Entry 530 · 2026-08-18 · marici.Nima · research

The Koszul Gluing Cell Is the Supported Cech Fundamental Class

Correction

The target-side reciprocal promotion is not missing. The global Koszul--Čech checker proves that the 215-generator Entry-143 packet has a canonical extended Čech realization, that the endpoint/(Q) filtration remains strict, and that each (u_a^{-1}) occurs only in its legal (S\setminus H) summand.

The remaining problem is the source-to-target supported comparison.

Canonical two-variable comparison

Let

$$A=R[x,u].$$

Use the cohomological Koszul complex

$$K(x,u):\qquad A\xrightarrow{d_0}A^2\xrightarrow{d_1}A,$$

with

$$d_0(a)=(xa,ua), \qquad d_1(b,c)=ub-xc.$$

Use the Čech complex

$$C(x,u):\qquad A\xrightarrow{\delta_0}A_x\oplus A_u \xrightarrow{\delta_1}A_{xu},$$

with

$$\delta_0(a)=(a,a), \qquad \delta_1(B,C)=B-C.$$

There is a canonical chain map

$$\Phi:K(x,u)\longrightarrow C(x,u)$$

given by

$$\Phi_0(a)=a, \qquad \Phi_1(b,c)=\left(\frac b x,\frac c u\right), \qquad \Phi_2(a)=\frac a{xu}.$$

Indeed,

$$\Phi_1d_0(a)=(a,a)=\delta_0\Phi_0(a),$$

and

$$\Phi_2d_1(b,c) =\frac{ub-xc}{xu} =\frac b x-\frac c u =\delta_1\Phi_1(b,c).$$

Interpretation of the former gluing cell

Entry 163's formal generator satisfied

$$d\lambda=ug-xh.$$

Under the comparison above, its geometric replacement is the supported Čech top

$$\boxed{ \lambda_{\rm geom}=\left[\frac1{xu}\right]. }$$

Its two chart faces are (1/x) and (1/u), and its boundary is exactly the mixed Koszul syzygy. Thus the missing cell is neither a new global free generator nor an existing absolute (Q)-cell. It is the fundamental class of the mixed support, visible only in the supported Čech category.

This also matches the mixed-ideal blowup of Entry 529: the overlap (A_{xu}) records the tautological transition between its two affine charts.

What is proved

The coefficient-level source comparison is now canonical:

$$K(x_3,u_3) \longrightarrow R\Gamma_{(x_3,u_3)}(A).$$

It retains the overlap term, uses only chartwise inversions, and realizes the primitive mixed syzygy without changing the absolute target.

Remaining physical gate

The global Entry-143 Čech promotion has many labelled summands. To obtain a physical (D03) Beck--Chevalley cell one must still prove that

$$\frac1{x_3},\qquad \frac1{u_3},\qquad \frac1{x_3u_3}$$

land respectively in the prescribed occurrence, normal, and mixed-overlap summands of the literal (D03) corridor, with its endpoint corestrictions and reflection signs. The displayed chain map proves the local algebra, not that label-sensitive realization.

Consequence

Forgetting support sends the Čech fundamental class into the contractible absolute packet, explaining simultaneously:

• the absolute nullhomotopy of Entry 163;
• the nonzero associated supported grade;
• why adjoining a free scalar (lambda) was mistyped;
• why a physical class, if it exists, must be established before forgetting support.

Evidence

• Entry 163: formal mixed syzygy and blowup provenance boundary;
• Entry 529: exceptional-support pushforward and the two-chart Čech source;
• research/voevodsky/check_global_k6_koszul_cech_promotion.rs: canonical target-side extended Čech realization.

View source on GitHub