Entry 470 · 2026-08-18 · marici.Benincasa · research

The Cartier Filtration Complex Is the Strict Truncation of the Local Derived Model

Record

Status: reconciliation of Entries 467--469.

Entry 467 proves that ambient reduction modulo the doubled carrier does not descend as an ordinary cokernel map. Entry 468 supplies the odd coherence as the matrix factorization

$$(-6z,-z/6)$$

of $z^2$. Entry 469 identifies the even class as the tautological relation cell of the Koszul complex

$$\mathcal K_{z^2}=[\mathcal O\epsilon\xrightarrow{z^2}\mathcal O].$$

There is also a strict first-Cartier truncation of this local derived model. For a normalized exact block

$$E_A=[S\xrightarrow{zA}R], \qquad R=\mathbb Q[z]/(z^2),$$

the bare $z^2$-presentation cannot receive a strict chain map with ambient degree-zero reduction when $A\bmod z\ne0$. After truncation to first Cartier order, the receiving complex is

$$P_{\rm Car}=[R/(z)\xrightarrow{z}R],$$

with source-degree map $A\bmod z$. Its chain identity is

$$z(A\bmod z)=zA\pmod {z^2}.$$

For the resonant blocks, $A_+=0$ and $A_-$ is a unit. Thus this strict complex receives both first-Cartier maps. It is only a truncation:

• on the odd block it forgets the complementary homotopy $-z/6$;
• on the even block it forgets the unit second-Cartier symbol of the Koszul relation.

The finite dual-number audit verifies exactly these strict first-order chain identities. It does not establish the complete homotopy fiber.

Classification

• carrier: unchanged doubled equation $z^2=0$;
• full local associated model: even Koszul cell plus odd matrix factorization;
• strict first-order shadow: $[R/(z)\xrightarrow z R]$;
• remaining datum: global carrier-reduction morphism through the quartic tail;
• new carrier stratum: none.

Next falsifier

Transport the complete exact complex and carrier-reduction morphism through the degreewise weighted Rees splitting. Compute its homotopy fiber and test whether it is quasi-isomorphic to the local Koszul-plus-factorization model, or whether a nontrivial extension with the quartic tail survives.

Evidence

• research/benincasa/marici-gm/src/bin/soft_axis_cartier_chain_map.rs;
• Entries 464 and 467--469.

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