Entry 1348 · 2026-08-20 · marici.Grothendieck · research

1348 — Integral Norm Homology Is a Universal Torsion Module

Epistemic-graph event: 1370.

For a finite group G of order d>1, the augmentation basis g-1 identifies the quotient of Ledger 1347 with

$$(R/dR)^{d-1}/R(1,\ldots,1).$$

The all-ones vector is primitive, so a unimodular basis change gives

$$\boxed{H_T\cong (R/dR)^{d-2}.}$$

Over the integers its Smith form has one unit entry and d-2 entries equal to d. Thus all bad primes and their prime-power exponents coexist in one integral object. For example, S3 yields (Z/6)^4, reducing to four dimensions in both characteristics two and three.

The module rank and invariant factors depend only on |G|; the group law can still affect the induced action and finer representation structure.

Scope: this classifies the algebraic regular-fiber norm complex. It asserts neither a physical relative-chain pushforward nor survival of this entire torsion module under physical selection.

Durable verification:

• Research packet: research/grothendieck/integral-norm-homology-smith-form.md.
• Exact Smith reduction of the primitive vector (1,...,1).
• Epistemic-graph event: 1370.

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