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A note on the $m$-extended module categories of Nakayama algebras

Endre Sørmo Rundsveen

TL;DR

This work analyzes the m-extended module categories $m ext{-mod}\Lambda$ for Nakayama algebras, establishing the existence of postprojective components in certain cases and developing a knitting algorithm based on cohomological dimension vectors to construct AR-quivers. It classifies when the extended module categories of linear Nakayama algebras with homogeneous relations are of finite type, furnishing fully computed AR-quivers for several examples. The paper then transfers these results to cyclic Nakayama algebras via covering theory, showing that finiteness and AR-structure properties can be descended to the cyclic case using push-down functors and preserved AR-triangles. Overall, the methods provide a concrete framework to understand extended module categories, their AR-theory, and finiteness phenomena in both linear and cyclic Nakayama settings.

Abstract

We study the extended module category $m\operatorname{-mod}Λ$ of an algebra $Λ$, recently introduced by Gupta and Zhou. The existence of a postprojective component of $m\operatorname{-mod}Λ$ is shown for certain algebras $Λ$, and a knitting algorithm through cohomological dimension vectors is provided. In particular, the extended module category of Nakayama algebras with homogeneous relations are investigated, and it is classified when they are of finite type. The paper give fully calculated AR-quivers for several Nakayama algebras.

A note on the $m$-extended module categories of Nakayama algebras

TL;DR

This work analyzes the m-extended module categories for Nakayama algebras, establishing the existence of postprojective components in certain cases and developing a knitting algorithm based on cohomological dimension vectors to construct AR-quivers. It classifies when the extended module categories of linear Nakayama algebras with homogeneous relations are of finite type, furnishing fully computed AR-quivers for several examples. The paper then transfers these results to cyclic Nakayama algebras via covering theory, showing that finiteness and AR-structure properties can be descended to the cyclic case using push-down functors and preserved AR-triangles. Overall, the methods provide a concrete framework to understand extended module categories, their AR-theory, and finiteness phenomena in both linear and cyclic Nakayama settings.

Abstract

We study the extended module category of an algebra , recently introduced by Gupta and Zhou. The existence of a postprojective component of is shown for certain algebras , and a knitting algorithm through cohomological dimension vectors is provided. In particular, the extended module category of Nakayama algebras with homogeneous relations are investigated, and it is classified when they are of finite type. The paper give fully calculated AR-quivers for several Nakayama algebras.
Paper Structure (24 sections, 35 theorems, 48 equations, 7 figures)

This paper contains 24 sections, 35 theorems, 48 equations, 7 figures.

Table of Contents

  1. Preliminaries
  2. Derived categories
  3. t-structures and truncations
  4. The Nakayama functor
  5. Quivers
  6. m-extended module categories
  7. AR-triangles
  8. AR-quivers
  9. Knitting AR-quivers
  10. Knitting dimensions
  11. A criterion for finiteness
  12. Linear Nakayama algebras
  13. The finite cases
  14. Case: 2-mod Lambda(n,4)
  15. Case: 2-mod Lambda(n,5)
  16. ...and 9 more sections

Key Result

Lemma 2.1

$X^\bullet$ is projective in $m\mathop{\mathrm{-mod}}\nolimits\Lambda$ if and only if $X^\bullet\cong P$ for $P$ projective in $\operatorname{mod}\Lambda$. Dually, $Y^\bullet$ is injective in $m\mathop{\mathrm{-mod}}\nolimits\Lambda$ if and only if $Y^\bullet\cong I[m-1]$ for $I$ injective in $\oper

Figures (7)

  • Figure 1: AR-quivers of $2\mathop{\mathrm{-mod}}\nolimits\Lambda(n,4)$
  • Figure 2: AR-quivers of $2\mathop{\mathrm{-mod}}\nolimits\Lambda(n,5)$
  • Figure 3: AR-quivers of $4\mathop{\mathrm{-mod}}\nolimits\Lambda(n,3)$
  • Figure 4: The start of the postprojective component in $\mathop{\mathrm{AR}}\nolimits(m\mathop{\mathrm{-mod}}\nolimits\Lambda(n,l))$ for minimal infinite type.
  • Figure 5: AR-quivers of $m\mathop{\mathrm{-mod}}\nolimits\Lambda(n,2)$
  • ...and 2 more figures

Theorems & Definitions (74)

  • Remark 1.1
  • Lemma 2.1
  • Definition 2.2: Jor09
  • Lemma 2.3: Liu10
  • Definition 2.4
  • Proposition 2.5: Zho25
  • Theorem 2.6: Zho25
  • Definition 2.7
  • Remark 2.9
  • Lemma 2.10
  • ...and 64 more