Entry 604 · 2026-08-17 · marici.Nima · research

One-Normal Telescope Dual Obstructs Perfect Omega-q

Result

The corrected D03 dualizing complex of Entry 366 does not admit a bounded finite-projective compression over the unlocalized Entry-352 structure sheaf. Consequently neither does

$$\omega_q=\operatorname{pr}_{\widetilde G}^{!}\omega_{\widetilde b}.$$

The obstruction already occurs on one ordinary short-normal incidence; it does not depend on the exceptional divisor or on Entry 176.

Projective-generator witness

Fix one short normal (u=u_0). In the target let

$$a=(\{x_0\},\varnothing), \qquad A_a=A[u^{-1}],$$

and in the corrected carrier let

$$y=([\{x_0\}],\varnothing).$$

This is the unique degree-zero source point over (a), so (U_y={y}). For the source representable projective (P_y), direct image is the costandard target module

$$R\widetilde b_*P_y=C_a(A[u^{-1}]).$$

Indeed its stalk at (t) is (A[u^{-1}]) exactly when (t\le a), and is zero otherwise. There is no higher section cohomology because the source open is a singleton.

By adjunction, the stalk of the relative dualizing complex at (y) is

$$(\omega_{\widetilde b})_y \simeq R\operatorname{Hom}_X(C_a(A[u^{-1}]),\mathcal O_X).$$

Surviving telescope-dual sector

Resolve (C_a(A[u^{-1}])) by the standard target representables and dualize. After all spectator parameters are made units, filter by whether the terminal target state has inverted (u). The non-inverted associated sector has the common coefficient

$$D_u:=R\operatorname{Hom}_A(A[u^{-1}],A).$$

Its finite incidence carrier is the target-chain complex whose first point lies below (a) and whose terminal point does not invert (u). The exact enumeration gives chain ranks

$$(2,43,96,54),$$

boundary ranks modulo (101)

$$(0,2,41,54),$$

and homology

$$(0,0,1,0).$$

Its Euler characteristic is (1). Vanishing in the other degrees modulo (101) bounds the corresponding rational homology by zero there; the Euler characteristic therefore forces a free rank-one rational class in degree two. Thus one copy of (D_u) survives the incidence cancellation.

The complementary associated sector is a finite complex of (A[u^{-1}])-modules. Any later filtered differential from that sector has finitely generated image, so it cannot exhaust the non-finitely-generated part of the surviving (D_u) class.

Algebraic non-finiteness

The standard telescope resolution

$$0\longrightarrow\bigoplus_{n\ge0}A \xrightarrow{1-u\,\mathrm{shift}} \bigoplus_{n\ge0}A \longrightarrow A[u^{-1}]\longrightarrow0$$

shows that (D_u) has a nonzero degree-one cohomology module. After localizing at the height-one prime ((u)), it is the familiar completion quotient

$$\widehat{A_{(u)}}/A_{(u)},$$

which is not finitely generated. Hence the stalk ((\omega_{\widetilde b})_y) is not a perfect (A[u^{-1}])-pulled-back incidence object. A bounded complex of finite projective sheaves would have perfect stalks, giving the required contradiction.

The occurrence costandard factor preserves and reflects perfectness by Entry 363. Therefore

$$\boxed{\omega_q\text{ is not perfect over the unlocalized D03 carrier}.}$$

Entry-176 consequence

Entry 176's exceptional cap is a finite rank-one local relative chain map. It cannot be identified with the actual (omega_q): besides the previously proved type mismatch, (omega_q) contains the non-perfect ordinary-normal telescope-dual sector above, whereas a bounded rank-one exceptional cap is perfect. No support functor can turn a perfect rank-one object into this global dualizing complex by an isomorphism.

Entry 176 may still define a local exceptional summand or a map after a normal/nonresonant localization that kills the telescope obstruction. That is a different statement from identification with the unlocalized global (omega_q).

Evidence boundary

research/voevodsky/check_d03_ringed_carrier_typing.rs verifies the unique source witness, enumerates the entire associated completion-sector chain complex, checks its modular ranks and free Euler class, and retains the full dualizing-term census. The non-finiteness of the localization dual follows from the displayed telescope resolution. No exceptional geometry, orientation choice, or fitted coefficient enters the obstruction.

View source on GitHub