arXiv subjects

Osamu Iyama

Publications and source records attributed to Osamu Iyama.

At least 19 recordsLinked to original sources

arXiv2026

Preprojective algebras and generalisations: A short survey

The preprojective algebra of a hereditary algebra $H$ can be defined as a certain orbit construction of the regular representation generated by the Auslander-Reiten translation. In this short survey, we will look at two important generalisations, namely, the contracted preprojective algebra and the total preprojective algebra. We will include several open problems and questions motivated by examples in the hope to stimulate future research on general orbit algebras of $H$-modules.

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arXiv2026

Lattice structure of Weyl groups via representation theory of preprojective algebras

This paper studies the combinatorics of lattice congruences of the weak order on a finite Weyl group $W$, using representation theory of the corresponding preprojective algebra $Π$. Natural bijections are constructed between important objects including join-irreducible congruences, join-irreducible (respectively, meet-irreducible) elements of $W$, indecomposable $τ$-rigid (respectively, $τ^-$-rigid) modules and layers of $Π$. The lattice-theoretically natural labelling of the Hasse quiver by join-irreducible elements of $W$ is shown to coincide with the algebraically natural labelling by layers of $Π$. We show that layers of $Π$ are nothing but bricks (or equivalently stones, or 2-spherical modules). The forcing order on join-irreducible elements of $W$ (arising from the study of lattice congruences) is described algebraically in terms of the doubleton extension order. We give a combinatorial description of indecomposable $τ^-$-rigid modules for type $A$ and $D$.

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arXiv2026

Left modularity and extremality for (some) infinite lattices

For some important families of complete infinite lattices, we study some generalizations of two fundamental notions which are mostly treated for finite lattices. Specifically, for well-separated $κ$-lattices, and also for weakly atomic completely semidistributive lattices, we generalize the notions of left modularity and extremality. These two families of lattices coincide if restricted to finite lattices, but are distinct when infinite lattices are also included. For both families, we prove that extremality and left modularity imply each other. Furthermore, for weakly atomic completely semidistributive lattices, we give several conceptual characterizations of left modular elements, and show that the set of left modular elements form a complete distributive sublattice. Our results, combined with some recent work on finite lattices, imply that the weakly atomic completely semidistributive lattices that are left modular (or extremal) generalize the semidistributive trim lattices; from finite to infinite lattices. We then apply our results to the lattice of torsion classes of finite dimensional algebras, which are known to fall in the intersection of the two families treated in our work. For an algebra $A$, we obtain that the lattice of torsion classes is left modular (equivalently, extremal) if and only if $A$ is brick-directed. This leads to an abundance of concrete examples and non-examples.

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arXiv2026

Fractionally Calabi-Yau algebras and cluster tilting

We show that the class of twisted fractionally Calabi-Yau algebras of finite global dimension coincides with the stable endomorphism algebras of $d$-cluster tilting modules over $d$-representation-finite algebras. This is an application of our main result stating that an algebra $A$ of finite global dimension is twisted fractionally Calabi-Yau if and only if there exists $i$ such that the replicated algebra $A^{(i)}$ is a higher Auslander algebra if and only if there exist infinitely many $i$ such that $A^{(i)}$ is a higher Auslander algebra. This gives a new connection between the study of higher Auslander-Reiten theory and twisted fractionally Calabi-Yau algebras, and provides a new construction of large classes of higher Auslander algebras and higher representation-finite algebras. We give several applications such as an explicit characterisation of twisted $\frac{n}{2}$-Calabi-Yau algebras, and a triangle equivalence between the bounded derived category of a twisted fractionally Calabi-Yau algebra of finite global dimension and the $\mathbb{Z}$-graded stable module category of an associated higher preprojective algebra.

