Pair-factor matrix factorization certificate (finite 3^m truncation)

Ring: R_m = Z/243Z[eta]/(eta(eta-3)). This is the finite approximation of the pair factor R = Z_3[eta]/(eta(eta-3)) used for machine verification.

Brute-force ring identities

statement	result
eta_eta_minus_3_zero	True
ker_eta_equals_I3	True
im_eta_equals_I0	True
ker_eta_minus_3_equals_I0	True
im_eta_minus_3_equals_I3	True
ann_I0_is_I3	True
ann_I3_is_I0	True

2-periodic MF data

For B0 = R/(eta), maps are ... -> R -(eta-3)-> R -(eta)-> R -> B0 -> 0.

• exact_B0_ending = True
• ker_map_eta_in_R = True
• im_eta_equals_I0 = True
• image_of_quotient_to_R_equals_I0 = True
• injection_RmodI3_to_I0_via_eta = True

For B3 = R/(eta-3), maps are ... -> R -(eta)-> R -(eta-3)-> R -> B3 -> 0.

• exact_B3_ending = True
• ker_map_eta_minus_3_in_R = True
• im_eta_minus_3_equals_I3 = True
• map_from_RmodI0_into_R_has_correct_rank = True

Categorical consequence

The ideal equalities imply ker(eta)=I3 and ker(eta-3)=I0, hence the resolutions are exact in the 2-periodic sense. From the standard hypersurface theorem (Buchweitz/Orlov), these give matrix factorizations (eta, eta-3) and (eta-3, eta) and the pair of nonfree CM modules B0 = R/(eta), B3 = R/(eta-3) with Omega(B0) ≅ B3, Omega(B3) ≅ B0 and Sigma^2 = Id.

Numerically on this truncation: - |R_m| = 59049 - |B0| = |R/I0| = 243 - |B3| = |R/I3| = 243

JSON outputs

Machine outputs are in hecke_pair_matrix_factorization.json.