§4 · Agent 4 · Sheaf–function dictionary · 2026-10-17T02:32:10Z

Lemma 4.5 · $\Csat=\cC_K$ if and only if every $\IC_\mu$ is a direct summand of a Satake object

proved audit: correct ★ key step

Second audit: correct. Re-derived (i)–(iii): thickness of D_Σ via the perverse long exact sequence and Serre-subcategory closure; (ii) by minimality; (iii)⇐ via Proposition I's standard-generation lemma (which needs H1–H3, assumed) and ⇒ via the semisimple decomposition. 'Direct summand' means a shifted summand, as Lemma 4.4 part 2 makes explicit.

The thick span of the Satake objects is controlled by perverse cohomology: a simple object that is never a constituent of any ${}^pH^i(V_\lambda\ast\IC_0)$ is not in $\Csat$, and when Satake objects are semisimple, $\Csat=\cC_K$ holds exactly when every $\IC_\mu$ is a (shifted) summand of one. This turns the categorical statement into a statement about multiplicity spaces, insensitive to the Grothendieck group.

Lemma 4.5.

Assume H1–H3 and D1 of Setup 4.1: the perverse heart $\mathcal P$ of $\cC_K$ has finite length, with simple objects $\IC_\mu$, $\mu\in\Lambda_S^+$.

For a set $\Sigma$ of simple objects let $\mathcal S_\Sigma\subseteq\mathcal P$ be the Serre subcategory of objects all of whose Jordan–Hölder constituents lie in $\Sigma$, and $\mathcal D_\Sigma=\{M\in\cC_K:{}^pH^i(M)\in\mathcal S_\Sigma\ \text{for all }i\}$. Then $\mathcal D_\Sigma$ is a thick subcategory of $\cC_K$.
If $\IC_\mu$ is a Jordan–Hölder constituent of no ${}^pH^i(V_\lambda\ast\IC_0)$, then $\IC_\mu\notin\Csat$.
Under D2 (the decomposition in Lemma 4.4, part 2): $\Csat=\cC_K$ if and only if every $\IC_\mu$ is a direct summand (up to shift) of some $V_\lambda\ast\IC_0$.

$0\in\mathcal D_\Sigma$, and shifts permute the ${}^pH^i$. For a cofibre sequence $A\to B\to C$, the long exact sequence $\cdots\to{}^pH^iA\to{}^pH^iB\to{}^pH^iC\to{}^pH^{i+1}A\to\cdots$ and the closure of $\mathcal S_\Sigma$ under subobjects, quotients and extensions give two-out-of-three. If $M$ is a retract of $N$, then ${}^pH^i(M)$ is a retract, hence a subobject, of ${}^pH^i(N)\in\mathcal S_\Sigma$.

Take $\Sigma=\{\IC_\nu:\nu\ne\mu\}$. Every $V_\lambda\ast\IC_0$ lies in $\mathcal D_\Sigma$, so $\Csat\subseteq\mathcal D_\Sigma$ by minimality of the thick span; but ${}^pH^0(\IC_\mu)=\IC_\mu\notin\mathcal S_\Sigma$.

($\Leftarrow$) A summand is a retract, so every $\IC_\mu$ lies in $\Csat$; this is the role of retracts, as in step 4 of Proposition I. Then $\cC_K=\Thick\{\IC_\mu\}$: by induction on $h(\mu)=\#\{\nu\le\mu\}$, the triangle

$$a_{\mu!}a_\mu^*\IC_\mu\to\IC_\mu\to i_{\mu*}i_\mu^*\IC_\mu,\qquad a_\mu^*\IC_\mu=\underline k[\delta(\mu)],\qquad i_{\mu*}i_\mu^*\IC_\mu\in\Thick\{\Delta_\nu:\nu<\mu\}$$

(Proposition I, Lemma "Standard generation") gives $\Thick\{\IC_\nu:\nu\le\mu\}=\Thick\{\Delta_\nu:\nu\le\mu\}$, and the standard objects $\Delta_\nu$ generate $\cC_K$ by the same lemma. ($\Rightarrow$) If $\IC_\mu$ is a summand of no $V_\lambda\ast\IC_0$, then by the decomposition of Lemma 4.4 it is a constituent of no ${}^pH^i(V_\lambda\ast\IC_0)$, and part 2 applies. $\square$

So "$\Csat=\cC_K$" says that for each $\mu$ the multiplicity spaces $E_{\mu,i}(\lambda)$ of Lemma 4.4 are nonzero for some $(\lambda,i)$; it is insensitive to the Grothendieck group (compare the remark on Thomason's theorem in "Three spans compared": thick span versus stable span). The function statement (SG) lives at the level of the Grothendieck group of Weil sheaves (Proposition 4.7).

