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Yann Palu

Publications and source records attributed to Yann Palu.

At least 19 recordsLinked to original sources
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 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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arXiv2024

Extriangulated ideal quotients, with applications to cluster theory and gentle algebras

We extend results of Brüstle-Yang on ideal quotients of 2-term subcategories of perfect derived categories of non-positive dg algebras to a relative setting. We find a new interpretation of such quotients: they appear as prototypical examples of a new construction of quotients of extriangulated categories by ideals generated by morphisms from injectives to projectives. We apply our results to Frobenius exact cluster categories and Higgs categories with suitable relative extriangulated structures, and to categories of walks related to gentle algebras. In all three cases, the extriangulated structures are well-behaved (they are 0-Auslander) and their quotients are equivalent to homotopy categories of two-term complexes of projectives over suitable finite-dimensional algebras.

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arXiv2023

Auslander--Reiten theory in extriangulated categories

The notion of an extriangulated category gives a unification of existing theories in exact or abelian categories and in triangulated categories. In this article, we develop Auslander--Reiten theory for extriangulated categories. This unifies Auslander--Reiten theories developed in exact categories and triangulated categories independently. We give two different sets of sufficient conditions on the extriangulated category so that existence of almost split extensions becomes equivalent to that of an Auslander--Reiten--Serre duality. We also show that existence of almost split extensions is preserved under taking relative extriangulated categories, ideal quotients, and extension-closed subcategories. Moreover, we prove that the stable category $\underline{\mathscr{C}}$ of an extriangulated category $\mathscr{C}$ is a $τ$-category if $\mathscr{C}$ has enough projectives, almost split extensions and source morphisms. This gives various consequences on $\underline{\mathscr{C}}$, including Igusa--Todorov's Radical Layers Theorem, Auslander--Reiten Combinatorics on dimensions of Hom-spaces, and Reconstruction Theorem of the associated completely graded category of $\underline{\mathscr{C}}$ via the complete mesh category of the Auslander--Reiten species of $\underline{\mathscr{C}}$. Finally we prove that any locally finite symmetrizable $τ$-quiver (=valued translation quiver) is an Auslander--Reiten quiver of some extriangulated category with sink morphisms and source morphisms.

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arXiv2023

Some applications of extriangulated categories

Extriangulated categories axiomatize extension-closed subcategories of triangulated categories and generalise both exact categories and triangulated categories. This survey article presents three applications of extriangulated categories to homotopical algebra, algebraic combinatorics and representation theory. The first shows that, via some generalised Hovey's correspondence, extriangulated categories easily give rise to model category structures with triangulated homotopy categories. As a second application, extriangulated structures play a fondamental role in the construction of polytopal realisations of $g$-vector fans. This allows for a generalisation of ABHY's construction appearing in the study of scattering amplitudes in theoretical physics. Lastly, extriangulated categories provide a convenient framework for studying mutations in representation theory and flips in algebraic combinatorics. In nice enough hereditary extriangulated categories, there is a well-behaved theory of mutation for silting objects, which encompass cluster tilting, two-term silting, relative tilting, mutation of maximal almost-rigid modules, flip of dissections and mutation of intermediate co-$t$-structures.

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arXiv2023

Hereditary extriangulated categories: Silting objects, mutation, negative extensions

In this article, we initiate the study of hereditary extriangulated categories. Many important categories arising in representation theory in connection with various theories of mutation are hereditary extriangulated. Special cases include homotopy categories of 2-term complexes with projective components, which are related to silting mutation, and cluster categories (with relevant relative extriangulated structures) where cluster tilting mutation take place. We prove that there is a theory of irreducible mutation for maximal rigid objects and subcategories in hereditary extriangulated categories of dominant dimension 1. Applied to the examples above, this recovers 2-term silting mutation in triangulated categories and cluster tilting mutation. By constructing suitable extriangulated categories, we also recover tau-tilting mutation for gentle algebras and flips for their non-kissing facets. Combined with results by Adachi-Tsukamoto and Pauksztello-Zvonareva, our mutation also provides mutation for intermediate co-t-structures. In spirit of our earlier work, we study negative extensions in hereditary extriangulated categories. We give sufficient conditions for the existence of universal balanced negative extensions. We explicitly compute certain, a priori non-universal, versions of negative extensions in hereditary categories constructed from triangulated categories with rigid subcategories. We discuss examples where these two constructions give the same delta-functors and where they disagree.

