Exceptional cycles in triangular matrix algebras

Abstract

An exceptional cycle in a triangulated category with Serre functor is a generalization of a spherical object. Suppose that $A$ and $B$ are Gorenstein algebras, given a perfect exceptional $n$-cycle $E_{\ast }$ in $K^b(A\mbox{-}{\rm proj})$ and a perfect exceptional $m$-cycle $F_{\ast }$ in $K^b(B\mbox{-}{\rm proj})$, we construct an $A$-$B$-bimodule $N$, and prove the product $E_{\ast }⊠F_{\ast }$ is an exceptional $(n+m-1)$-cycle in $K^b(Λ\mbox{-}{\rm proj})$, where $\hat{I}›=\left(\begin{array}{cc}A& N\\ 0& B\end{array}\right)$. Using this construction, one gets many new exceptional cycles which is unknown before for certain class of algebras.

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