Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Isomorphic associated graded objects need not give isomorphic filtered objects
Statement refuted
The assertion “isomorphic associated graded objects imply isomorphic filtered objects” is false. Let A=ℤ/4 and B=(ℤ/2)². Set for p<0, , , and , for p≥1. These filtered objects have isomorphic graded families and nonisomorphic underlying groups.
Facts & Assumptions
Given: The displayed filtrations on A=ℤ/4 and B=(ℤ/2)².
The graded pieces are (Associated graded object of a filtered object).
The indicated groups exist with ordinary subgroup quotients and explicit residues (Abelian-group model for spectral-sequence computations).
Counterexample
The sets {0,2} and (ℤ/2)×0 are subgroups by the additions in [F2], so the specified families are finite increasing filtrations. Their degree-zero pieces are isomorphic via [k]₂↦[2k]₄ for A and [k]₂↦([k]₂,0) for B. The first map is well defined because 2(k+2l)≡2k mod 4, and its two images are exactly 0,2.
The degree-one quotient of A maps to ℤ/2 by : the two cosets are {0,2} and {1,3}, so this is an additive bijection. For B the map is an additive bijection. All other graded pieces are zero, since consecutive filtration pieces agree. Thus the two graded families are isomorphic in each degree.
Every element of B is killed by 2, whereas [1]₄ in A is not: 2[1]₄=[2]₄≠0. Any additive isomorphism f:A→B would satisfy f([2]₄)=2f([1]₄)=0, contrary to injectivity. Thus no underlying or filtered-group isomorphism exists.
Source notes
Weibel, §5.2 bounded abutment and extension interpretation; the two finite-group filtrations and all quotient calculations are supplied explicitly here.