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arXiv2022

Non-kissing complexes and tau-tilting for gentle algebras

We interpret the support $τ$-tilting complex of any gentle bound quiver as the non-kissing complex of walks on its blossoming quiver. Particularly relevant examples were previously studied for quivers defined by a subset of the grid or by a dissection of a polygon. We then focus on the case when the non-kissing complex is finite. We show that the graph of increasing flips on its facets is the Hasse diagram of a congruence-uniform lattice. Finally, we study its $\mathbf{g}$-vector fan and prove that it is the normal fan of a non-kissing associahedron.

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arXiv2022

Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors

We show that the mesh mutations are the minimal relations among the $\boldsymbol{g}$-vectors with respect to any initial seed in any finite type cluster algebra. We then use this algebraic result to derive geometric properties of the $\boldsymbol{g}$-vector fan: we show that the space of all its polytopal realizations is a simplicial cone, and we then observe that this property implies that all its realizations can be described as the intersection of a high dimensional positive orthant with well-chosen affine spaces. This sheds a new light on and extends earlier results of N. Arkani-Hamed, Y. Bai, S. He, and G. Yan in type $A$ and of V. Bazier-Matte, G. Douville, K. Mousavand, H. Thomas and E. Yildirim for acyclic initial seeds. Moreover, we use a similar approach to study the space of polytopal realizations of the $\boldsymbol{g}$-vector fans of another generalization of the associahedron: non-kissing complexes (a.k.a. support $τ$-tilting complexes) of gentle algebras. We show that the space of realizations of the non-kissing fan is simplicial when the gentle bound quiver is brick and $2$-acyclic, and we describe in this case its facet-defining inequalities in terms of mesh mutations. Along the way, we prove algebraic results on $2$-Calabi-Yau triangulated categories, and on extriangulated categories that are of independent interest. In particular, we prove, in those two setups, an analogue of a result of M. Auslander on minimal relations for Grothendieck groups of module categories.

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arXiv2021

Positive and negative extensions in extriangulated categories

We initiate the study of derived functors in the setting of extriangulated categories. By using coends, we adapt Yoneda's theory of higher extensions to this framework. We show that, when there are enough projectives or enough injectives, thus defined extensions agree with the ones defined earlier via projective or injective resolutions. For categories with enough projective or enough injective morphisms, we prove that these are right derived functors of the $\operatorname{Hom}$-bifunctor in either argument. Since $\operatorname{Hom}$ is only half-exact in each argument, it is natural to expect "negative extensions", i.e. its left derived functors, to exist and not necessarily vanish. We define negative extensions with respect to the first and to the second argument and show that they give rise to universal $δ$-functors, when there are enough projective or injective morphisms, respectively. In general, they are not balanced. However, for topological extriangulated categories, the existence of a balanced version of negative extensions follows from combining the work of Klemenc on exact $\infty$-categories with results of the second and third authors. We discuss various criteria under which one has the balance of the above bifunctors on the functorial or on the numerical levels. This happens, in particular, in the cases of exact or triangulated categories, and also in the case of the category of $n$-term complexes with projective components over a finite-dimensional algebra. Given a connected sequence of functors on an extriangulated category $(\mathcal{C},\mathbb{E},\mathfrak{s})$, we determine the maximal relative extriangulated structure, with respect to which the sequence is a $δ$-functor. We also find several equivalent criteria for the existence of enough projective or injective morphisms in a given extriangulated category.

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arXiv2018

Non-kissing and non-crossing complexes for locally gentle algebras

Starting from a locally gentle bound quiver, we define on the one hand a simplicial complex, called the non-kissing complex. On the other hand, we construct a punctured, marked, oriented surface with boundary, endowed with a pair of dual dissections. From those geometric data, we define two simplicial complexes: the accordion complex, and the slalom complex, generalizing work of A. Garver and T. McConville in the case of a disk. We show that all three simplicial complexes are isomorphic, and that they are pure and thin. In particular, there is a notion of mutation on their facets, akin to $τ$-tilting mutation. Along the way, we also construct inverse bijections between the set of isomorphism classes of locally gentle bound quivers and the set of homeomorphism classes of punctured, marked, oriented surfaces with boundary, endowed with a pair of dual dissections.

