Lemma 10.2 for a basic representation-finite algebra #
lem:mod-four-comparison (integral-deformations.tex), for A basic and
representation-finite over an algebraically closed field of characteristic two, A⁽²⁾ the
projective corner of the prime-field mesh category and A⁽⁴⁾ = 𝒪P (tab₂): for a normalized
Hochschild two-cocycle μ̄ of A⁽²⁾ of internal degree q > 0, 2μ̃ is a coboundary in the
normalized Hochschild complex of A⁽⁴⁾; and for every mesh weight W (values in any abelian
group, e.g. ℤ^r for simultaneous integer weights) and every character χ of μ̄, the
coboundary can be chosen homogeneous of weight χ for the restricted ray weights (hvW).
The instance arguments are those of the Auslander–Reiten quiver (S.hasAtMostOneArrow,
S.finiteDimensional_meshHom, S.prime_fd).
Lemma 10.2 (lem:mod-four-comparison) for a basic representation-finite algebra in
characteristic two (ModFourStmt): for a normalized Hochschild two-cocycle μ̄ of A⁽²⁾ of
internal degree q > 0, 2μ̃ is a coboundary in the normalized Hochschild complex of A⁽⁴⁾.
Lemma 10.2 in every simultaneous weight (ModFourWeightStmt): for any mesh weight W
(values in any abelian group) and any character χ of μ̄, the normalized coboundary can be
chosen homogeneous of weight χ for the restricted ray weights hvW.
Proposition 10.3 (prop:integral-bockstein) for a basic representation-finite algebra
in characteristic two (BocksteinStmt): the connecting map HH¹(A⁽²⁾)_q → HH²(A⁽²⁾)_q is
surjective for q > 0, with β(δ̄) represented by (dδ)/2 for any lift δ.
Proposition 10.3 in every simultaneous weight (BocksteinWeightStmt).