arXiv subjects

Erlend D. Børve

Publications and source records attributed to Erlend D. Børve.

9 recordsLinked to original sources

arXiv2026

Splitting torsion classes in higher homological algebra

Let $d \geqslant 1$ be an integer, $\mathscr{M}$ a suitable $d$-abelian category. There is a notion of higher torsion classes in $\mathscr{M}$, also known as $d$-torsion classes. We generalise a classic theorem of Hoshino by providing criteria for a $d$-torsion class $\mathscr{U}$ to be splitting, notably that $\mathscr{U}$ is stable under $τ_d^-$, the inverse $d$-Auslander--Reiten translation. We apply the criteria to show that under additional assumptions, which are satisfied in Dynkin type $A$, the elements of the spine of the lattice of $d$-torsion classes are precisely the splitting $d$-torsion classes. The spine consists of the elements which belong to a chain of maximal length.

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arXiv2026

Picture groups and green sequences from the perspective of Ringel--Hall algebras

Let $k$ be an algebraically closed field and let $Λ$ be a finite-dimensional associative $k$-algebra. We apply Joyce and Brideland's notion of Ringel--Hall algebra to prove results about picture spaces and picture groups. For instance, we construct a faithful group functor for $Λ$, i.e. a faithful functor from the $τ$-cluster morphism category of $Λ$ into a groupoid. Consequently, by results of Hanson--Igusa, the picture space of $Λ$ is locally CAT(0) provided that the $τ$-cluster morphism category of $Λ$ admits compatibility of last factors. We also show that green sequences are in bijection with certain positive expressions in the picture group, which generalizes a result of Igusa--Todorov beyond hereditary $k$-algebras.

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arXiv2026

$g$-vector fans and picture categories for 0-Auslander extriangulated categories

We extend the notion of $\mathbf{g}$-vector fan so that it is defined for a Hom-finite Krull-Schmidt 0-Auslander $k$-linear extriangulated category $\mathcal{C}$ with a projective silting object $T$. Moreover, we show that the $\mathbf{g}$-vector fan admits an admissible partition, in the sense of the second-named author, which is induced by thick subcategories. One can thus define the picture category of $\mathcal{C}$. We establish a bijection between thick subcategories of $\mathcal{C}$ generated by presilting objects containing all projective-injective objects and $τ$-perpendicular subcategories of the endomorphism $k$-algebra of $T$. This shows that our construction unifies all previous constructions of picture categories and $τ$-cluster morphism categories of finite-dimensional algebras. We introduce morphisms of partitioned fans to provide a common framework for the functorial relationships between picture categories of different algebras and categories.

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arXiv2026

Magnitude of module categories

We define an invariant of the module category of a representation-finite algebra by the magnitude of its Auslander algebra. This invariant will be called the magnitude of the module category. For bound path algebras, it can be computed as the Euler characteristic of the Auslander--Reiten quiver, in a suitable sense. To aid the computation of our invariant, we define the Auslander--Reiten--Euler characteristic of a translation quiver. We build on classical results in Auslander--Reiten theory to determine the magnitude of module categories of biserial algebras, hereditary path algebras, radical square zero bound path algebras, and self-injective bound path algebras. In these cases, we express our invariant in terms of other known quantities, notably the rank of the Grothendieck group and Coxeter numbers of Dynkin quivers. Based on our calculations and results, we obtain a conjectural characterisation of representation-finite biserial algebras in terms of the magnitude of the module category and the rank of the Grothendieck group.

