On a derived equivalence classification of weakly symmetric algebras of Euclidean type

We consider a family $({\mathrm{\Lambda }}_{\lambda }{)}_{\lambda \in {\mathbb{k}}^{\times }}$ of weakly symmetric algebras of Euclidean type over an algebraically closed field $\mathbb{k}$, where ${\mathbb{k}}^{\times }:=\mathbb{k}\setminus \{0\}$, and for each $\lambda \in {\mathbb{k}}^{\times }$, ${\mathrm{\Lambda }}_{\lambda }$ is socle equivalent to ${\mathrm{\Lambda }}_1$, which is the trivial extension of a hereditary algebra of type $\tilde{A}$. We compute their Hochschild cohomologies, which shows that ${\mathrm{\Lambda }}_{\lambda }$ is not derived equivalent to ${\mathrm{\Lambda }}_1$ if $\lambda \mathrm{\ne }1$. This means that a derived equivalence classification of weakly symmetric algebras of Euclidean type given by Bocian–Holm–Skowro{' n}ski is not complete.