Higher hereditary algebras and toric Fano stacks of Picard number one or two
Authors
Ryu Tomonaga
Abstract
We prove the existence and give a classification of all $d$-tilting bundles (and thus geometric Helices) consisting of line bundles on $d$-dimensional smooth toric Fano DM stacks of Picard number one or two. Here, a $d$-tilting bundle is a tilting bundle whose endomorphism algebra has global dimension $d$ or less.
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 endomorphisms algebras become $d$-representation infinite algebras of type $\tilde{A}$. Conversely, all such algebras arise in this way. In this sense, we can think of smooth toric Fano DM stacks with Picard number one as geometric models of higher representation infinite algebras of type $\tilde{A}$. Using this geometric model, we give a new combinatorial description to $d$-APR tilting modules of them.
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 a non-commutative crepant resolution (NCCR) of a certain Gorenstein toric singularity with divisor class group of rank one and $I'$ corresponds to a cut of the quiver of this NCCR. Moreover, the endomorphism algebras of these $d$-tilting bundles also become $d$-representation infinite algebras.
