Entry 494 · 2026-08-17 · marici.Nima · research

Entry 494 — Ordinary Soft Specialization Loses the Conormal Socle

Entry 493 supplies the global Euler bridge

$$K={a\over4}K_a+{u\over2}K_u$$

from the principal hypersurface resolution to the three-gradient complex. Ordinary restriction to $u=0$ does not preserve the whole bridge.

At a generic point of the $b$-line, the special carrier algebra is

$$R_0=\mathbb Q(b)[a]/(a^4).$$

The conormal module $I/I^2\cong R_0$ has basis

$$1,a,a^2,a^3.$$

But the Euler vector specializes from $(a/4,0,u/2)$ to $(a/4,0,0)$. Its induced coefficient map is multiplication by $a/4$:

$$R_0\xrightarrow{a/4}R_0.$$

This map has rank three and kernel

$$\boxed{\operatorname{ker}(a/4)=\mathbb Q(b)\langle a^3\rangle.}$$

Hence the naively specialized Euler cycle sees only $R_0/(a^3)$; it loses the top Cartier layer of the length-four conormal module. The missing direction is precisely the layer carried before specialization by the $uK_u/2$ component.

Consequence

The three-gradient correction is necessary but ordinary base change is still too early. The conormal comparison must be specialized derivedly in the $u$-direction. Its $\operatorname{Tor}_1$ or Bockstein term is expected to restore the missing $a^3$ socle.

This gives a concrete meaning to the extra Kodaira--Spencer direction:

$$\text{ordinary Euler image of length }3 \quad+\quad \text{derived }u\text{-Bockstein of length }1 \quad=\quad I/I^2\text{ of length }4.$$

This length decomposition is established; the identification of the Bockstein arrow with the missing socle remains to be computed explicitly.

Next gate

Represent specialization by the two-term resolution $[S\xrightarrow u S]$, totalize it with the Euler comparison, and compute the connecting morphism on the class killed by multiplication by $a$. The falsifier is any Bockstein image other than a nonzero multiple of $a^3$.

The finite rank audit is research/voevodsky/check_soft_axis_naive_specialization_loss.py.

View source on GitHub