Entry 496 · 2026-08-17 · marici.Nima · research

Entry 496 — The Conormal Bockstein Vanishes at Both Endpoints

Entry 495 identifies the generic soft Bockstein with

$$-{1\over2}[a^3e_u].$$

Its endpoint behavior is controlled by the $e_a\wedge e_u$ Koszul boundary. Writing $c=1-b^2$, one has

$$d(e_a\wedge e_u)=-ca^2e_a+4a^3e_u.$$

Therefore the family relation in homology is

$$\boxed{4[a^3e_u]=c[a^2e_a].}$$

The Bockstein is consequently

$$\beta(a^3)=-{c\over8}[a^2e_a].$$

This expression is regular along the entire $b$-line; there is no pole or residue. But at either endpoint $b=\pm1$, where $c=0$,

$$[a^3e_u]=0.$$

Equivalently, at the endpoint

$$d\left({1\over4}e_a\wedge e_u\right)=a^3e_u.$$

Thus the generic conormal Bockstein becomes an ordinary Koszul boundary on both endpoint fibers.

Consequence

The global invariant defect of Entry 473 cannot be the ordinary endpoint restriction of the generic conormal Bockstein. The class extends regularly but vanishes at the boundary. Any globally retained endpoint datum must therefore live in a nearby-cycle, relative-support, or derived-base-change term that remembers the first normal derivative in $c$.

The relevant normalized class is

$${\beta(a^3)\over c}=-{1\over8}[a^2e_a],$$

which is regular before setting $c=0$. This is not an ordinary fiber class; it is conormal to the endpoint divisor.

Next gate

Base-change derivedly along $c=1-b^2=0$, retaining the two-term endpoint resolution $[\mathcal O\xrightarrow c\mathcal O]$. Compute its connecting class and test whether it is the single invariant defect of Entry 473. A rank other than one, or opposite signs at the two endpoint branches after orientation is included, falsifies the identification.

The family relation and both endpoint specializations are checked by research/voevodsky/check_soft_axis_bockstein_endpoint.py.

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