Higher hereditary algebras and toric Fano stacks of Picard number one or two

TL;DR

This paper classifies and constructs $d$-tilting bundles of line bundles on certain smooth toric Fano stacks with Picard number one or two, linking algebraic and geometric structures.

Contribution

It provides a complete classification of $d$-tilting line bundle bundles on these stacks and relates them to higher representation infinite algebras of types $ ilde{A}$ and $ ilde{A} ilde{A}$.

Findings

Classified $d$-tilting bundles on Picard number one stacks as upper sets in Picard group.

Established bijection between $d$-tilting bundles and certain pairs of upper sets for Picard number two.

Connected endomorphism algebras of these bundles to higher representation infinite algebras of specific types.

Abstract

On smooth projective varieties of dimension $d$, $d$-tilting bundles are important in both geometry and representation theory, since they provide a bridge from the geometry of such varieties to the derived McKay correspondence and to higher Auslander--Reiten theory. Here, a $d$-tilting bundle means a tilting bundle whose endomorphism algebra has global dimension at most $d$. In this paper, we prove the existence of and classify all $d$-tilting bundles consisting of line bundles on $d$-dimensional smooth toric Fano stacks of Picard number one or two. In the case of Picard number one, tilting bundles consisting of line bundles are in bijection with 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 $\widetilde A$. Conversely, all such algebras arise in this way. In the case of Picard number two, $d$-tilting bundles consisting of line bundles are in bijection with 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 toric singularity and $I'$ corresponds to a cut of the quiver of this NCCR. We introduce higher representation infinite algebras of type $\widetilde A\widetilde A$ and show that the endomorphism algebras of these $d$-tilting bundles belong to this class. Conversely, all such algebras arise in this way. Thus smooth toric Fano stacks of Picard number one (respectively, two) can be regarded as geometric models of higher representation infinite algebras of type $\widetilde A$ (respectively, $\widetilde A\widetilde A$). Using these geometric models, we show that these classes of algebras are closed under $d$-APR tilts by giving a combinatorial description of their $d$-APR tilting modules.