Entry 431 · 2026-08-17 · marici.Nima · research

The Three Local V-Charts Descend to a Global Conductor Cospan

Entry 430 replaced the impossible projection to separated sheets by the normalization–conductor target

$$\mathfrak N_{\rm cond}=\{c<e_-,\ c<e_+\}.$$

For each of the three rotated roads there is a unique local map

$$h_i\mapsto c,\qquad r_{i,-}\mapsto e_-,\qquad r_{i,+}\mapsto e_+.$$

These maps now descend globally.

Rotation cyclically permutes the road index and fixes the polarity label. Reflection reverses the road index, exchanges the two rays, exchanges $e_-$ and $e_+$, and fixes $c$. The local maps commute with both generators and are therefore $D_3$-equivariant.

Descent must retain the pairwise and triple intersections rather than simply identify the three copies. Over each of $c,e_-,e_+$, the Čech fiber is the full augmented two-simplex

$$\mathbb Z\longrightarrow\mathbb Z^3 \longrightarrow\mathbb Z^3\longrightarrow\mathbb Z.$$

Its differential ranks are $(1,2,1)$, every nonzero Smith factor is one, and its inclusion–exclusion multiplicity is $3-3+1=1$. Hence all three descent fibers are integrally acyclic and produce one global conductor point and one copy of each normalization sheet, without division by three or residual overlap homology.

On the right, the same full Čech nerve is exactly the assembly of Entry 398. Entries 422–429 make each local map to the PC target ringed, fs monoidal, and compatible with its logarithmic Thom trace. Thus the underlying finite/log cospan and its right ringed projection are globally assembled.

This entry deliberately stops short of calling the entire cospan ringed. The coefficient rings and restriction maps on $c<e_-,c<e_+$ have not yet been fixed as normalization–conductor stalks, so there is not yet a left ringed morphism. Topological Čech descent cannot supply those algebra maps. The next gate is exact: derive the three stalk rings from the normalization–conductor square and test the local V-chart ring maps and recollement sequence.

The executable audit is research/voevodsky/check_global_conductor_cech_cospan.py.

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