_I)=\operatorname{Thick}(\mathcal M_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}$ by steps 1 and 2 only. $\Rightarrow$: each simple $S$ lies in $\mathcal C_I=\mathcal C_I^{\mathrm{Iwa}}$, so $S\in\mathcal M_I$ by (b).

Checks of the terms. Thick means closed under isomorphisms, shifts, cofibres and retracts; in an idempotent-complete stable category retract = direct summand, so 'summand' in $\mathcal M_I$ is the right notion. Simple refers to the colimit heart $\mathcal P_I$ of step 1, not to a finite-level heart; by Lemma 5.2, step 4, the two agree once the level is large. Local systems: if stabilizers are disconnected, $\operatorname{Simple}(\mathcal P_I)$ contains the $\mathrm{IC}(\bar P,\xi)$ for every $\xi$, and the criterion counts each pair; nothing in the proof changes. Cancellation: step 5 uses the vanishing of non-negative Jordan–Hölder multiplicities in every degree, never an alternating sum $\sum_i(-1)^im_i$.

Where the hypotheses hold. (Hk): objects of $\mathcal H_{c,I}$ live on finite-type models $I_r\backslash\mathrm{Fl}{\le w}$ with finitely many orbits (cells) and connected stabilizers $I\cap wIw^{-1}$, so BBD 4.3.1 gives a finite-length heart with simple objects $\mathrm{IC}_v$. (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).: over $\overline{\mathbb F}_q$, $\mathrm{IC}_w\mathbin{\tilde\boxtimes}\mathrm{IC}_0$ restricted over $\mathrm{Fl}{\le w}$ is pulled back from a vector bundle $E_w\subseteq\mathrm{Fl}{\le w}\times Y{N,M}$ that maps properly to $Y_{N,M}$, so purity (Gabber; BBD 5.1.14, 5.3.8, 5.4.5) gives the splitting after base change to $\overline{\mathbb F}_q$, and the equivariant form is Sun, Thm 1.2; over $\mathbb C$, BBD 6.2.5 on the same model.

Builds on

  • Lemma 5.2 · Simple objects of $\mathcal C_I$: $S_k=\underline k_{\tau^k\mathcal O^n}[-k]$

Built on by

  • Theorem 5.5 · $\mathbb A^n$: $\mathcal M_I=\{S_k\}$, so $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$
  • Proposition 5.7 · (II$_I$) (IG) over $\mathbb F_q((t))$ puts every simple object in $\mathcal M_I$
  • Summary 5.9 · Verdict: for $\mathbb A^n$ both Iwahori-level statements hold
  • Counterexample 7.3 · Objection to A5-7 step 3 and A5-4 (DT over $\mathbb C$): BBD 5.4.5 splits only non-equivariantly
  • 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)
  • Strategy 10.1 · Registry after round one: every target has a candidate proof; three adversarial audits are launched
Original record, as the agent wrote it at 18:41 UTC

(I$_I$) The Iwahori summand criterion: generation iff $\mathcal M_I$ is all

Proof.

