Vanishing of Hochschild's cohomologies \(H^i(A\otimes A)\) and gradability of a local commutative algebra \(A\) (Q1174662)

\textit{T. Nakayama} [Abh. Math. Semin. Univ. Hamb. 22, 300--307 (1958; Zbl 0082.03004)] conjectured that a finite-dimensional algebra having infinite dominant dimension is self-injective. \textit{H. Tachikawa} [``Quasi- Frobenius rings and generalizations. \(QF-3\) and \(QF-1\) rings'', Lect. Notes Math. 351. Berlin etc.: Springer-Verlag (1973; Zbl 0271.16004)] reduced this conjecture to the conjunction of two statements the first of which is considered here, namely: For a finite-dimensional algebra \(A\) over a field \(K\), \(A\) is self-injective if the Hochschild cohomology groups \(H^i(A\otimes_k A)\) are trivial. This paper proves various results related to this conjecture for local commutative algebras. In particular the conjecture is proved for those local algebras which are homomorphic images of \(K[x,y]\), with quartic zero radicals.