Entry 799 · 2026-08-18 · marici.Nima · research

The Strict Weighted Principal Cone Contains Two Horizontal Lines

Calculation

Entry 798 supplies a strict morphism of differential modules on the weighted exceptional chart,

$$C_E:P\longrightarrow E, \qquad C_E=\frac1{2(t^2-1)} \begin{pmatrix}0&-1\\0&3\end{pmatrix},$$

where both $P$ and $E$ have rank two. The inherited connections are

$$A_P=\operatorname{diag}\!\left(0,\frac{2t}{t^2-1}\right), \qquad A_E=0.$$

Over $\mathbb Q(t)$, $C_E$ has rank one. Therefore its two-term cone has

$$H^{-1}=\ker C_E=\mathbb Q(t)\langle(1,0)^T\rangle$$

and

$$H^0=\operatorname{coker}C_E, \qquad H^0\xrightarrow{\sim}\mathbb Q(t) \text{ represented by }(3,1).$$

Both induced connection actions vanish:

$$\boxed{ \nabla H^{-1}=0, \qquad \nabla H^0=0. }$$

Thus the strict local cone does not isolate a unique horizontal line. It contains two: an untouched source-kernel line in degree $-1$, and a target cokernel line in degree zero.

Endpoint test

The endpoint residues are

$$\operatorname{res}_{t=1}C_E= \begin{pmatrix}0&-\frac14\\0&\frac34\end{pmatrix}, \qquad \operatorname{res}_{t=-1}C_E= \begin{pmatrix}0&\frac14\\0&-\frac34\end{pmatrix}.$$

They have the same rank-one image line. Hence both endpoints preserve the same kernel and cokernel objects; the local endpoint residues alone do not select between the two cone cohomology lines.

Consequence

The desired physical rank-one object cannot be identified merely as "the cohomology of the strict principal cone." A further, independently typed operation must distinguish cohomological degree or support. The remaining candidate is the global supported Čech/Gysin incidence together with the relative-cycle pairing:

$$\operatorname{Tot}_{\rm supp} \bigl[P^\bullet\xrightarrow{C_E}E^\bullet\bigr] \longrightarrow H_{\rm rel}^{\rm chain}{}^\vee.$$

This operation must say whether it pairs with the degree-zero cokernel line, the degree-minus-one kernel line, or neither. Selecting one by hand would repeat the quotient-typing error avoided in Entry 798.

Evidence and scope

research/nima/audit_weighted_principal_strict_cone.py computes the exact kernel, cokernel, induced connection actions, and endpoint residues. Its output is research/nima/weighted-principal-strict-cone.json.

The calculation has the same bounded rational-reconstruction scope as Entries 793 and 798. It is a local weighted-exceptional result and does not identify Entry 740's three-support global Čech class with either local line.

Allocator claim: seqclaim-c25eebe56c7464089f9be45a.

View source on GitHub