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On compatibility of Koszul- and higher preprojective gradings

Darius Dramburg, Mads Hustad Sandøy

TL;DR

The article clarifies when a higher preprojective grading on a Koszul or almost Koszul algebra $R$ is compatible with the natural Koszul structure, in the two main settings: $n$-representation finite and acyclic $n$-representation infinite. Using graded Morita theory, the Wedderburn–Malcev framework, and the Grant–Iyama construction, it shows that in the RF case, a (possibly almost) Koszul grading on $\\Pi_{n+1}(A)$ forces $A$ to be Koszul and that the preprojective grading is a cut of the Koszul grading; similarly in the RI case, a Koszul grading on $\\Pi_{n+1}(A)$ implies $A$ is Koszul and the preprojective grading is a cut, up to certain finiteness assumptions. The results yield that, under suitable hypotheses, one can move to a grading where the two gradings commute (a genuine $\\mathbb{Z}^2$-grading) and that $n$-APR tilting preserves Koszulity for $n$-RI algebras. These findings connect structural gradings, preprojective constructions, and homological properties, providing a framework for constructing and recognizing compatible gradings and for transferring Koszulity across higher preprojective algebras.

Abstract

We investigate compatibility of gradings for an almost Koszul or Koszul algebra $R$ that is also the higher preprojective algebra $Π_{n+1}(A)$ of an $n$-hereditary algebra $A$. For an $n$-representation finite algebra $A$, we show that $A$ must be Koszul if $Π_{n+1}(A)$ can be endowed with an almost Koszul grading. For an acyclic basic $n$-representation infinite algebra $A$, we show that $A$ must be Koszul if $Π_{n+1}(A)$ can be endowed with a Koszul grading. From this we deduce that a higher preprojective grading of an (almost) Koszul algebra $R = Π_{n+1}(A)$ is, in both cases, isomorphic to a cut of the (almost) Koszul grading. Up to a further assumption on the tops of the degree $0$ subalgebras for the different gradings, we also show a similar result without the basic assumption in the $n$-representation infinite case. As an application, we show that $n$-APR tilting preserves the property of being Koszul for $n$-representation infinite algebras.

On compatibility of Koszul- and higher preprojective gradings

TL;DR

The article clarifies when a higher preprojective grading on a Koszul or almost Koszul algebra is compatible with the natural Koszul structure, in the two main settings: -representation finite and acyclic -representation infinite. Using graded Morita theory, the Wedderburn–Malcev framework, and the Grant–Iyama construction, it shows that in the RF case, a (possibly almost) Koszul grading on forces to be Koszul and that the preprojective grading is a cut of the Koszul grading; similarly in the RI case, a Koszul grading on implies is Koszul and the preprojective grading is a cut, up to certain finiteness assumptions. The results yield that, under suitable hypotheses, one can move to a grading where the two gradings commute (a genuine -grading) and that -APR tilting preserves Koszulity for -RI algebras. These findings connect structural gradings, preprojective constructions, and homological properties, providing a framework for constructing and recognizing compatible gradings and for transferring Koszulity across higher preprojective algebras.

Abstract

We investigate compatibility of gradings for an almost Koszul or Koszul algebra that is also the higher preprojective algebra of an -hereditary algebra . For an -representation finite algebra , we show that must be Koszul if can be endowed with an almost Koszul grading. For an acyclic basic -representation infinite algebra , we show that must be Koszul if can be endowed with a Koszul grading. From this we deduce that a higher preprojective grading of an (almost) Koszul algebra is, in both cases, isomorphic to a cut of the (almost) Koszul grading. Up to a further assumption on the tops of the degree subalgebras for the different gradings, we also show a similar result without the basic assumption in the -representation infinite case. As an application, we show that -APR tilting preserves the property of being Koszul for -representation infinite algebras.

Paper Structure

This paper contains 17 sections, 32 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. The Wedderburn-Malcev theorem
  4. Graded algebras
  5. Koszul algebras
  6. Almost Koszul algebras
  7. (n+1)-preprojective algebras
  8. The Grant-Iyama construction
  9. Higher preprojective cuts
  10. Reduction to the graded basic case
  11. The n-representation finite case
  12. The n-representation infinite case
  13. Graded radicals and Graded Simples
  14. The graded radicals
  15. Nilpotency of generators
  16. ...and 2 more sections

Key Result

Theorem 2.1

malcev1942representation Let $A$ be a finite dimensional algebra such that $A/\mathop{\mathrm{rad}}\nolimits(A)$ is separable. Then there exists a semisimple subalgebra $S \subseteq A$ such that $A = S \oplus \mathop{\mathrm{rad}}\nolimits(A)$. Furthermore, for any semisimple subalgebra $T \subseteq

Theorems & Definitions (77)

  • Theorem 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Corollary 2.7
  • proof
  • Theorem 2.8
  • Corollary 2.9
  • ...and 67 more