This project is maintained by ysykimura
2025年11月19日(水) 14:30~15:30, 16:00~17:00
i-Siteなんば S1 (およびZoomのハイブリッド形式の予定)
朝永 龍 氏 (東京大学)
Higher hereditary algebras and toric Fano stacks of Picard number one or two
A $d$-tilting bundle is a tilting bundle whose endomorphism algebra has global dimension $d$ or less. Studying $d$-tilting bundles on $d$-dimensional smooth projective (stacky) varieties is important both for geometry and representation theory. In this talk, we show the existence and give a classification of $d$-tilting bundles consisting of line bundles on $d$-dimensional smooth toric Fano DM stacks of Picard number one or two. In the case of Picard number one, tilting bundles consisting of line bundles correspond bijectively to non-trivial upper sets in its Picard group equipped with a certain partial order. Moreover, all of them are $d$-tilting bundles and their endomorphism algebras are $d$-representation infinite algebras of type $\tilde{A}$. We explain that such toric stacks can be viewed as geometric models of higher representation infinite algebras of type $\tilde{A}$. In the case of Picard number two, $d$-tilting bundles consisting of line bundles correspond bijectively to pairs $(I,I’)$ where $I$ and $I’$ are non-trivial upper sets in certain partially ordered sets. Here, $I$ corresponds to an NCCR of a certain toric singularity and $I’$ corresponds to a cut of the quiver of this NCCR. Moreover, the endomorphism algebras of these $d$-tilting bundles are $d$-representation infinite algebras.