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arXiv2026

$M$-TF equivalences on the real Grothendieck groups

For an abelian length category $\mathcal{A}$ with only finitely many isoclasses of simple objects, we have the wall-chamber structure and the TF equivalence on the dual real Grothendieck group $K_0(\mathcal{A})_\mathbb{R}^*=\operatorname{Hom}_\mathbb{R}(K_0(\mathcal{A})_\mathbb{R},\mathbb{R})$, which are defined by semistable subcategories and semistable torsion pairs in $\mathcal{A}$ associated to elements $θ\in K_0(\mathcal{A})_\mathbb{R}^*$. In this paper, we introduce the $M$-TF equivalence for each object $M \in \mathcal{A}$ as a systematic way to coarsen the TF equivalence. We show that the set $Σ(M)$ of closures of $M$-TF equivalence classes is a rational generalized fan in $K_0(\mathcal{A})_\mathbb{R}^*$ which is finite and complete. More precisely, we show that $Σ(M)$ is the normal generalized fan of the Newton polytope $\mathrm{N}(M)$ in $K_0(\mathcal{A})_\mathbb{R}$. When $\mathcal{A}$ is the category of finitely generated modules over a finite dimensional algebra $A$, $Σ(M)$ can be regarded as a completion of a certain coarsening of the $g$-fan of $A$.

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arXiv2026

Triangle equivalences between Gorenstein tiled orders and incidence algebras of posets

We prove that for any $\mathbb{N}$-graded Gorenstein tiled order $A$, the stable category $\underline{\mathrm{CM}}^{\mathbb{Z}}A$ is triangle equivalent to the perfect derived category of the incidence algebra of a finite poset $\mathbb{V}_A^{op}$. Moreover, for a finite poset $P$, we prove that the incidence algebra of $P$ can be realized as the endomorphism algebra of a standard tilting object if and only if $P$ is either empty or has the maximum. We also study the behaviors of the corresponding poset under graded Morita equivalences and coverings of a Gorenstein tiled order. Finally, we classify Gorenstein tiled orders $A$ satisfying $|\mathbb{V}_A^{op}|\leq 3$.

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arXiv2026

Cohen-Macaulay representations of Artin-Schelter Gorenstein algebras of dimension one

Tilting theory is one of the central tools in modern representation theory, in particular in the study of Cohen-Macaulay representations. We study Cohen-Macaulay representations of $\mathbb N$-graded Artin-Schelter Gorenstein algebras $A$ of dimension one, without assuming the connectedness condition. This framework covers a broad class of noncommutative Gorenstein rings, including classical $\mathbb N$-graded Gorenstein orders. We prove that the stable category $\underline{\mathsf{CM}}_0^{\mathbb Z}A$ admits a silting object if and only if $A_0$ has finite global dimension. In this case we give such a silting object explicitly. Assuming that $A$ is ring-indecomposable, we further show that $\underline{\mathsf{CM}}_0^{\mathbb Z}A$ admits a tilting object if and only if either $A$ is Artin-Schelter regular or the average Gorenstein parameter of $A$ is non-positive. These results generalize those of Buchweitz, Iyama, and Yamaura. We give two proofs of the second result: one via Orlov-type semiorthogonal decompositions, and the other via a direct calculation. As an application, we show that for a Gorenstein tiled order $A$, the category $\underline{\mathsf{CM}}^{\mathbb Z}A$ is equivalent to the derived category of the incidence algebra of an explicitly constructed poset. We also apply our results and Koszul duality to study smooth noncommutative projective quadric hypersurfaces $\mathsf{qgr}\,B$ of arbitrary dimension. We prove that $\mathsf{D}^{\mathrm b}(\mathsf{qgr}\,B)$ admits an explicitly constructed tilting object, which contains the tilting object of $\underline{\mathsf{CM}}^{\mathbb Z}B$ due to Smith and Van den Bergh as a direct summand via Orlov's semiorthogonal decomposition.

