Derived equivalences for \(\Phi \)-Auslander-Yoneda algebras (Q2849026)

In the present paper the authors introduce a family of Yoneda algebras in triangulated categories and use this family in order to produce derived equivalences for these algebras.NEWLINENEWLINEA subset \(\Phi\subseteq \mathbb{N}\) is called admisible if \(0\in \Phi\) and if \(i+j+k\in\Phi\) then \(i+j\in \Phi\) if and only if \(j+k\in\Phi\). If \(\mathcal{T}\) is a triangulated \(R\)-category over a commutative Artin ring \(R\) and \(0\in \Phi\subseteq \mathbb{N}\) then the authors introduce a functor \(E^\Phi_\mathcal{T}(-,-):\mathcal{T}\times\mathcal{T}\to R\mathrm{-Mod}\) defined by \(E^\Phi_\mathcal{T}(X,Y)=\bigoplus_{i\in \Phi}\mathrm{Hom}_\mathcal{T}(X,Y[i])\) which is defined on the canonical way homomorphisms. Then the \(R\)-algebra \(E^\Phi_\mathcal{T}(X)=E^\Phi_\mathcal{T}(X,X)\) is called the \(\Phi\)-Auslander-Yoneda algebra of \(X\) in \(\mathcal{T}\). It is proved that these algebras can be very useful in order to produce derived equivalences. For instance, if \(A\) is a self-injective Artin algebra then for every module \(X\) and for every admissible set \(\Phi\) the \(\Phi\)-admisible algebras of \(A\oplus X\) and \(A\oplus \Omega_A(X)\) are derived equivalent, where \(\Omega\) is the Heller loop operator (Corollary 3.12).