Exceptional cycles for perfect complexes over gentle algebras

Abstract

Exceptional cycles in a triangulated category $\mathcal T$ with Serre duality, introduced by N. Broomhead, D. Pauksztello, and D. Ploog, have a notable impact on the global structure of $\mathcal T$. In this paper we show that if $\mathcal T$ is homotopy-like, then any exceptional $1$-cycle is indecomposable and at the mouth; and any object in an exceptional $n$-cycle with $n\ge 3$ is at the mouth. Let $A$ be an indecomposable gentle $k$-algebra with $A\ne k$. The Hom spaces between string complexes at the mouth are explicitly determined. The main result classifies "almost all" the exceptional cycles in $K^b(A\mbox{-}{\rm proj})$, using characteristic components and their AG-invariants, except those exceptional $1$-cycles which are band complexes. Namely, the mouth of a characteristic component $C$ of $K^b(A\mbox{-}{\rm proj})$ forms a unique exceptional cycle in $C$, up to an equivalent relation $\approx$; if the quiver of $A$ is not of type $A_3$, this gives all the exceptional $n$-cycle in $K^b(A\mbox{-}{\rm proj})$ with $n\ge 2$, up to $\approx$; and a string complex is an exceptional $1$-cycle if and only if it is at the mouth of a characteristic component with {\rm AG}-invariant $(1, m)$. However, a band complex at the mouth is possibly not an exceptional $1$-cycle.