Entry 492 · 2026-08-17 · marici.Nima · research

Entry 492 — The Gradient Homotopy Complex Does Not Supply the Conormal Cell

Entry 491 proposed recovering the even conormal cell as the degree-minus-one fiber of the lifted carrier map into the two-term gradient complex of Entry 487. The proposed identification is not well typed.

Two different Koszul constructions

Entry 487 uses

$$G_K=\left[\mathcal O^{\oplus2} \xrightarrow{(K_a,K_b)}\mathcal O/(K)\right]$$

to nullhomotope the residues of the complete exact operators. Indeed

$$H_p=(0,3m/2),\qquad H_q=(-3m/2,0)$$

map to $3mK_b/2$ and $-3mK_a/2$. This is the correct complex for the gradient obstruction.

But its degree-minus-one kernel contains, for every coefficient $f$,

$$f(-K_b,K_a).$$

Thus the unquotiented two-term kernel is a coefficient-sized gradient syzygy module. It cannot canonically equal the single principal conormal line.

The conormal cell instead comes from the derived self-intersection of the principal hypersurface. If $S$ is the ambient ring, $R=S/(K)$, and $I=(K)$, tensoring the resolution

$$[S\xrightarrow{K}S]$$

with $R$ gives

$$[R\xrightarrow{0}R], \qquad H^{-1}\cong R\cong I/I^2.$$

Therefore

$$\boxed{ \text{gradient nullhomotopy data}\ne \text{principal conormal data}. }$$

Consequence

Entry 491's flatness result remains valid: the ordinary generic even source is $(\mathbb Q[u]/(u^2))^{\oplus2}$. What fails is only its proposed next identification. Taking the homotopy fiber against $G_K$ without retaining the hypersurface resolution would mix the universal gradient syzygies with the desired Cartier relation.

The derived carrier comparison must therefore be a bicomplex (or an equivalent iterated fiber) retaining both structures:

1. the principal complex $[S\xrightarrow K S]$, which supplies $I/I^2$;
2. the gradient complex, which supplies the canonical homotopies of Entry 487.

No additional carrier geometry is required; the correction is entirely in the coefficient complex.

Next gate

Construct the comparison as an iterated fiber: first lift ordinary carrier reduction through the gradient homotopies, then base-change the resulting map along the principal hypersurface resolution. Test whether the even fiber is exactly one copy of $I/I^2$, with no residual gradient-syzygy class.

The symbolic type check is research/voevodsky/check_soft_axis_gradient_vs_conormal.py.

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