Builds on

Setup 4.1 · Set-up of the sheaf–function dictionary over $\F_q(\!(t)\!)$: Weil objects, trace functions, obligations D0–D5
Lemma 4.4 · Expansion of $\tr(V_\lambda\ast\IC_0)$ in the IC basis: the coefficients are traces of Frobenius on multiplicity spaces

Built on by

Theorem 4.11 · $\SL_2/\SO_2$: the sign-twisted IC sheaves are not in the Satake span, so $\Csat\ne\cC_K$ over $\overline{\F}_q$
Theorem 4.6 · Theorem: (SG) over $\F_q(\!(t)\!)$ implies $\Csat=\cC_K$ over $\overline{\F}_q$
Proposition 4.7 · The converse fails in general: $\Csat=\cC_K$ gives detection by Hecke translates, not spherical generation
Strategy 8.1 · Strategy for $E_6/F_4$: the fibre over an uncovered orbit lies in the open Schubert cell
Audit 7.4 · Audit of the thick-span criterion and the one-way transfer theorem; the converse is correctly refused
Lemma 8.7 · Detection lemma: a stalk class in degree $\ge-\delta(\mu)$ forces $\IC_\mu$ to split off
Summary 8.10 · Verdict for $E_6/F_4$: $\Csat=\cC_K$, and the claimed divergence from the function level
Correction 10.8 · Correction: the square-class automorphism multiplies $\tr_n\IC_{\chi,\nu}$ by $\chi(-1)$
Theorem (repair of A4-11) X2.4 · Repair of A4-11: $\mathrm{IC}_{\mathrm{sgn},\mu}\notin\mathcal C_K^{\mathrm{Sat}}$ in both $\mathbb F_q$-forms, $q$ odd

Original record, as the agent wrote it at 02:32 UTC

Thick span equals the whole category iff every IC is a direct summand of a Satake object

Lemma A4-5 (answer to (c)). Assume H1–H3 and D1 (perverse heart $\mathcal P$ of $\mathcal C$ of finite length, simple objects $\mathrm{IC}_\mu$, $\mu\in\Lambda_S^+$). (i) For a set $\Sigma$ of simples let $\mathcal S_\Sigma\subseteq\mathcal P$ be the Serre subcategory of objects all of whose Jordan–Hölder constituents lie in $\Sigma$, and $\mathcal D_\Sigma:=\{M\in\mathcal C:{}^p\mathcal H^i(M)\in\mathcal S_\Sigma\ \forall i\}$. Then $\mathcal D_\Sigma$ is a thick subcategory of $\mathcal C$. (ii) If $\mathrm{IC}_\mu$ is not a Jordan–Hölder constituent of any ${}^p\mathcal H^i(V_\lambda\ast\mathrm{IC}_0)$, then $\mathrm{IC}_\mu\notin\mathcal C^{\mathrm{Sat}}$. (iii) Under D2 (A4-4.2): $\mathcal C^{\mathrm{Sat}}=\mathcal C\iff$ every $\mathrm{IC}_\mu$ is a direct summand of some $V_\lambda\ast\mathrm{IC}_0$.

