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

Ryu Tomonaga

TL;DR

The paper classifies all $d$-tilting bundles formed from line bundles on $d$-dimensional smooth toric Fano DM stacks with Picard numbers $1$ or $2$, linking these tiltings to high-dimensional representation theory. Using a Beilinson-type theorem for $G$-graded dg rings and Gale duality, the authors translate geometric data into combinatorial invariants—upper sets and cuts—and establish bijections between tilting bundles and these invariants. In the Picard number $1$ case, tilting bundles correspond to non-trivial upper sets in the Picard group, and their endomorphism algebras are $d$-representation infinite of type $ ilde{A}$; in Picard number $2$, tilting bundles correspond to pairs of upper sets, with NCCRs and quiver cuts guiding the $d$-RI structure. The work thus provides geometric models for higher representation infinite algebras and a new combinatorial perspective on $d$-APR tilting mutations, enriching both toric geometry and higher Auslander–Reiten theory.

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.

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

TL;DR

The paper classifies all -tilting bundles formed from line bundles on -dimensional smooth toric Fano DM stacks with Picard numbers or , linking these tiltings to high-dimensional representation theory. Using a Beilinson-type theorem for -graded dg rings and Gale duality, the authors translate geometric data into combinatorial invariants—upper sets and cuts—and establish bijections between tilting bundles and these invariants. In the Picard number case, tilting bundles correspond to non-trivial upper sets in the Picard group, and their endomorphism algebras are -representation infinite of type ; in Picard number , tilting bundles correspond to pairs of upper sets, with NCCRs and quiver cuts guiding the -RI structure. The work thus provides geometric models for higher representation infinite algebras and a new combinatorial perspective on -APR tilting mutations, enriching both toric geometry and higher Auslander–Reiten theory.

Abstract

We prove the existence and give a classification of all -tilting bundles (and thus geometric Helices) consisting of line bundles on -dimensional smooth toric Fano DM stacks of Picard number one or two. Here, a -tilting bundle is a tilting bundle whose endomorphism algebra has global dimension 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 -tilting bundles and their endomorphisms algebras become -representation infinite algebras of type . 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 . Using this geometric model, we give a new combinatorial description to -APR tilting modules of them. In the case of Picard number two, -tilting bundles consisting of line bundles correspond bijectively to pairs , where and are non-trivial upper sets in certain partially ordered sets. Here, corresponds to a non-commutative crepant resolution (NCCR) of a certain Gorenstein toric singularity with divisor class group of rank one and corresponds to a cut of the quiver of this NCCR. Moreover, the endomorphism algebras of these -tilting bundles also become -representation infinite algebras.

Paper Structure

This paper contains 14 sections, 34 theorems, 78 equations.

Table of Contents

  1. Back grounds from tilting theory for toric stacks and higher Auslander-Reiten theory
  2. The case of Picard number one
  3. The case of Picard number two
  4. Preliminaries
  5. Higher representation infinite algebras
  6. Smooth toric Fano stacks
  7. Combinatorics
  8. Combinatorics of upper sets
  9. Combinatorics of cuts
  10. Cut-upper set correspondence
  11. Starting from $G$
  12. Beilinson-type theorem for $G$-graded dg rings
  13. Tilting theory for smooth toric Fano stacks of Picard number one
  14. Tilting theory for smooth toric Fano stacks of Picard number two

Key Result

Theorem 3

(Theorem classfitiltrk1) Let $\mathcal{X}$ be a $d$-dimensional smooth toric Fano DM stack with Picard number one. In the above notations, we have a bijection between the following two sets. A bijection from (1) to (2) is given by $I\mapsto I\cap(I^c+\vec{p})$ where $\vec{p}=\sum_{i=0}^d\vec{x_i}$.

Theorems & Definitions (74)

  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • Definition 1.4
  • Remark 1.5
  • Proposition 1.6
  • ...and 64 more