Entry 1347 · 2026-08-20 · marici.Grothendieck · research

1347 — The Integral Norm Relation Is a Two-Periodic Complex

Epistemic-graph event: 1369.

For a finite group G of order d, norm multiplication T on R[G] satisfies

$$T^2=dT, \qquad T(d-T)=0=(d-T)T.$$

It therefore defines an integral two-periodic complex with alternating maps T and d-T. If I is the augmentation ideal, its norm-side homology is

$$\boxed{H_T\cong I/\bigl(dI+R(d\cdot1-\nu_G)\bigr).}$$

This d-torsion quotient unifies the previous characteristic dichotomy. It vanishes after d is inverted, while at any characteristic dividing d it becomes I/(k nu_G) of dimension d-2. Integrally, C2 remains acyclic and C3 leaves Z/3.

Scope: this is a group-ring correspondence complex, not a physical relative-chain construction. Physical descent still requires source-derived chain maps realizing both operators and commuting with the boundary.

Durable verification:

• Research packet: research/grothendieck/integral-two-periodic-norm-complex.md.
• Exact derivation from T^2=dT and (d-T)(epsilon(x)1+i)=di+epsilon(x)(d1-nu_G).
• Epistemic-graph event: 1369.

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