arXiv subjects

Johanne Haugland

Publications and source records attributed to Johanne Haugland.

9 recordsLinked to original sources

arXiv2026

Higher Koszul algebras and the $\textbf{(Fg)}$-condition

Determining when a finite dimensional algebra satisfies the finiteness property known as the $(\textbf{Fg})$-condition is of fundamental importance in the celebrated and influential theory of support varieties. We give an answer to this question for higher Koszul algebras, generalizing a result by Erdmann and Solberg. This allows us to establish a strong connection between the $(\textbf{Fg})$-condition and higher homological algebra, which significantly extends the classes of algebras for which it is known whether the $(\textbf{Fg})$-condition is satisfied. In particular, we show that the condition holds for an important class of algebras arising from consistent dimer models.

math.RT↗

arXiv2026

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

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.

math.RT↗

arXiv2026

Higher torsion classes, $τ_d$-tilting theory and silting complexes

Initiated in work by Adachi, Iyama and Reiten, the area known as $τ$-tilting theory plays a fundamental role in contemporary representation theory. In this paper we explore a higher-dimensional analogue of this theory, formulated with respect to the higher Auslander-Reiten translation $τ_d$. In particular, we associate to any functorially finite $d$-torsion class a maximal $τ_d$-rigid pair and a $(d+1)$-term silting complex. In the case $d=1$, the notions of maximal $τ_d$-rigid and support $τ$-tilting pairs coincide, and our theory recovers the classical bijections. However, the proof strategies for $d>1$ differ significantly. As an intermediate step, we prove that a $d$-cluster tilting subcategory of a module category induces a $d$-cluster tilting subcategory of the category of $(d+1)$-term complexes, producing novel examples of $d$-exact categories. We introduce the notion of a $d$-torsion class in the exact setup, and use this to obtain the aforementioned $(d+1)$-term silting complex. We moreover apply our theory to study $d$-APR tilting modules and slices. To illustrate our results, we provide explicit combinatorial descriptions of maximal $τ_d$-rigid pairs and $(d+1)$-term silting complexes for higher Auslander and higher Nakayama algebras.

math.RT↗

arXiv2024

A characterisation of higher torsion classes

Let $\mathcal{A}$ be an abelian length category containing a $d$-cluster tilting subcategory $\mathcal{M}$. We prove that a subcategory of $\mathcal{M}$ is a $d$-torsion class if and only if it is closed under $d$-extensions and $d$-quotients. This generalises an important result for classical torsion classes. As an application, we prove that the $d$-torsion classes in $\mathcal{M}$ form a complete lattice. Moreover, we use the characterisation to classify the $d$-torsion classes associated to higher Auslander algebras of type $\mathbb{A}$, and give an algorithm to compute them explicitly. The classification is furthermore extended to the setup of higher Nakayama algebras.

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arXiv2023

The category of extensions and a characterisation of $n$-exangulated functors

Additive categories play a fundamental role in mathematics and related disciplines. Given an additive category equipped with a biadditive functor, one can construct its category of extensions, which encodes important structural information. We study how functors between categories of extensions relate to those at the level of the original categories. When the additive categories in question are $n$-exangulated, this leads to a characterisation of $n$-exangulated functors. Our approach enables us to study $n$-exangulated categories from a $2$-categorical perspective. We introduce $n$-exangulated natural transformations and characterise them using categories of extensions. Our characterisations allow us to establish a $2$-functor between the $2$-categories of small $n$-exangulated categories and small exact categories. A similar result with no smallness assumption is also proved. We employ our theory to produce various examples of $n$-exangulated functors and natural transformations. Although the motivation for this article stems from representation theory and the study of $n$-exangulated categories, our results are widely applicable: several require only an additive category equipped with a biadditive functor with no extra assumptions; others can be applied by endowing an additive category with its split $n$-exangulated structure.

math.CT↗

arXiv2023

The category of extensions and idempotent completion

Building on previous work, we study the splitting of idempotents in the category of extensions $\mathbb{E}\operatorname{-Ext}(\mathcal{C})$ associated to a pair $(\mathcal{C},\mathbb{E})$ of an additive category and a biadditive functor to the category of abelian groups. In particular, we show that idempotents split in $\mathbb{E}\operatorname{-Ext}(\mathcal{C})$ whenever they do so in $\mathcal{C}$, allowing us to prove that idempotent completions and extension categories are compatible constructions in a $2$-category-theoretic sense. Furthermore, we show that the exact category obtained by first taking the idempotent completion of an $n$-exangulated category $(\mathcal{C},\mathbb{E},\mathfrak{s})$, in the sense of Klapproth-Msapato-Shah, and then considering its category of extensions is equivalent to the exact category obtained by first passing to the extension category and then taking the idempotent completion. These two different approaches yield a pair of $2$-functors each taking small $n$-exangulated categories to small idempotent complete exact categories. The collection of equivalences that we provide constitutes a $2$-natural transformation between these $2$-functors. Similar results with no smallness assumptions and regarding weak idempotent completions are also proved.

math.CT↗

arXiv2022

The role of gentle algebras in higher homological algebra

We investigate the role of gentle algebras in higher homological algebra. In the first part of the paper, we show that if the module category of a gentle algebra $Λ$ contains a $d$-cluster tilting subcategory for some $d \geq 2$, then $Λ$ is a radical square zero Nakayama algebra. This gives a complete classification of weakly $d$-representation finite gentle algebras. In the second part, we use a geometric model of the derived category to prove a similar result in the triangulated setup. More precisely, we show that if $\mathcal{D}^b(Λ)$ contains a $d$-cluster tilting subcategory that is closed under $[d]$, then $Λ$ is derived equivalent to an algebra of Dynkin type $A$. Furthermore, our approach gives a geometric characterization of all $d$-cluster tilting subcategories of $\mathcal{D}^b(Λ)$ that are closed under $[d]$.

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arXiv2021

Auslander-Reiten triangles and Grothendieck groups of triangulated categories

We prove that if the Auslander-Reiten triangles generate the relations for the Grothendieck group of a Hom-finite Krull-Schmidt triangulated category with a (co)generator, then the category has only finitely many isomorphism classes of indecomposable objects up to translation. This gives a triangulated converse to a theorem of Butler and Auslander-Reiten on the relations for Grothendieck groups. Our approach has applications in the context of Frobenius categories.

math.CT↗

arXiv2020

The Grothendieck Group of an $n$-exangulated Category

We define the Grothendieck group of an $n$-exangulated category. For $n$ odd, we show that this group shares many properties with the Grothendieck group of an exact or a triangulated category. In particular, we classify dense complete subcategories of an $n$-exangulated category with an $n$-(co)generator in terms of subgroups of the Grothendieck group. This unifies and extends results of Thomason, Bergh--Thaule, Matsui and Zhu--Zhuang for triangulated, $(n+2)$-angulated, exact and extriangulated categories, respectively. We also introduce the notion of an $n$-exangulated subcategory and prove that the subcategories in our classification theorem carry this structure.

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