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Maximilian Kaipel

Publications and source records attributed to Maximilian Kaipel.

8 recordsLinked to original sources
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 $\tau$-perpendicular subcategories of the endomorphism $k$-algebra of $T$. This shows that our construction unifies all previous constructions of picture categories and $\tau$-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

Exceptional versus $\tau$-exceptional sequences for the Auslander algebra of $K[x]/(x^t)$

For $\mathcal{A}_t$, the Auslander algebra of $K[x]/(x^t)$, it is shown that every complete exceptional sequence of $\mathcal{A}_t$-modules is a complete $\tau$-exceptional sequence. Moreover, it is established that the mutation of complete $\tau$-exceptional sequences generalises the mutation of complete exceptional sequences in the category of $\mathcal{A}_t$-modules.

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arXiv2025

Bricks and $\tau$-tilting theory under base field extensions

Let $K:k$ be a field extension and let $\Lambda$ be a finite-dimensional $k$-algebra. We investigate the relationship between $\Lambda$ and $\Lambda_K = \Lambda \otimes_k K$ with particular emphasis on various aspects of $\tau$-tilting theory and bricks. We show that many types of objects for $\Lambda$ lift injectively to the same type of object for $\Lambda_K$, and many common constructions in $\tau$-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 $\tau$-cluster morphism category $\mathfrak{W}(\Lambda)$ of $\Lambda$ to the $\tau$-cluster morphism category $\mathfrak{W}(\Lambda_K)$ of $\Lambda_K$. In particular, this establishes a faithful functor from $\mathfrak{W}(\Lambda)$ 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 $\tau$-tilting finiteness under base field extension.

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arXiv2025

Mutating ordered $\tau$-rigid modules with applications to Nakayama algebras

A mutation operation for $\tau$-exceptional sequences of modules over any finite-dimensional algebra was recently introduced, generalising the mutation for exceptional sequences of modules over hereditary algebras. We interpret this mutation in terms of TF-ordered $\tau$-rigid modules, which are in bijection with $\tau$-exceptional sequences. As an application we show that the mutation is transitive for Nakayama algebras, by providing an explicit combinatorial description of mutation over this class of algebras.

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arXiv2024

$\tau$-cluster morphism categories of factor algebras

We take a novel lattice-theoretic approach to the $\tau$-cluster morphism category $\mathfrak{T}(A)$ of a finite-dimensional algebra $A$ and define the category via the lattice of torsion classes $\mathrm{tors } A$. Using the lattice congruence induced by an ideal $I$ of $A$ we establish a functor $F_I: \mathfrak{T}(A) \to \mathfrak{T}(A/I)$. If $\mathrm{tors } A$ is finite, $F_I$ is a regular epimorphism in the category of small categories and we characterise when $F_I$ is full and faithful. The construction is purely combinatorial, meaning that the lattice of torsion classes determines the $\tau$-cluster morphism category up to equivalence.

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arXiv2023

The category of a partitioned fan

In this paper, we introduce the notion of an admissible partition of a simplicial polyhedral fan and define the category of a partitioned fan as a generalisation of the $\tau$-cluster morphism category of a finite-dimensional algebra. This establishes a complete lattice of categories around the $\tau$-cluster morphism category, which is closely tied to the fan structure. We prove that the classifying spaces of these categories are cube complexes, which reduces the process of determining if they are $K(\pi,1)$ spaces to three sufficient conditions. We characterise when these conditions are satisfied for fans in $\mathbb{R}^2$ and prove that the first one, the existence of a certain faithful functor, is satisfied for hyperplane arrangements whose normal vectors lie in the positive orthant. As a consequence we obtain a new infinite class of algebras for which the $\tau$-cluster morphism category admits a faithful functor and for which the cube complexes are $K(\pi,1)$ spaces. In the final section we also offer a new algebraic proof of the relationship between an algebra and its $g$-vector fan.

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