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Higher Koszul duality and connections with $n$-hereditary algebras

Johanne Haugland, Mads Hustad Sandøy

TL;DR

The paper develops a higher Koszul duality framework by introducing $n$-$T$-Koszul algebras and their duals, connecting Koszul theory with higher Auslander–Reiten theory. It proves a Beilinson–Ginzburg–Soergel–type duality: $\mathcal{D}^b(\mathrm{gr}\,\Lambda) \simeq \mathcal{D}^{\mathrm{perf}}(\mathrm{Gr}\,\Lambda^!)$, under suitable coherence assumptions, and shows that the tilting object $\widetilde{T}$ yields a bridge between $n$-representation theory and $n$-$T$-Koszulity via $B=\mathrm{End}_{\mathrm{gr}\Lambda}(\widetilde{T})$ which is $(na-1)$-representation infinite when $\Lambda$ is $n$-$T$-Koszul. A central payoff is the characterization: $A$ is $n$-representation infinite iff its trivial extension $\Delta A$ is $(n+1)$-Koszul w.r.t. $A$, and in the tame case the derived categories of graded modules over $\Delta A$ and the associated $(n+1)$-preprojective algebra $\Pi_{n+1}A$ are equivalent, linking graded $T$-Koszulity with higher representation theory. In the $n$-representation finite setting, the notion of almost $n$-$T$-Koszulity yields a parallel characterization connecting finiteness phenomena to endomorphism algebras, broadening the scope of Koszul-like dualities in higher dimensions.

Abstract

We establish a connection between two areas of independent interest in representation theory, namely Koszul duality and higher homological algebra. This is done through a generalization of the notion of $T$-Koszul algebras, for which we obtain a higher version of classical Koszul duality. Our approach is motivated by and has applications for $n$-hereditary algebras. In particular, we characterize an important class of $n$-$T$-Koszul algebras of highest degree $a$ in terms of $(na-1)$-representation infinite algebras. As a consequence, we see that an algebra is $n$-representation infinite if and only if its trivial extension is $(n+1)$-Koszul with respect to its degree $0$ part. Furthermore, we show that when an $n$-representation infinite algebra is $n$-representation tame, then the bounded derived categories of graded modules over the trivial extension and over the associated $(n+1)$-preprojective algebra are equivalent. In the $n$-representation finite case, we introduce the notion of almost $n$-$T$-Koszul algebras and obtain similar results.

Higher Koszul duality and connections with $n$-hereditary algebras

TL;DR

The paper develops a higher Koszul duality framework by introducing --Koszul algebras and their duals, connecting Koszul theory with higher Auslander–Reiten theory. It proves a Beilinson–Ginzburg–Soergel–type duality: , under suitable coherence assumptions, and shows that the tilting object yields a bridge between -representation theory and --Koszulity via which is -representation infinite when is --Koszul. A central payoff is the characterization: is -representation infinite iff its trivial extension is -Koszul w.r.t. , and in the tame case the derived categories of graded modules over and the associated -preprojective algebra are equivalent, linking graded -Koszulity with higher representation theory. In the -representation finite setting, the notion of almost --Koszulity yields a parallel characterization connecting finiteness phenomena to endomorphism algebras, broadening the scope of Koszul-like dualities in higher dimensions.

Abstract

We establish a connection between two areas of independent interest in representation theory, namely Koszul duality and higher homological algebra. This is done through a generalization of the notion of -Koszul algebras, for which we obtain a higher version of classical Koszul duality. Our approach is motivated by and has applications for -hereditary algebras. In particular, we characterize an important class of --Koszul algebras of highest degree in terms of -representation infinite algebras. As a consequence, we see that an algebra is -representation infinite if and only if its trivial extension is -Koszul with respect to its degree part. Furthermore, we show that when an -representation infinite algebra is -representation tame, then the bounded derived categories of graded modules over the trivial extension and over the associated -preprojective algebra are equivalent. In the -representation finite case, we introduce the notion of almost --Koszul algebras and obtain similar results.

Paper Structure

This paper contains 11 sections, 43 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Conventions and notation
  3. Preliminaries
  4. Graded algebras, modules and extensions
  5. Graded algebras as preadditive categories
  6. Graded Frobenius algebras
  7. Tilting subcategories, equivalences and Serre functors
  8. Higher Koszul duality
  9. On $n$-hereditary algebras
  10. Higher Koszul duality and $n$-representation infinite algebras
  11. Higher almost Koszulity and $n$-representation finite algebras

Key Result

Theorem 1

Let $\Lambda=\oplus_{i \geq 0}\Lambda_i$ be a finite dimensional graded Frobenius algebra of highest degree $a \geq 1$ with $\mathop{\mathrm{gl.\!dim}}\nolimits \Lambda_0 < \infty$. Consider a basic graded $\Lambda$-module $T$ which is concentrated in degree $0$ and a tilting module over $\Lambda_0$

Theorems & Definitions (92)

  • Theorem 1: see \ref{['Characterization']}
  • Corollary 2: see \ref{['motivating result']}
  • Theorem 3: see \ref{['nrepfinchar']}
  • Corollary 4: see \ref{['cor sec 7']}
  • Lemma 2.1: See Nastasescu-Van Oystaeyen*Corollary 2.4.7
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 82 more