{
  "A": {
    "complex": "C = [Z^2 --(2, 3)--> Z] in degrees 0, 1, standard bases",
    "H0": "Z · (3, -2)",
    "H1": "0",
    "canonical_basis_of_det^{-1}(C)": "e1 ∧ e2 (x) f^{-1}",
    "canonical_isomorphism": "det^{-1}(C) ≅ det H^0 (x) det^{-1} H^1 from 0 → H^0 → Z^2 → Z → H^1 → 0",
    "lift_of_f": [
      -1,
      1
    ],
    "det[k; s]": 1,
    "image_of_the_basis": "+(3, -2), a generator of det H^0",
    "accepted_as_a_canonical_nonzero_element": true,
    "rule": "a basis of the determinant of a perfect complex, carried by a canonical isomorphism, is accepted at once; no explicit vector is demanded"
  },
  "B": {
    "data": "only the equality of characteristic ideals (6) = (6) between Z/6 and a module with the same Fitting ideal; no map, no line",
    "generators_of_the_ideal": [
      -6,
      6
    ],
    "a_distinguished_generator_exists": false,
    "PASS_granted": false,
    "rule": "an ideal equality names no element: over Lambda the ambiguity is Lambda^x; PASS is refused until a map such as (2.9) singles one out"
  },
  "agrees": true
}