arXiv subjects

Sondre Kvamme

Publications and source records attributed to Sondre Kvamme.

16 recordsLinked to original sources

arXiv2026

Higher extension closure and $d$-exact categories

We prove that any weakly idempotent complete $d$-exact category is equivalent to a $d$-cluster tilting subcategory of a weakly idempotent complete exact category, and that any weakly idempotent complete algebraic $(d+2)$-angulated category is equivalent to a $d$-cluster tilting subcategory of an algebraic triangulated category closed under $d$-shifts. Furthermore, we show that the ambient exact category of a $d$-cluster tilting subcategory is unique up to exact equivalence, assuming it is weakly idempotent complete. This follows from the inclusion of the $d$-cluster tilting subcategory satisfying a universal property. As a consequence of our theory we also get that any $d$-torsion class is $d$-cluster tilting in an extension-closed subcategory, and we recover the fact that any $d$-wide subcategory is $d$-cluster tilting in a unique wide subcategory. In the last part of the paper we rectify the description of the $d$-exact structure of a $d$-cluster tilting subcategory of a non-weakly idempotent complete exact category.

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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.

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arXiv2024

A functorial approach to monomorphism categories II: Indecomposables

We investigate the (separated) monomorphism category $\operatorname{mono}(Q,Λ)$ of a quiver $Q$ over an Artin algebra $Λ$. We construct an epivalence from $\overline{\operatorname{mono}}(Q,Λ)$ to $\operatorname{rep}(Q,\overline{\operatorname{mod}}\, Λ)$, where $\operatorname{mod}Λ$ is the category of finitely generated modules and $\overline{\operatorname{mod}}\, Λ$ and $\overline{\operatorname{mono}}(Q,Λ)$ denote the respective injectively stable categories. Furthermore, if $Q$ has at least one arrow, then we show that this is an equivalence if and only if $Λ$ is hereditary. In general, it induces a bijection between indecomposable objects in $\operatorname{rep}(Q,\overline{\operatorname{mod}}\, Λ)$ and non-injective indecomposable objects in $\operatorname{mono}(Q,Λ)$. We show that the generalized Mimo-construction, an explicit minimal right approximation into $\operatorname{mono}{(Q,Λ)}$, gives an inverse to this bijection. Using this, we describe the indecomposables in the monomorphism category of a radical-square-zero Nakayama algebra, and give a bijection between the indecomposables in the monomorphism category of two artinian uniserial rings of Loewy length $3$ with the same residue field. These results are proved using free monads on an abelian category, in order to avoid the technical combinatorics arising from quiver representations. The setup also specializes to representations of modulations. In particular, we obtain new results on the singularity category of the algebras $H$ which were introduced by Geiss, Leclerc, and Schröer in order to extend their results relating cluster algebras and Lusztig's semicanonical basis to symmetrizable Cartan matrices. We also recover results on the $ι$quivers algebras which were introduced by Lu and Wang to realize $ι$quantum groups via semi-derived Hall algebras.

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arXiv2024

The left heart and exact hull of an additive regular category

Quasi-abelian categories are abundant in functional analysis and representation theory. It is known that a quasi-abelian category $\mathcal{E}$ is a cotilting torsionfree class of an abelian category. In fact, this property characterizes quasi-abelian categories. This ambient abelian category is derived equivalent to the category $\mathcal{E}$, and can be constructed as the heart $\mathcal{LH}(\mathcal{E})$ of a $\operatorname{t}$-structure on the bounded derived category $\operatorname{D^b}(\mathcal{E})$ or as the localization of the category of monomorphisms in $\mathcal{E}.$ However, there are natural examples of categories in functional analysis which are not quasi-abelian, but merely one-sided quasi-abelian or even weaker. Examples are the category of $\operatorname{LB}$-spaces or the category of complete Hausdorff locally convex spaces. In this paper, we consider additive regular categories as a generalization of quasi-abelian categories that covers the aforementioned examples. Additive regular categories can be characterized as those subcategories of abelian categories which are closed under subobjects. As for quasi-abelian categories, we show that such an ambient abelian category of an additive regular category $\mathcal{E}$ can be found as the heart of a $\operatorname{t}$-structure on the bounded derived category $\operatorname{D^b}(\mathcal{E})$, or as the localization of the category of monomorphisms of $\mathcal{E}$. In our proof of this last construction, we formulate and prove a version of Auslander's formula for additive regular categories. Whereas a quasi-abelian category is an exact category in a natural way, an additive regular category has a natural one-sided exact structure. Such a one-sided exact category can be 2-universally embedded into its exact hull. We show that the exact hull of an additive regular category is again an additive regular category.

