arXiv subjects

Endre S. Rundsveen

Publications and source records attributed to Endre S. Rundsveen.

3 recordsLinked to original sources

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.

math.RT↗

arXiv2025

$τ_d$-tilting theory for linear Nakayama algebras

Support $τ$-tilting pairs, functorially finite torsion classes and $2$-term silting complexes are three much studied concepts in the representation theory of finite-dimensional algebras, which moreover turn out to be connected via work of Adachi, Iyama and Reiten. We investigate their higher-dimensional analogues via $τ_d$-rigid pairs, $d$-torsion classes and $(d+1)$-term silting complexes as well as the connections between these three concepts. Our work is done in the setting of truncated linear Nakayama algebras $Λ(n,l)=\mathbf{k} \mathbb{A}_{n}/\mathrm{rad}{\mathbf{k} \mathbb{A}_{n}}^l$ admitting a $d$-cluster tilting module. More specifically, we classify $τ_d$-rigid pairs $(M,P)$ of $Λ(n,l)$ with $|M|+|P|=n$ via an explicit combinatorial description and show that they can be characterized by a certain maximality condition as well as by giving rise to a $(d+1)$-term silting complex in $\mathrm{K}^b(\mathrm{proj}(Λ(n,l)))$. We also describe all $d$-torsion classes of $Λ(n,l)$. Finally, we compare our results to the classical case $d=1$ and investigate mutation with a special emphasis on the case where $d$ equals the global dimension of $Λ$.

math.RT↗

arXiv2024

Universal Extensions and Ext-Orthogonal Complements of Torsion Classes

We show that torsion pairs in Krull--Schmidt abelian categories induce an equivalence between the subcategory of torsion-free objects admitting universal extensions to the torsion subcategory, and a quotient of the ext-orthogonal complement of the torsion subcategory. This generalize an equivalence described by Bauer--Botnan--Oppermann--Steen for tilting-torsion pairs and by Buan--Zhou for functorially finite torsion pairs. The result also provides a more direct proof of the functorially finite case, not relying on the machinery of two-term silting complexes. We illustrate the equivalence in the special case of tube categories. After first appearing on the arXiv, the author learnt that the equivalence has previously been described by Demonet--Iyama.

math.RT↗