Galois Cohomology Does Not Truncate: Total Collapse of the Finite-Precision Herr Complex and the Exact Euler-Characteristic Barrier

Abstract

Fix a prime p. The integral cyclotomic period ring truncates to the finite rings Λ_N,m = (Z/p^N)[T]/(T^m) with φ(T) = (1+T)^p − 1 and γ(T) = (1+T)^c − 1: this tower is the inevitable home of any attempt to compute local Galois cohomology by exact finite arithmetic on the integral side of the period ring. We prove that the attempt computes nothing beyond its own residue fibre. For every free (φ, γ)-module D over Λ_N,m the Herr complex of the submodule T·D is acyclic, and reduction modulo T is a quasi-isomorphism onto the Koszul complex of the pair of commuting matrices (A_φ, A_γ) over Z/p^N. The cohomology tower is therefore constant in the window m: stabilization is immediate at the first admissible window, uniformly in p, N, the rank, and the module, and the essential image of the theory is exactly the Koszul cohomology of commuting matrix pairs over Z/p^N — no more, no less. Every such complex has Euler characteristic zero, while Tate’s formula gives every nonzero finite Galois module the Euler characteristic −[K : Q_p]·ℓ(V) < 0: consequently no object of the truncated integral layer, at any precision (N, m) and any rank, has Herr cohomology abstractly isomorphic to H•(G_K, V). Already for the trivial module the truncated layer reproduces H^0 and H^1 of Z/p^N exactly and then manufactures a phantom H^2 of full length N. The entire local Euler characteristic of G_K is created by inverting T; no part of it exists on the integral side of the boundary. Every statement, down to the individual graded blocks, is verified by exhaustive exact computation over Z/p^N.