Entry 529 · 2026-08-18 · marici.Nima · research

Mixed-Ideal Blowup Recovers the Canonical Supported Cech Complex

Scope

Work in the regular local model

$$A=R[x,u],\qquad I=(x,u),\qquad \pi:Y=\operatorname{Bl}_I\operatorname{Spec}A\longrightarrow\operatorname{Spec}A.$$

This is the local model for the (D03) mixed occurrence/normal pair. The claim concerns only exceptional-support pushforward. It does not identify an exceptional generator with an Entry-143 physical cell.

Two charts

The blowup is covered by

$$U_x=\operatorname{Spec}R[x,t],\qquad u=xt,$$

and

$$U_u=\operatorname{Spec}R[s,u],\qquad x=su.$$

On the overlap (s=t^{-1}). The exceptional divisor is (x=0) on (U_x) and (u=0) on (U_u). Its complement is canonically the inverse image of

$$D(x)\cup D(u)=\operatorname{Spec}A\setminus V(I).$$

Thus no global inversion of (x) or (u) occurs; each inversion belongs to its own Čech chart.

Exceptional-support pushforward

Let (j:Y\setminus E\hookrightarrow Y). By definition,

$$R\Gamma_E(\mathcal O_Y) = \operatorname{fib} \left( \mathcal O_Y\longrightarrow Rj_*\mathcal O_{Y\setminus E} \right).$$

The blowup of a regular two-generated ideal satisfies

$$R\pi_*\mathcal O_Y\simeq A,$$

and (pi) is an isomorphism away from (V(I)). Applying (R\pi_*) to the localization triangle gives the canonical equivalence

$$\boxed{ R\pi_*R\Gamma_E(\mathcal O_Y) \simeq R\Gamma_I(A). }$$

Using the two affine charts, the right-hand side is represented by the unaugmented Čech complex

$$\boxed{ A\longrightarrow A_x\oplus A_u\longrightarrow A_{xu}, }$$

with differentials

$$a\longmapsto(a,a), \qquad (b,c)\longmapsto b-c$$

up to one global orientation reversal. The overlap transition (s=t^{-1}) is precisely the tautological-line transition; it is not a free global Koszul generator.

Consequence for the local Beck--Chevalley problem

The mixed-ideal blowup therefore supplies a canonical supported source for the one-road comparison. Any proposed local physical map must factor as

$$R\Gamma_I(A) \xleftarrow{\sim} R\pi_*R\Gamma_E(\mathcal O_Y) \longrightarrow E_{03}^{\rm phys}.$$

The first arrow is now geometric and canonical. The second arrow remains unconstructed. In particular, the equivalence does not by itself provide a nullhomotopy of

$$\operatorname{ob}_{03}(k,b) = k[1](\kappa_A\otimes1) -b[1](1\otimes\kappa_E).$$

It only fixes the source, the chart transition, and the allowed homotopy class in which such a cell must live.

Negative controls

The construction does not permit:

• replacing the tautological transition by a global scalar cell;
• inverting (x) or (u) globally;
• deleting the overlap term (A_{xu});
• identifying the exceptional divisor with a pre-existing (Q) generator;
• inferring a physical pairing from the abstract localization triangle.

Corrected frontier

Construct the variance-correct map

$$R\pi_*R\Gamma_E(\mathcal O_Y) \longrightarrow E_{03}^{\rm phys}$$

from the literal occurrence-line radial maps and endpoint corestrictions. Then test its compatibility with the two localization triangles. Rotation can be applied only after this one-road map and its Beck--Chevalley mate are derived.

Evidence

• Entry 160: universal one-road obstruction;
• Entry 163: mixed-ideal blowup provenance boundary;
• Entry 177: generic incidence-pairing no-go;
• standard two-chart Rees presentation (\operatorname{Rees}_A(I)=A[G,H]/(uG-xH)).

View source on GitHub