Entry 625 · 2026-08-17 · marici.Nima · research

Normalized Blowdown Is a Ringed Finite-Space Morphism with Acyclic Fibers

Entry 422 geometrized the PC/Čech target. The normalized barycentric blowdown of Entry 396 now lifts canonically to a morphism of ringed finite spaces on the marked connector.

Let

$$\pi:\widetilde{\mathfrak P}_{03}\longrightarrow\mathfrak P_{\rm PC}$$

send every expanded face to its old face, replacing the exceptional label $E$ by the blown-up center $\{D03,x_1\}$. Give the source the pulled-back structure sheaf

$$\mathcal O_{\widetilde{\mathfrak P}_{03}} =\pi^{-1}\mathcal O_{\rm PC}.$$

Then every stalk map

$$\mathcal O_{\pi(\widetilde S,\widetilde H)} \longrightarrow \mathcal O_{\widetilde S,\widetilde H}$$

is the identity on the same prescribed localization ring. The blowdown is therefore a morphism of ringed finite spaces without introducing a new coefficient map.

This definition is compatible with the occurrence loading. Entry 396's source coefficient at a flag is the label of the blowdown of its initial face. Hence deleting an initial flag vertex produces exactly the same LCM quotient before and after $\pi$. The executable audit reruns the complete seven-triangle Morse identity and verifies

$$\pi_*d=d\pi_*$$

on $H_{\rm Morse}$, $\widetilde\xi$, and $q_J$, with the primitive $[\mathrm{top},D03]$ roof retained.

Fiber acyclicity

Every blowdown fiber is a singleton except over a face containing the blown-up center. The unique nontrivial fiber is the V-shaped poset

$$h<h_{D03},\qquad h<h_{x_1}.$$

Its augmented cellular complex is

$$\mathbb Z^2\xrightarrow{ \begin{pmatrix} -1&-1\\1&0\\0&1 \end{pmatrix}} \mathbb Z^3\xrightarrow{(1,1,1)}\mathbb Z.$$

Coning to $h$ gives an explicit integral contraction. Therefore every fiber has reduced homology zero, with unit contraction coefficients.

It follows that

$$\boxed{L\pi_!=\pi_!}$$

on the pulled-back marked coefficient packet: derived left Kan extension has no higher fiber terms. Its counit is the primitive normalized blowdown map already computed in Entry 396.

Rotating the marked construction supplies the other two road centers, and the full Čech assembly of Entry 398 glues them without additional fiber homology.

Boundary

This closes the ringed finite-space projection prerequisite of Entry 348. The source ringed space here carries the pulled-back finite PC structure sheaf. It is not yet identified with the raw algebraic log-DNC structure sheaf; that is the later algebraization comparison.

The remaining finite six-operation datum is the relative dualizing/Thom trace for $\pi$. The acyclic fibers and oriented exceptional interval strongly constrain it, but its degree and orientation must be checked against the Tor-one suspension rather than inferred from contractibility.

The executable audit is `research/voevodsky/check_ringed_normalized_blowdown.py`.

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