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arXiv2025

Bricks and $τ$-tilting theory under base field extensions

Let $K:k$ be a field extension and let $Λ$ be a finite-dimensional $k$-algebra. We investigate the relationship between $Λ$ and $Λ_K = Λ\otimes_k K$ with particular emphasis on various aspects of $τ$-tilting theory and bricks. We show that many types of objects for $Λ$ lift injectively to the same type of object for $Λ_K$, and many common constructions in $τ$-tilting theory commute with the process of extending the base field. One of our main applications is the construction of a faithful functor from the $τ$-cluster morphism category $\mathfrak{W}(Λ)$ of $Λ$ to the $τ$-cluster morphism category $\mathfrak{W}(Λ_K)$ of $Λ_K$. In particular, this establishes a faithful functor from $\mathfrak{W}(Λ)$ to a group whenever $k$ is of characteristic zero which has many important consequences. In the appendix, E. J. Hanson shows the analogous result whenever $k$ is a finite field. Moreover, we give some nontrivial examples to illustrate the behaviour of $τ$-tilting finiteness under base field extension.

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arXiv2025

$τ$-tilting finiteness and $\mathbf{g}$-tameness: Incidence algebras of posets and concealed algebras

We prove that any $τ$-tilting finite incidence algebra of a finite poset is representation-finite, and that any $\mathbf{g}$-tame incidence algebra of a finite simply connected poset is tame. As the converse of these assertions are known to hold, we obtain characterizations of $τ$-tilting finite incidence algebras and $\mathbf{g}$-tame simply connected incidence algebras. Both results are proved using the theory of concealed algebras. The former will be deduced from the fact that tame concealed algebras are $τ$-tilting infinite, and to prove the latter, we show that wild concealed algebras are not $\mathbf{g}$-tame. We conjecture that any incidence algebra of a finite poset is wild if and only if it is not $\mathbf{g}$-tame, and prove a result showing that there are relatively few possible counterexamples. In the appendix, we determine the representation type of a $τ$-tilting reduction of a concealed algebra of hyperbolic type.

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arXiv2024

Silting reduction and picture categories of 0-Auslander extriangulated categories

Let $\mathcal{C}$ be an extriangulated category and let $\mathcal{R}\subseteq \mathcal{C}$ be a rigid subcategory. Generalizing Iyama--Yang silting reduction, we devise a technical condition $\textbf{(gCP)}$ on $\mathcal{R}$ which is sufficient for the Verdier quotient $\mathcal{C}/\mathrm{thick}(\mathcal{R})$ to be equivalent to an ideal quotient. In particular, the Verdier quotient $\mathcal{C}/\mathrm{thick}(\mathcal{R})$ will admit an extriangulation in such a way that the localization functor $L_{\mathcal{R}}\colon \mathcal{C} \rightarrow \mathcal{C}/\mathrm{thick}(\mathcal{R})$ is extriangulated. When $\mathcal{C}$ is 0-Auslander, the condition $\textbf{(gCP)}$ holds for all rigid subcategories $\mathcal{R}$ admitting Bongartz completions. Furthermore, we prove that the Verdier quotient $\mathcal{C}/\mathrm{thick}(\mathcal{R})$ then remains 0-Auslander. As an application, we define the picture category of a connective $0$-Auslander exact dg category $\mathscr{A}$ with Bongartz completions, which generalizes the notion of $τ$-cluster morphism category. We show that the picture category of $\mathscr{A}$ is a cubical category, in the sense of Igusa. The picture group of $\mathscr{A}$ is defined as the fundamental group of its picture category. When $H_0\mathscr{A}$ is $\mathbf{g}$-finite, the picture group of $\mathscr{A}$ is finitely presented.

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arXiv2024

Two-term silting and $τ$-cluster morphism categories

We generalise $τ$-cluster morphism categories to the setting of non-positive dg algebras with finite dimensional cohomology in all degrees. The compatibility of silting reduction with support $τ$-tilting reduction will be an essential ingredient when linking our definition to those of Buan--Marsh and Buan--Hanson.

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arXiv2023

A theorem of Retakh for exact $\infty$-categories and higher extension functors

We define extension $\infty$-categories for exact $\infty$-categories in terms of bifibrations. Extension $\infty$-categories are invariant when passing to the stable hull, and consequently we show that they form an $Ω$-spectrum, generalizing a theorem of Retakh. Finally, we show that the homotopy groups of extension $\infty$-categories are naturally isomorphic to the higher extension groups of the extriangulated category given by the homotopy category.

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