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arXiv2017

Mutation via Hovey twin cotorsion pairs and model structures in extriangulated categories

We give a simultaneous generalization of exact categories and triangulated categories, which is suitable for considering cotorsion pairs, and which we call extriangulated categories. Extension-closed, full subcategories of triangulated categories are examples of extriangulated categories. We give a bijective correspondence between some pairs of cotorsion pairs which we call Hovey twin cotorsion pairs, and admissible model structures. As a consequence, these model structures relate certain localizations with certain ideal quotients, via the homotopy category which can be given a triangulated structure. This gives a natural framework to formulate reduction and mutation of cotorsion pairs, applicable to both exact categories and triangulated categories. These results can be thought of as arguments towards the view that extriangulated categories are a convenient setup for writing down proofs which apply to both exact categories and (extension-closed subcategories of) triangulated categories.

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arXiv2015

Algebras of finite representation type arising from maximal rigid objects

We give a complete classification of all algebras appearing as endomorphism algebras of maximal rigid objects in standard 2-Calabi-Yau categories of finite type. Such categories are equivalent to certain orbit categories of derived categories of Dynkin algebras. It turns out that with one exception, all the algebras that occur are $2$-Calabi-Yau-tilted, and therefore appear in an earlier classification by Bertani-Økland and Oppermann. We explain this phenomenon by investigating the subcategories generated by rigid objects in standard 2-Calabi-Yau categories of finite type.

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arXiv2014

From triangulated categories to module categories via homotopical algebra

The category of modules over the endomorphism algebra of a rigid object in a Hom-finite triangulated category C has been given two different descriptions: On the one hand, as shown by Osamu Iyama and Yuji Yoshino, it is equivalent to an ideal quotient of a subcategory of C. On the other hand, Aslak Buan and Robert Marsh proved that this module category is also equivalent to some localisation of C. In this paper, we give a conceptual interpretation, inspired from homotopical algebra, of this double description. Our main aim, yet to be acheived, is to generalise Buan-Marsh's result to the case of Hom-infinite cluster categories. We note that, contrary to the more common case where a model category is a module category whose homotopy category is triangulated, we consider here some triangulated categories whose homotopy categories are module categories.

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arXiv2014

Nearly Morita equivalences and rigid objects

If two cluster-tilting objects of an acyclic cluster category are related by a mutation, then their endomorphism algebras are nearly-Morita equivalent [Buan-Marsh-Reiten], i.e. their module categories are equivalent "up to a simple module". This result has been generalised by D. Yang, using a result of P-G. Plamondon, to any simple mutation of maximal rigid objects in a 2-Calabi--Yau triangulated category. In this paper, we investigate the more general case of any mutation of a (non-necessarily maximal) rigid object in a triangulated category with a Serre functor. In that setup, the endomorphism algebras might not be nearly-Morita equivalent and we obtain a weaker property that we call pseudo-Morita equivalence. Inspired by results of Buan-Marsh, we also describe our result in terms of localisations.

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arXiv2013

Coloured quivers for rigid objects and partial triangulations: The unpunctured case

We associate a coloured quiver to a rigid object in a Hom-finite 2-Calabi--Yau triangulated category and to a partial triangulation on a marked (unpunctured) Riemann surface. We show that, in the case where the category is the generalised cluster category associated to a surface, the coloured quivers coincide. We also show that compatible notions of mutation can be defined and give an explicit description in the case of a disk. A partial description is given in the general 2-Calabi-Yau case. We show further that Iyama-Yoshino reduction can be interpreted as cutting along an arc in the surface.

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arXiv2011

Cluster characters II: A multiplication formula

Let $\mathcal{C}$ be a Hom-finite triangulated 2-Calabi-Yau category with a cluster tilting object. Under some constructibility assumptions on $\mathcal{C}$ which are satisfied for instance by cluster categories, by generalized cluster categories and by stable categories of modules over a preprojective algebra, we prove a multiplication formula for the cluster character associated with any cluster tilting object. This formula generalizes those obtained by Caldero-Keller for representation finite path algebras and by Xiao-Xu for finite-dimensional path algebras. It is analogous to a formula obtained by Geiss-Leclerc-Schröer in the context of preprojective algebras.

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