arXiv subjects

Chrysostomos Psaroudakis

Publications and source records attributed to Chrysostomos Psaroudakis.

At least 19 recordsLinked to original sources

arXiv2026

Equivariant recollements and singular equivalences

In this paper we investigate equivariant recollements of abelian (resp. triangulated) categories. We first characterize when a recollement of abelian (resp. triangulated) categories induces an equivariant recollement, i.e. a recollement between the corresponding equivariant abelian (resp. triangulated) categories. We further investigate singular equivalences in the context of equivariant abelian recollements. In particular, we characterize when a singular equivalence induced by the quotient functor in an abelian recollement lifts to a singular equivalence induced by the equivariant quotient functor. As applications of our results: (i) we construct equivariant recollements for the derived category of a quasi-compact, quasi-separated scheme where the action comes from a subgroup of the automorphism group of the scheme and (ii) we establish new singular equivalences between certain skew group algebras.

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arXiv2025

Intrinsic homological algebra for triangulated categories

We propose a new framework for the study of homological properties for (compactly generated) triangulated categories such as regularity, finiteness of global or finitistic dimension, gorensteinness or injective generation and the relation between them. Our approach focuses on distinguished, intrinsically defined, subcategories and our main tool is the new notion of far-away orthogonality. We observe that these homological properties generalise previously studied properties on derived categories of modules over rings, and we use the generality of our theory to also examine those same attributes for the homotopy category of injectives and the big singularity category (in the sense of Krause) of an Artin algebra, as well as the derived category of a non-positive differential graded algebra. Finally, using our theory we recover and generalise various results in the theory of recollements of triangulated categories.

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arXiv2025

Monomial arrow removal and the finitistic dimension conjecture

In this paper, we introduce the monomial arrow removal operation for bound quiver algebras, and show that it is a novel reduction technique for determining the finiteness of the finitistic dimension. Our approach first develops a general method within the theory of abelian category cleft extensions. We then demonstrate that the specific conditions of this method are satisfied by the cleft extensions arising from strict monomial arrow removals. This crucial connection is established through the application of non-commutative Gröbner bases in the sense of Green. The theory is illustrated with various concrete examples.

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arXiv2025

Injective generation for graded rings

In this paper we investigate injective generation for graded rings. We first examine the relation between injective generation and graded injective generation for graded rings. We then reduce the study of injective generation for graded rings to the study of injective generation for certain Morita context rings and we provide sufficient conditions for injective generation of the latter. We then provide necessary and sufficient conditions so that injectives generate for tensor rings and for trivial extension rings. We provide two proofs for the class of tensor rings, the one uses covering theory and the other uses the framework of cleft extensions of module categories. We finally prove injective generation for twisted tensor products of finite dimensional 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

A higher dimensional Auslander-Iyama-Solberg correspondence

In this paper, we prove a higher dimensional version of Auslander-Iyama-Solberg correspondence. Iyama and Solberg have shown a bijection between $n$-minimal Auslander-Gorenstein algebras and $n$-precluster tilting modules. If $A$ is an $n$-minimal Auslander-Gorenstein algebra, then the pair $(A,P)$ is a relative $(n+1)$-Auslander-Gorenstein pair in the sense of the authors, where $P$ is the minimal faithful projective-injective left $A$-module. We establish a higher dimensional Auslander-Iyama-Solberg, where $P$ is replaced by any self-orthogonal module $Q$ having finite projective and injective dimension. This new correspondence provides a bijection between relative Auslander--Gorenstein pairs and a new class of objects that generalise precluster tilting modules. This way, we obtain a new correspondence coming from the modular representation theory of general linear groups.

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arXiv2024

Relative Auslander--Gorenstein Pairs

In this paper, we introduce and study relative Auslander--Gorenstein pairs. This consists of a finite-dimensional Gorenstein algebra together with a self-orthogonal module that provides a further homological feature of the algebra in terms of relative dominant dimension. These pairs will be called relative Auslander pairs whenever the algebra in question has finite global dimension. We characterize relative Auslander pairs by the existence and uniqueness of tilting-cotilting modules having higher values relative dominant and codominant dimension with respect to the self-orthogonal module. The same characterisation remains valid for relative Auslander--Gorenstein pairs if the self-orthogonal module has injective or projective dimension at most one. Our relative approach generalises and unifies the known results from the literature, for instance, the characterization of minimal Auslander--Gorenstein algebras. As an application of our methods, we prove that for any relative Auslander pair pieces of the module category of the endomorphism algebra of the self-orthogonal module can be identified with pieces of the module category of the endomorphism algebra of the unique tilting-cotilting module associated with the relative Auslander pair. We provide explicit examples of relative Auslander pairs.

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arXiv2023

Lifting recollements of abelian categories and model structures

We use Quillen model structures to show a systematic method to lift recollements of hereditary abelian model categories to recollements of their associated homotopy categories. To that end, we use the notion of Quillen adjoint triples and we investigate transfers of abelian model structures along adjoint pairs. Applications include liftings of recollements of module categories to their derived counterpart, liftings to homotopy categories that provide models for stable categories of Gorenstein projective and injective modules and liftings to homotopy categories of n-morphism categories over Iwanaga-Gorenstein rings.

