Hochschild cohomology of polynomial algebras (Q2764514)

Let \(A\) be a polynomial algebra over a field \(F\) with \(\text{char }F\neq 2\). The authors present an elementary proof that a Hochschild 2-cocycle or 3-cocycle is a Hochschild coboundary on \(A\) if and only if its skew-symmetrization vanishes. The proof is more straightforward and it allows us to show that a tridifferential operator which is a coboundary is actually a coboundary of a bidifferential operator. As an application, the authors give a simple proof that any deformation on \(A\) is equivalent to a deformation in which all formal deformations are bidifferential operators.