# Exceptional cycles for perfect complexes over gentle algebras

## TL;DR

This paper classifies exceptional cycles in the homotopy category of perfect complexes over gentle algebras, revealing their structure and explicit descriptions, especially focusing on those at the mouths of characteristic components.

## Contribution

It provides a classification of most exceptional cycles in $K^b(A$-proj), using AG-invariants and characteristic components, with explicit descriptions for gentle algebras.

## Key findings

- Exceptional 1-cycles are indecomposable and at the mouth in homotopy-like categories.
- Hom spaces between string complexes at the mouth are explicitly determined.
- Most exceptional cycles are classified via characteristic components and AG-invariants.

## 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.