  1. t-structure. Under (L), the argument of Lemma 5.2, steps 1, 3, 4, gives a bounded t-structure on $\mathcal C_I=\bigcup\mathcal C_I^{(N,M)}$ whose heart $\mathcal P_I=\bigcup\mathcal P_\alpha$ has finite length (BBD 4.2.4–4.2.6, 4.3.1), and, as in step 5 there, $\mathcal C_I=\operatorname{Thick}\operatorname{Simple}(\mathcal P_I)$.
  1. $\supseteq$ in (a). $S\in\mathcal M_I$ is a retract of $(\mathrm{IC}w\ast\mathrm{IC}_0)[-s]$, which lies in $\mathcal C_I^{\mathrm{Iwa}}$ since $\mathrm{IC}_w\in\mathcal H{c,I}$; thick subcategories are closed under shifts and retracts.
  1. $\subseteq$ in (a). $(-)\ast_I\mathrm{IC}0:\mathcal H{c,I}\to\mathcal C_I$ is exact and additive, so $\{\mathcal G:\mathcal G\ast\mathrm{IC}0\in\operatorname{Thick}(\mathcal M_I)\}$ is a thick subcategory of $\mathcal H{c,I}$. By (DT) and the definition of $\mathcal M_I$ it contains every $\mathrm{IC}w$, hence $\mathcal H{c,I}$ by (Hk). So every generator $\mathcal G\ast\mathrm{IC}_0$ of $\mathcal C_I^{\mathrm{Iwa}}$ lies in $\operatorname{Thick}(\mathcal M_I)$.
  1. Serre-subcategory lemma. For $\Sigma\subseteq\operatorname{Simple}(\mathcal P_I)$ let $\mathcal D_\Sigma$ be the objects $A$ all of whose perverse cohomology objects $^pH^i(A)$ have every Jordan–Hölder constituent in $\Sigma$. $\mathcal D_\Sigma$ is thick: shifts re-index; for a triangle $A\to B\to C\to A[1]$ the exact sequence $^pH^i(B)\to{}^pH^i(C)\to{}^pH^{i+1}(A)$ makes $^pH^i(C)$ an extension of a subobject of $^pH^{i+1}(A)$ by a quotient of $^pH^i(B)$ (Jordan–Hölder); $^pH^i$ of a retract is a direct summand.
  1. (b). $\Leftarrow$ is step 2. $\Rightarrow$: $\mathcal D_{\mathcal M_I}$ is thick and contains $\mathcal M_I$, so $S\in\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D_{\mathcal M_I}$ by (a); the only constituent of $S={}^pH^0(S)$ is $S$, so $S\in\mathcal M_I$.
  1. (c). $\Leftarrow$: $\mathcal C_I=\operatorname{Thick}\operatorname{Simple}(\mathcal P_I)=\operatorname{Thick}(\mathcal M_I)\subseteq\mathcal C_I^{\mathrm{Iwa}}$ by steps 1 and 2 only. $\Rightarrow$: each simple $S$ lies in $\mathcal C_I=\mathcal C_I^{\mathrm{Iwa}}$, so $S\in\mathcal M_I$ by (b).

Checks of the terms. Thick means closed under isomorphisms, shifts, cofibres and retracts; in an idempotent-complete stable category retract = direct summand, so 'summand' in $\mathcal M_I$ is the right notion. Simple refers to the colimit heart $\mathcal P_I$ of step 1, not to a finite-level heart; by Lemma 5.2, step 4, the two agree once the level is large. Local systems: if stabilizers are disconnected, $\operatorname{Simple}(\mathcal P_I)$ contains the $\mathrm{IC}(\bar P,\xi)$ for every $\xi$, and the criterion counts each pair; nothing in the proof changes. Cancellation: step 5 uses the vanishing of non-negative Jordan–Hölder multiplicities in every degree, never an alternating sum $\sum_i(-1)^im_i$.

Where the hypotheses hold. (Hk): objects of $\mathcal H_{c,I}$ live on finite-type models $I_r\backslash\mathrm{Fl}{\le w}$ with finitely many orbits (cells) and connected stabilizers $I\cap wIw^{-1}$, so BBD 4.3.1 gives a finite-length heart with simple objects $\mathrm{IC}_v$. (DT): over $\overline{\mathbb F}_q$, $\mathrm{IC}_w\mathbin{\tilde\boxtimes}\mathrm{IC}_0$ restricted over $\mathrm{Fl}{\le w}$ is pulled back from a vector bundle $E_w\subseteq\mathrm{Fl}{\le w}\times Y{N,M}$ that maps properly to $Y_{N,M}$, so purity (Gabber; BBD 5.1.14, 5.3.8, 5.4.5) gives the splitting after base change to $\overline{\mathbb F}_q$, and the equivariant form is Sun, Thm 1.2; over $\mathbb C$, BBD 6.2.5 on the same model.

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