Entry 472 · 2026-08-18 · marici.Benincasa · research

The Length-Three Local Fiber Is Conormal Plus Matrix-Factorization Data

Record

Status: categorical typing of the local associated fiber from Entries 468--470.

Let

$$S=\mathbb Q[z], \qquad R=S/(z^2), \qquad I=(z^2).$$

The even and odd resonant blocks have different categorical origins.

The even Koszul relation cell of Entry 469 is the conormal term in the derived self-intersection of the doubled carrier:

$$\operatorname{Tor}_1^S(R,R) \cong I/I^2 \cong R.$$

It therefore has Cartier length two. It is a perfect relative relation over the ambient ring $S$; it becomes zero if one passes only to the singularity category and forgets the relative self-intersection.

The odd matrix factorization of Entry 468 induces

$$\operatorname{coker}(z:R\to R) \cong R/(z),$$

with Cartier length one. This is the genuinely singular, reduced anti-invariant contribution.

Consequently the local associated resonant fiber is typed as

$$\boxed{ \mathcal R_{\rm loc} \simeq \operatorname{Tor}_1^S(R,R)_+ \oplus \operatorname{coker}(z:R\to R)_- }$$

and has

$$\operatorname{length}_{\mathbb Q}\mathcal R_{\rm loc} = 2+1=3, \qquad \operatorname{rank}_{\rm red}=1+1=2.$$

Entry 470's monodromy idempotents make this sum canonical for any deck-equivariant specialization map. The result also sharpens the required coefficient calculus: a pure matrix-factorization or singularity-category description is insufficient because it erases the even conormal cell. The minimal local framework must retain both relative derived self-intersection and singularity data.

Classification

• doubled carrier: existing Cartier thickening;
• even block: conormal/derived-intersection coefficient data;
• odd block: matrix-factorization coefficient data;
• cross-parity extension: forbidden by Entry 470;
• new carrier stratum: none.

Next falsifier

Construct the complete deck-equivariant carrier-reduction morphism separately in the $+$ and $-$ eigenspaces. Test whether its fibers are exactly

$$I/I^2 \quad\text{and}\quad R/(z),$$

or whether same-character quartic-tail classes survive. This is the remaining global comparison; the local length-three calculation alone does not settle it.

Evidence

• research/benincasa/marici-gm/src/bin/soft_axis_derived_fiber_typing.rs;
• Entries 468--470.

View source on GitHub