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arXiv2024

Auslander's formula and correspondence for exact categories

The Auslander correspondence is a fundamental result in Auslander-Reiten theory. In this paper we introduce the category $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ of admissibly finitely presented functors and use it to give a version of Auslander correspondence for any exact category $\mathcal{E}$. An important ingredient in the proof is the localization theory of exact categories. We also investigate how properties of $\mathcal{E}$ are reflected in $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$, for example being (weakly) idempotent complete or having enough projectives or injectives. Furthermore, we describe $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ as a subcategory of $\operatorname{mod}(\mathcal{E})$ when $\mathcal{E}$ is a resolving subcategory of an abelian category. This includes the category of Gorenstein projective modules and the category of maximal Cohen-Macaulay modules as special cases. Finally, we use $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ to give a bijection between exact structures on an idempotent complete additive category $\mathcal{C}$ and certain resolving subcategories of $\operatorname{mod}(\mathcal{C})$.

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arXiv2024

An introduction to monomorphism categories

This manuscript was written for the Proceedings of the ICRA 2022 in Buenos Aires. It can be divided into four parts: The first part is an introduction to the theory of monomorphism categories, including a short survey on some representation theoretic results. The second part is a summary of some recent results on monomorphism categories based on joint work with Nan Gao, Julian Külshammer and Chrysostomos Psaroudakis. It also includes a new result, namely a classification of all monomorphism categories of finite type over cyclic abelian groups. The third part concerns the relationship between submodule categories and $p$-valuated abelian groups. The last part contains a proof of wildness of a certain monomorphism category, rectifying a statement in the literature.

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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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arXiv2022

$n\mathbb{Z}$-cluster tilting subcategories for Nakayama algebras

$n\mathbb{Z}$-cluster tilting subcategories are an ideal setting for higher dimensional Auslander-Reiten theory. We give a complete classification of $n\mathbb{Z}$-cluster tilting subcategories of module categories of Nakayama algebras. In particular, we show that there are three kinds of Nakayama algebras that admit $n\mathbb{Z}$-cluster tilting subcategories: finite global dimension, selfinjective and non-Iwanaga-Gorenstein. Only the selfinjective ones can admit more than one $n\mathbb{Z}$-cluster tilting subcategory. It has been shown by the second author, that each such $n\mathbb{Z}$-cluster tilting subcategory induces an $n\mathbb{Z}$-cluster tilting subcategory of the corresponding singularity category. For each Nakayama algebra in our classification, we describe its singularity category, the canonical functor from its module category to its singularity category, and provide a complete comparison of $n\mathbb{Z}$-cluster tilting subcategories in the module category and the singularity category. This relies heavily of results by Shen, who described the singularity categories of all Nakayama algebras.

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arXiv2021

$d\mathbb{Z}$-cluster tilting subcategories of singularity categories

For an exact category $\mathcal{E}$ with enough projectives and with a $d\mathbb{Z}$-cluster tilting subcategory, we show that the singularity category of $\mathcal{E}$ admits a $d\mathbb{Z}$-cluster tilting subcategory. To do this we introduce cluster tilting subcategories of left triangulated categories, and we show that there is a correspondence between cluster tilting subcategories of $\mathcal{E}$ and $\underline{\mathcal{E}}$. We also deduce that the Gorenstein projectives of $\mathcal{E}$ admit a $d\mathbb{Z}$-cluster tilting subcategory under some assumptions. Finally, we compute the $d\mathbb{Z}$-cluster tilting subcategory of the singularity category for a finite-dimensional algebra which is not Iwanaga-Gorenstein.

