Unit-normalized representatives of Hochschild two-classes #
Every Hochschild two-cocycle μ of internal degree q is cohomologous, by the coboundary of the
degree-q one-cochain h(a) = a μ(1, 1), to a cocycle vanishing when either argument is 1
(exists_unit_normalized). So statements about the classes of HH²(Λ)_q may be checked on
unit-normalized cocycles; this is how the Bockstein statement of Proposition 10.3
(BocksteinStmt) covers every class of HH²(A⁽²⁾)_q.
theorem
MultiplicativeBasisDeformations.GExt.isHCocycle_hcobdry
{R : Type u_1}
[CommRing R]
{Λ : Type u_2}
[Ring Λ]
[Algebra R Λ]
(h : Λ →ₗ[R] Λ)
:
IsHCocycle (hcobdry h)
theorem
MultiplicativeBasisDeformations.GExt.exists_unit_normalized
{R : Type u_1}
[CommRing R]
{Λ : Type u_2}
[Ring Λ]
[Algebra R Λ]
{GΛ : GAlg Λ}
{μ : Λ →ₗ[R] Λ →ₗ[R] Λ}
(hμ : IsHCocycle μ)
{q : ℤ}
(hd : HDeg GΛ μ q)
:
Every Hochschild two-cocycle of internal degree q is cohomologous to a unit-normalized
one: subtract the coboundary of h(a) = a μ(1, 1), of degree q.