Non-semistable Exceptional Objects in Hereditary Categories

George Dimitrov, Ludmil Katzarkov

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

For a given stability condition σ on a triangulated category, we define a σ -exceptional collection as an Ext-exceptional collection, whose elements are σ -semistable with phases contained in an open interval of length 1. If a full σ -exceptional collection exists, then σ is generated by this collection in a procedure described by E. Macrì. Constructing σ -exceptional collections in hereditary hom -finite categories, we introduce certain conditions on the Ext-nontrivial couples (exceptional objects with Ext1(X,Y)≠ 0, Ext1(Y,X)≠ 0). We study exceptional representations of the quivers and show that the needed conditions do hold in Q1= Q2= and show that the needed conditions do hold in Repk(Q1), Repk(Q2). Finally, any σ ∈ Stab(Db(Q1)) is shown to admit a full σ -exceptional collection. This implies that the space Stab(Db(Q1)) is connected.

Original languageEnglish (US)
Pages (from-to)6293-6377
Number of pages85
JournalInternational Mathematics Research Notices
Volume2016
Issue number20
DOIs
StatePublished - Jan 1 2016
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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