arXiv subjects

Marvin Plogmann

Publications and source records attributed to Marvin Plogmann.

3 recordsLinked to original sources

arXiv2026

Spherical modules and the Auslander--Gorenstein condition for Auslander--Yoneda algebras

For a finite dimensional algebra $A$ of finite global dimension we study the Auslander--Yoneda algebra defined as the Yoneda algebra of the direct sum of all indecomposable $A$-modules. We show that the Auslander--Yoneda algebra is an Auslander--Gorenstein algebra if and only if every indecomposable left and right $A$-module is spherical in the sense of Auslander and Bridger. This motivates the study of spherical algebras defined by the condition that every indecomposable module is spherical. We characterize spherical algebras by a certain natural pair of subcategories being a split torsion pair. Moreover, we prove that representation-finite algebras which are spherical are directed and give a full classification of spherical Nakayama algebras. Furthermore, we show that replicated algebras of hereditary algebras are spherical. As a final application of the new notion of spherical algebras, we give a negative answer to a question of Venjakob on Auslander regular algebras in general, but show that there is a positive answer when assuming that every indecomposable left $A$-module is spherical.

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arXiv2026

Complicial simple-minded collections

We consider the problem of characterizing derived endomorphism algebras of simple objects in length categories up to quasi-isomorphism. We give such a characterization for module categories, abelian categories, exact categories, as well as, for certain differential graded analogues of them. It turns out that the property of being $d$-complicial ($d\geq 1$), in the sense of Lurie, of the involved simple-minded collections plays a central role. We also explain how this characterization can be interpreted as a coherent generation property for any minimal $A_{\infty}$-model of the derived endomorphism algebra. Along the way, we propose a notion of length exact differential graded categories and explain how they relate to length abelian $d$-truncated differential graded categories, generalizing results of Enomoto.

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arXiv2025

On the Auslander--Reiten Theory for Extended Hearts of Proper Connective DG Algebras

We prove that, for a proper connective dg algebra $A$ with cohomology concentrated in degrees between $1-d$ and $0$, the extended heart $\mathcal{D}^{fd}(A)^{(-d,0]}\subseteq \mathcal{D}^{fd}(A)$ is an extriangulated category with almost-split conflations. We also prove a version of the 1st Brauer--Thrall Conjecture in this context.

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