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arXiv2022

Partial Serre duality and cocompact objects

A successful theme in the development of triangulated categories has been the study of compact objects. A weak dual notion called 0-cocompact objects was introduced in arXiv:1801.07995, motivated by the fact that sets of such objects cogenerate co-t-structures, dual to the t-structures generated by sets of compact objects. In the present paper, we show that the notion of 0-cocompact objects also appears naturally in the presence of certain dualities. We introduce "partial Serre duality", which is shown to link compact to 0-cocompact objects. We show that partial Serre duality gives rise to an Auslander--Reiten theory, which in turn implies a weaker notion of duality which we call "non-degenerate composition", and throughout this entire hierarchy of dualities the objects involved are 0-(co)compact. Furthermore, we produce explicit partial Serre functors for multiple flavors of homotopy categories, thus illustrating that this type of duality, as well as the resulting 0-cocompact objects, are abundant in prevalent triangulated categories.

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arXiv2021

Homological invariants of the arrow removal operation

In this paper we show that Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology are invariants under the arrow removal operation for a finite dimensional algebra.

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arXiv2021

Exact categories, big Cohen-Macaulay modules and finite representation type

One of the first remarkable results in the representation theory of artin algebras, due to Auslander and Ringel-Tachikawa, is the characterization of when an artin algebra is representation-finite. In this paper, we investigate aspects of representation-finiteness in the general context of exact categories in the sense of Quillen. In this framework, we introduce "big objects" and prove an Auslander-type "splitting-big-objects" theorem. Our approach generalises and unifies the known results from the literature. As a further application of our methods, we extend the theorems of Auslander and Ringel-Tachikawa to arbitrary dimension, i.e. we characterise when a Cohen-Macaulay order over a complete regular local ring is of finite representation type.

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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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arXiv2020

Ladders of recollements of abelian categories

Ladders of recollements of abelian categories are introduced, and used to address three general problems. Ladders of a certain height allow to construct recollements of triangulated categories, involving derived categories and singularity categories, from abelian ones. Ladders also allow to tilt abelian recollements, and ladders guarantee that properties like Gorenstein projective or injective are preserved by some functors in abelian recollements. Breaking symmetry is crucial in developing this theory.

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arXiv2018

Change of rings and singularity categories

We investigate the behavior of singularity categories and stable categories of Gorenstein projective modules along a morphism of rings. The natural context to approach the problem is via change of rings, that is, the classical adjoint triple between the module categories. In particular, we identify conditions on the change of rings to induce functors between the two singularity categories or the two stable categories of Gorenstein projective modules. Moreover, we study this problem at the level of `big singularity categories' in the sense of Krause. Along the way we establish an explicit construction of a right adjoint functor between certain homotopy categories. This is achieved by introducing the notion of 0-cocompact objects in triangulated categories and proving a dual version of Bousfield's localization lemma. We provide applications and examples illustrating our main results.

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arXiv2018

Reduction techniques for the finitistic dimension

In this paper we develop new reduction techniques for testing the finiteness of the finitistic dimension of a finite dimensional algebra over a field. Viewing the latter algebra as a quotient of a path algebra, we propose two operations on the quiver of the algebra, namely arrow removal and vertex removal. The former gives rise to cleft extensions and the latter to recollements. These two operations provide us new practical methods to detect algebras of finite finitistic dimension. We illustrate our methods with many examples.

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arXiv2018

Recollements of abelian categories and ideals in heredity chains - a recursive approach to quasi-hereditary algebras

Recollements of abelian categories are used as a basis of a homological and recursive approach to quasi-hereditary algebras. This yields a homological proof of Dlab and Ringel's characterisation of idempotent ideals occuring in heredity chains, which in turn characterises quasi-hereditary algebras recursively. Further applications are given to hereditary algebras and to Morita context rings.

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arXiv2017

Ladders of compactly generated triangulated categories and preprojective algebras

In this paper we characterize when a recollement of compactly generated triangulated categories admits a ladder of some height going either upwards or downwards. As an application, we show that the derived category of the preprojective algebra of Dynkin type $\mathbb{A}_n$ admits a periodic infinite ladder, where the one outer term in the recollement is the derived category of a differential graded algebra.

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arXiv2017

Realisation functors in tilting theory

Derived equivalences and t-structures are closely related. We use realisation functors associated to t-structures in triangulated categories to establish a derived Morita theory for abelian categories with a projective generator or an injective cogenerator. For this purpose we develop a theory of (noncompact, or large) tilting and cotilting objects that generalises the preceding notions in the literature. Within the scope of derived Morita theory for rings we show that, under some assumptions, the realisation functor is a derived tensor product. This fact allows us to approach a problem by Rickard on the shape of derived equivalences. Finally, we apply the techniques of this new derived Morita theory to show that a recollement of derived categories is a derived version of a recollement of abelian categories if and only if there are tilting or cotilting t-structures glueing to a tilting or a cotilting t-structure. As a further application, we answer a question by Xi on a standard form for recollements of derived module categories for finite dimensional hereditary algebras.

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