# Verdict: does recollement need finite unions of orbits?

Origin: Toan, 2026-10-20 08:47, on `solve-maths/satake-two-levels/props/summand.html`, hypothesis H2.

| statement | status | steps |
|---|---|---|
| I. H2 for \((Z,D)\) iff \(Y_N\setminus Z_N\), \(Z_N\setminus D_N\) quasi-compact for all \(N\) | proved for \(\mathbb A^n\); general \(X\) under (P) | A1-2–A1-4, A2-3, A2-6, A2-13 |
| II. \(\{0\}\subseteq t^m\mathcal O^n\): one orbit, no recollement | proved (in-run audit) | A1-5, A2-5 |
| III. Finiteness is for the induction; symmetric finite unions are cylinders | proved (in-run audit) | A1-7, A1-8, A2-9–A2-11 |

## Proposition I: recollement criterion (proved for \(\mathbb A^n\); general \(X\) under (P))
**Assumes.** Levels \(Y_N=\varprojlim_M Y_{N,M}\); constructible = pulled back from a level. For general \(X\): (P) transitions with affine-space fibres, finitely presented ind-transitions.
**Statement.** For closed \(L^+G\)-stable \(D\subseteq Z\subseteq LX\), with \(Z_N=Z\cap Y_N\), \(D_N=D\cap Y_N\), the following are equivalent: (i) \(r_Z^*,r_{Z*},j^*,j_!,j_*,i^*,i_*,i^!\) preserve the constructible categories and \(\mathcal C_D\to\mathcal C_Z\to\mathcal C_U\) is a recollement; (ii) the triangle exists for \(\underline k_{Z_N}\) for all \(N\); (iii) \(Y_N\setminus Z_N\) and \(Z_N\setminus D_N\) are quasi-compact for all \(N\) ("cylinders"); (iv) \(Z_N\), \(D_N\) have finitely many irreducible components, each of finite codimension.
**Proof.** Cylinder ⇔ quasi-compact complement [[A1-2]]; level-wise BBD 1.4 + base change along levels [[A1-3]], Verdier sequence [[A2-3]]; necessity by stalk supports [[A1-4]]; any closed \(Z\) [[A2-13]]; codimension form [[A2-6]].

## Proposition II: a finite union with no recollement (proved)
**Statement.** For \(X=\mathbb A^n\), \(D=\{0\}\subseteq Z=t^m\mathcal O^n\) is a single orbit, not a cylinder; \(\ker(j^*:\mathcal C_Z\to\mathcal C_U)=0\) while \(\mathcal C_{\{0\}}\ne0\); \(\delta_0\), \(j_!\underline k_U\notin\mathcal C\). So "finite union of orbits" is not sufficient; \(t^{m+1}\mathcal O^n\subseteq t^m\mathcal O^n\) (infinitely many orbits, a cylinder) shows it is not necessary.
**Proof.** [[A1-5]], [[A2-5]]. Proposition IV's "H2 is empty" should read "H2 fails at \(\{0\}\)".

## Proposition III: what finiteness is for (proved)
**Statement.** (a) In filtration.html and Proposition I, finiteness is used for the maximal orbit and termination of the inductions on \(n(Z)\), \(h(\mu)\) (W1–W7), for finite support H1(b) (S1–S2), and, through H2, for recollement (R1–R6). (b) For symmetric \(X\), every closed finite union of orbits is a cylinder (Chen–Yi v4, Thm. 36 with A.1(2),(4),(5) and Prop. 17), so R1–R6 follow from Proposition I. (c) For \(\mathbb A^n\), \(\mathrm{IC}_0\notin\operatorname{Thick}\{\Delta_m,\delta_0\}\) in any ambient category with stalks: the failure is the infinitely many orbits, not the recollement.
**Proof.** [[A1-8]], [[A2-10]]; [[A2-11]]; [[A1-7]], [[A2-9]].

## Gaps
- General \(X\): (P) (affine-space fibres), or the \(*\)-colimit definition, for which the transfer is heuristic. BBD §2.2 not checked (scan).

## Proposed edit (H2 on summand.html and filtration.html)
Add after H2: "Recollement itself needs only that \(D\) and \(Z\) be cut out at a finite jet level (\(Z\setminus D\) quasi-compact); for symmetric \(X\) closed finite unions of orbits are such (Chen–Yi Thm. 36). Finiteness is used for the induction (H1)." In affine.html / Proposition IV: "H2 is empty" → "H2 fails at \(\{0\}\)".