Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Modules determined by their Newton polytopes

Peigen Cao

TL;DR

The paper introduces Newton polytopes $\mathcal{N}(U)$ of quotient-dimension vectors to distinguish modules beyond their dimension vectors, and proves that for finite-dimensional basic algebras, sharing a Newton polytope forces equivalence in the associated torsion- and mutation-theoretic data. The main theorem shows that indecomposable $\tau$-rigid modules and left finite bricks are uniquely determined by their Newton polytopes, and that $U\oplus V$ is $\tau$-rigid when $U,V$ are $\tau$-rigid. A corollary for $\tau$-tilting finite algebras asserts bricks are uniquely determined by Newton polytopes, highlighting the power of polyhedral invariants to classify modules beyond dimension vectors.

Abstract

In the $τ$-tilting theory, there exist two classes of foundamental modules: indecomposable $τ$-rigid modules and left finite bricks. In this paper, we prove the indecomposable $τ$-rigid modules and the left finite bricks are uniquely determined by their Newton polytopes spanned by the dimensional vectors of their quotient modules. This is a kind of generalization of Gabriel's result that the indecomposable modules over path algebras of Dynkin quivers are uniquely determined by their dimensional vectors.

Modules determined by their Newton polytopes

TL;DR

The paper introduces Newton polytopes of quotient-dimension vectors to distinguish modules beyond their dimension vectors, and proves that for finite-dimensional basic algebras, sharing a Newton polytope forces equivalence in the associated torsion- and mutation-theoretic data. The main theorem shows that indecomposable -rigid modules and left finite bricks are uniquely determined by their Newton polytopes, and that is -rigid when are -rigid. A corollary for -tilting finite algebras asserts bricks are uniquely determined by Newton polytopes, highlighting the power of polyhedral invariants to classify modules beyond dimension vectors.

Abstract

In the -tilting theory, there exist two classes of foundamental modules: indecomposable -rigid modules and left finite bricks. In this paper, we prove the indecomposable -rigid modules and the left finite bricks are uniquely determined by their Newton polytopes spanned by the dimensional vectors of their quotient modules. This is a kind of generalization of Gabriel's result that the indecomposable modules over path algebras of Dynkin quivers are uniquely determined by their dimensional vectors.
Paper Structure (3 sections, 10 theorems, 8 equations)

This paper contains 3 sections, 10 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. $\tau$-tilting theory
  3. Modules determined by their Newton polytopes

Key Result

Theorem 1.1

Let $K$ be an algebraically closed field and $A=KQ$ the path algebra of a Dynkin quiver $Q$. Then the dimensional vector $\underline{\dim}(N)$ of an indecomposable module $N$ in $\operatorname{\mathsf{mod}}\nolimits A$ gives a positive root in the root system $\Phi(Q)$ associated to $Q$ and this cor

Theorems & Definitions (19)

  • Theorem 1.1: Gabriel's theorem
  • Example 1.2
  • Definition 1.3: Newton polytopes of modules
  • Remark 1.4
  • Definition 1.5: Left finite modules
  • Theorem 1.6: Theorem \ref{['thm:main']}
  • Corollary 1.7: Corollary \ref{['cor:main']}
  • Definition 2.1
  • Theorem 2.2: air_2014*Proposition 1.2 (b) and Theorem 2.7
  • Proposition 2.3: air_2014*Proposition 2.9 and Theorem 2.10
  • ...and 9 more