Exceptional cycles in triangular matrix algebras

TL;DR

This paper constructs new exceptional cycles in the derived categories of triangular matrix algebras from known cycles in Gorenstein algebras, expanding the understanding of their structure.

Contribution

It introduces a method to generate exceptional cycles in triangular matrix algebras from existing cycles in component Gorenstein algebras, revealing new algebraic structures.

Findings

Construction of exceptional cycles in triangular matrix algebras

Generation of previously unknown exceptional cycles

Extension of the theory of exceptional objects in derived categories

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 $E_*$ in $K^b(A\mbox{-}{\rm proj})$ and a perfect exceptional $m$-cycle $F_*$ in $K^b(B\mbox{-}{\rm proj})$, we construct an $A$-$B$-bimodule $N$, and prove the product $E_*\boxtimes F_*$ is an exceptional $(n+m-1)$-cycle in $K^b(\Lambda\mbox{-}{\rm proj})$, where $\Lambda=\begin{pmatrix}A & N\\ 0 & B \end{pmatrix}$. Using this construction, one gets many new exceptional cycles which is unknown before for certain class of algebras.