$d$-abelian quotients of $(d+2)$-angulated categories
| Autor: | Peter Jørgensen, Karin Marie Jacobsen |
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| Rok vydání: | 2017 |
| Předmět: |
Subcategory
Pure mathematics 16G10 18E10 18E30 18E35 Algebra and Number Theory Endomorphism Quotient category Triangulated category 010102 general mathematics Mathematics::General Topology 01 natural sciences Morphism Mathematics::Category Theory 0103 physical sciences FOS: Mathematics 010307 mathematical physics Ideal (ring theory) 0101 mathematics Abelian group Representation Theory (math.RT) Mathematics::Representation Theory Mathematics - Representation Theory Quotient Mathematics |
| DOI: | 10.48550/arxiv.1712.07851 |
| Popis: | Let ${\mathscr T}$ be a triangulated category. If $T$ is a cluster tilting object and $I = [ \operatorname{add} T ]$ is the ideal of morphisms factoring through an object of $\operatorname{add} T$, then the quotient category ${\mathscr T} / I$ is abelian. This is an important result of cluster theory, due to Keller-Reiten and K\"{o}nig-Zhu. More general conditions which imply that ${\mathscr T} / I$ is abelian were determined by Grimeland and the first author. Now let ${\mathscr T}$ be a suitable $( d+2 )$-angulated category for an integer $d \geqslant 1$. If $T$ is a cluster tilting object in the sense of Oppermann-Thomas and $I = [ \operatorname{add} T ]$ is the ideal of morphisms factoring through an object of $\operatorname{add} T$, then we show that ${\mathscr T} / I$ is $d$-abelian. The notions of $( d+2 )$-angulated and $d$-abelian categories are due to Geiss-Keller-Oppermann and Jasso. They are higher homological generalisations of triangulated and abelian categories, which are recovered in the special case $d = 1$. We actually show that if $\Gamma = \operatorname{End}_{ \mathscr T }T$ is the endomorphism algebra of $T$, then ${\mathscr T} / I$ is equivalent to a $d$-cluster tilting subcategory of $\operatorname{mod} \Gamma$ in the sense of Iyama; this implies that ${\mathscr T} / I$ is $d$-abelian. Moreover, we show that $\Gamma$ is a $d$-Gorenstein algebra. More general conditions which imply that ${\mathscr T} / I$ is $d$-abelian will also be determined, generalising the triangulated results of Grimeland and the first author. Comment: 19 pages. This is the final accepted version, which has been accepted for publication in the Journal of Algebra |
| Databáze: | OpenAIRE |
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