Derived equivalence of algebras (Q1368266)

Let \(A\) be a connected finite-dimensional \(k\)-algebra and \(D=\Hom_k(-,k)\) the ordinary duality. Then \(DA\) is an \((A,A)\)-bimodule and injective on each side. Let \(\widehat A\) be the repetitive algebra of \(A\), that is \(\widehat A\) is the infinite-dimensional algebra \((A\oplus DA)^{(\mathbb{Z})}\), where multiplication is given by \(((a_i,\phi_i))((b_j,\psi_j))=((a_ib_i,a_{i+1}\psi_i+\phi_ib_i))\). Let \(\nu\colon\widehat A\to\widehat A\) be the Nakayama automorphisms, that is \((\nu((a_i,\phi_i)))_m=(a_{m-1},\phi_{m-1})\). The algebras \(R^m_A=\widehat A/(\nu^m)\) are finite-dimensional self-injective algebras. For example \(R^1_A\) is the trivial extension of \(A\) by \(DA\). The main result of the paper says, that the algebras \(R^m_A\) and \(R^m_B\) are derived equivalent, provided \(A\) and \(B\) are derived equivalent. The proof is based on Rickard's theorem on derived equivalent algebras [\textit{J. Rickard}, J. Lond. Math. Soc., II. Ser. 39, No. 3, 436-456 (1989; Zbl 0642.16034)]. Since derived equivalent self-injective algebras are stable equivalent, it follows that \(A\) and \(B\) derived equivalent implies that \(R^m_A\) and \(R^m_B\) are stable equivalent.