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arXiv2021

Axiomatizing Subcategories of Abelian Categories

We investigate how to characterize subcategories of abelian categories in terms of intrinsic axioms. In particular, we find intrinsic axioms which characterize generating cogenerating functorially finite subcategories, precluster tilting subcategories, and cluster tilting subcategories of abelian categories. As a consequence we prove that any $d$-abelian category is equivalent to a $d$-cluster tilting subcategory of an abelian category, without any assumption on the categories being projectively generated.

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arXiv2020

A functorial approach to monomorphism categories for species I

We introduce a very general extension of the monomorphism category as studied by Ringel and Schmidmeier which in particular covers generalised species over locally bounded quivers. We prove that analogues of the kernel and cokernel functor send almost split sequences over the path algebra and the preprojective algebra to split or almost split sequences in the monomorphism category. We derive this from a general result on preservation of almost split morphisms under adjoint functors whose counit is a monomorphism. Despite of its generality, our monomorphism categories still allow for explicit computations as in the case of Ringel and Schmidmeier.

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arXiv2019

A generalization of the Nakayama functor

In this paper we introduce a generalization of the Nakayama functor for finite-dimensional algebras. This is obtained by abstracting its interaction with the forgetful functor to vector spaces. In particular, we characterize the Nakayama functor in terms of an ambidextrous adjunction of monads and comonads. In the second part we develop a theory of Gorenstein homological algebra for such Nakayama functor. We obtain analogues of several classical results for Iwanaga-Gorenstein algebras. One of our main examples is the module category $Λ\text{-}\operatorname{Mod}$ of a $k$-algebra $Λ$, where $k$ is a commutative ring and $Λ$ is finitely generated projective as a $k$-module.

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arXiv2019

Co-Gorenstein algebras

We review the theory of Co-Gorenstein algebras, which was introduced by Beligiannis in the article "The Homological Theory of Contravariantly Finite Subcategories: Gorenstein Categories, Auslander-Buchweitz Contexts and (Co-)Stabilization". We show a connection between Co-Gorenstein algebras and the Nakayama and Generalized Nakayama conjecture.

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arXiv2019

Gorenstein projective objects in functor categories

Let $k$ be a commutative ring, let $\mathcal{C}$ be a small, $k$-linear, Hom-finite, locally bounded category, and let $\mathcal{B}$ be a $k$-linear abelian category. We construct a Frobenius exact subcategory $\mathcal{GP}(\mathcal{G}_P\operatorname{proj}(\mathcal{B}^{\mathcal{C}}))$ of the functor category $\mathcal{B}^{\mathcal{C}}$, and we show that it is a subcategory of the Gorenstein projective objects $\mathcal{GP}(\mathcal{B}^{\mathcal{C}})$ in $\mathcal{B}^{\mathcal{C}}$. Furthermore, we obtain criteria for when $\mathcal{GP}(\mathcal{G}_P\operatorname{proj}(\mathcal{B}^{\mathcal{C}}))=\mathcal{GP}(\mathcal{B}^{\mathcal{C}})$. We show in examples that this can be used to compute $\mathcal{GP}(\mathcal{B}^{\mathcal{C}})$ explicitly.

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arXiv2018

An introduction to higher Auslander-Reiten theory

This article consists of an introduction to Iyama's higher Auslander-Reiten theory for Artin algebras from the viewpoint of higher homological algebra. We provide alternative proofs of the basic results in higher Auslander-Reiten theory, including the existence of $d$-almost-split sequences in $d$-cluster-tilting subcategories, following the approach to classical Auslander-Reiten theory due to Auslander, Reiten, and Smalø. We show that Krause's proof of Auslander's defect formula can be adapted to give a new proof of the defect formula for $d$-exact sequences. We use the defect formula to establish the existence of morphisms determined by objects in $d$-cluster-tilting subcategories.

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arXiv2016

Projectively generated $d$-abelian categories are $d$-cluster tilting

Building on work of Jasso, we prove that any projectively generated $d$-abelian category is equivalent to a $d$-cluster tilting subcategory of an abelian category with enough projectives. This supports the claim that $d$-abelian categories are good axiomatizations of $d$-cluster tilting subcategories.

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