Proof. (i) $0\in\mathcal D_\Sigma$; shifts permute the ${}^p\mathcal H^i$; for a cofibre sequence $A\to B\to C$ the long exact sequence $\cdots\to{}^p\mathcal H^i A\to{}^p\mathcal H^iB\to{}^p\mathcal H^iC\to{}^p\mathcal H^{i+1}A\to\cdots$ and closure of $\mathcal S_\Sigma$ under extensions and subquotients give two-out-of-three; a retract $M$ of $N$ has ${}^p\mathcal H^i(M)$ a retract, hence a subobject, of ${}^p\mathcal H^i(N)\in\mathcal S_\Sigma$. (ii) Take $\Sigma=\{\mathrm{IC}_\nu:\nu\ne\mu\}$: every $V_\lambda\ast\mathrm{IC}_0\in\mathcal D_\Sigma$, so $\mathcal C^{\mathrm{Sat}}\subseteq\mathcal D_\Sigma$ by minimality of the thick span, and ${}^p\mathcal H^0(\mathrm{IC}_\mu)=\mathrm{IC}_\mu\notin\mathcal S_\Sigma$. (iii) $\Leftarrow$: a summand is a retract, so $\mathrm{IC}_\mu\in\mathcal C^{\mathrm{Sat}}$ (this is the role of retracts; cf. Prop. I step 4). Then $\mathcal C=\mathrm{Thick}\{\mathrm{IC}_\mu\}$: by induction on $h(\mu)$, the triangle $a_{\mu!}a_\mu^*\mathrm{IC}_\mu\to\mathrm{IC}_\mu\to i_{\mu*}i_\mu^*\mathrm{IC}_\mu$ with $a_\mu^*\mathrm{IC}_\mu=\underline k[\delta(\mu)]$ and $i_\mu^*\mathrm{IC}_\mu\in\mathrm{Thick}\{\Delta_\nu:\nu<\mu\}$ (Prop. I Lemma "Standard generation") gives $\mathrm{Thick}\{\mathrm{IC}_\nu:\nu\le\mu\}=\mathrm{Thick}\{\Delta_\nu:\nu\le\mu\}$, and the $\Delta_\nu$ generate $\mathcal C$ (Prop. I Lemma). $\Rightarrow$: if $\mathrm{IC}_\mu$ is a summand of no $V_\lambda\ast\mathrm{IC}_0$, then by A4-4.2 it is a constituent of no ${}^p\mathcal H^i(V_\lambda\ast\mathrm{IC}_0)$, and (ii) applies. $\square$

So "$\mathcal C^{\mathrm{Sat}}=\mathcal C$" is a statement about the multiplicity spaces $E_{\mu,i}(\lambda)$ of A4-4 being nonzero for some $(\lambda,i)$, for each $\mu$; it is insensitive to $K_0$ (compare the Thomason remark on thick-spans.html: thick span vs stable span). The function statement (SG) lives at the level of $K_0$ of Weil sheaves (A4-7).

Sources

/Users/miltonlin/Projects/Trees/subpages/case-study-1-ramifiedrelative/proof_graphs/spherical-generation/pdf/proposition-I.tex (Lemma Standard generation, Step 4)
/Users/miltonlin/Projects/Trees/subpages/case-study-1-ramifiedrelative/proof_graphs/spherical-generation/thick-spans.html (Remark after Lemma small)
sources/bbd-perverse.pdf 1.3 (perverse cohomology long exact sequence)

Table 1. Page production by session
	tokens	US $
This conversation: the search, the audit, the pages and propositions, and the VI characteristic-p run01:44–05:19 UTC	802 M	$490
Setup pages16:17–19:42 UTC	37 M	$19
Shareable animation17:16–17:22 UTC	1 M	$1
Proposition IIb: figures and the linear-algebra lemma19:48–20:05 UTC	14 M	$7
Audit pass: the repairs by Agents X1–X4, their re-checks and Propositions VI–VII (with the audit skill)19:35–02:12 UTC	275 M	$113
Proposition IX: converse search, verdict and trace formalism19:56–02:12 UTC	199 M	$106
Audit pass on the VI characteristic-p run; the simplified VIb page and the VI corrections05:21–07:33 UTC	268 M	$107
Removing the detail switch; Proposition IX restated; the hub and verdict reviewed and simplified05:34–15:56 UTC	38 M	$20
C_K for the categorical statements, in the Iwahori run's conversation18:27–18:32 UTC	9 M	$3
Total	1,642 M	$866

Lemma 4.5 · \(\Csat=\cC_K\) if and only if every \(\IC_\mu\) is a direct summand of a Satake object

Builds on

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Thick span equals the whole category iff every IC is a direct summand of a Satake object

Sources