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arXiv2025

Silting correspondences and Calabi-Yau dg algebras

This paper is devoted to studying two important classes of objects in triangulated categories; silting objects and $d$-cluster tilting objects, and their correspondences. First, we introduce the notion of $d$-silting objects as a generalization tilting objects whose endomorphism algebras have global dimension at most $d$. For a smooth dg algebra $A$ and its $(d+1)$-Calabi-Yau completion $Π$, we show that the induction functor gives an embedding from the poset $\operatorname{silt}^dA$ of $d$-silting objects of $A$ to the poset $\operatorname{silt}Π$ of silting objects of $Π$. Moreover, when $H^0Π$ is finite dimensional, this functor identifies the Hasse quiver of $\operatorname{silt}^dA$ as a full subquiver of the Hasse quiver of $\operatorname{silt}Π$. In this case, we also prove that each $d$-silting object $P$ of $A$ gives a $d$-cluster tilting subcategory of $\operatorname{per} A$ as the $ν[-d]$-orbit of $P$. Secondly, for a connective Calabi-Yau dg algebra $Π$, we study the map from $\operatorname{silt}Π$ to the set $d\text{-}\operatorname{ctilt}\mathcal{C}(Π)$ of $d$-cluster tilting objects in the cluster category $\mathcal{C}(Π)$. We call $Π$ $\mathcal{F}$-liftable if the induced map $\operatorname{silt}Π\cap\mathcal{F}\to d\text{-}\operatorname{ctilt}\mathcal{C}(Π)$ is bijective, where $\mathcal{F}$ is the fundamental domain in $\operatorname{per}Π$. We prove that $\mathcal{F}$-liftable Calabi-Yau dg algebras $Π$ such that $H^0Π$ is hereditary are precisely the Calabi-Yau completions of hereditary algebras. As an application, we obtain counter-examples to an open question posed in [IYa1]. We also study Calabi-Yau dg algebras such that the map $\operatorname{silt}Π\to d\text{-}\operatorname{ctilt}\mathcal{C}(Π)$ is surjective, which we call liftable. We explain our results by polynomial dg algebras and Calabi-Yau completions of type $A_2$.

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arXiv2025

Total preprojective algebras

We introduce total preprojective algebras $Ψ$ of path algebras of Dynkin quivers $kQ$, and prove that they are isomorphic to $2$-Auslander algebras of preprojective algebras $Π$ of $kQ$. In particular, $Ψ$ has global dimension $3$ and dominant dimension $3$. We also describe $Ψ$ as a tensor algebra of a certain explicit bimodule over the Auslander algebra of $kQ$. As an application, we give a presentation of $Ψ$ by explicit quivers with relations. More generally, we introduce total $(d+1)$-preprojective algebras of $d$-representation finite algebras, and give all the corresponding results.

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arXiv2025

Tilting theory for hypersurface singularities of dimension one

Any $\mathbb{N}$-graded commutative Gorenstein ring $R$ of Krull dimension one with $R_0$ a field admits a standard silting object $V$ in the stable category $\underline{\mathrm{CM}}_0^{\mathbb{Z}}R$, and the object $V$ is tilting if and only if the $a$-invariant $a$ is non-negative, as shown by Buchweitz, the first author, and Yamaura. In this article, under the additional assumption that $R$ is a hypersurface singularity, we prove that endomorphism algebra of $V$ is Iwanaga-Gorenstein of self-injective dimension at most $2$, and we give its explicit presentation in terms of a quiver with relation. In the case of where $a$ is negative, we prove that the dg endomorphism algebra of $V$ is Gorenstein, and we give its explicit presentation in terms of a dg path algebra. We explain our results by several examples including numerical semigroup algebras generated by two elements. Moreover, for each finite and countable Cohen-Macaulay representation type, we include the Auslander-Reiten quiver of the category $\mathrm{CM}_0^{\mathbb{Z}}R$ with the position of the standard silting object. As a step of the proof of our results, we give a characterization of Gorensteinness of homologically finite dg algebras in terms of Serre functors.

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arXiv2025

Brick-splitting Torsion Pairs and Left Modularity

We introduce the notion of brick-splitting torsion pairs as a modern analogue and generalization of the classical notion of splitting torsion pairs. A torsion pair is called brick-splitting if any given brick is either torsion or torsion-free with respect to that torsion pair. After giving some properties of these pairs, we fully characterize them in terms of some lattice-theoretical properties, including left modularity. This leads to the notion of brick-directed algebras, which are those for which there does not exist any cycle of non-zero non-isomorphisms between bricks. This class of algebras is a novel generalization of representation-directed algebras. We show that brick-directed algebras have many interesting properties and give several characterizations of them. In particular, we prove that a brick-finite algebra is brick-directed if and only if the lattice of torsion classes is left modular (or equivalently, extremal). We also give a characterization of brick-directed algebras in terms of their wall-and-chamber structure, as well as of a certain Newton polytope associated to them. Moreover, we introduce an explicit construction of an abundance of brick-directed algebras, both of the tame and wild representation types.

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arXiv2025

Fans and polytopes in tilting theory III: Classification of convex $g$-fans of rank 3

The $g$-fan $Σ(A)$ of a finite dimensional algebra $A$ is a non-singular fan in its real Grothendieck group, defined by tilting theory. If the union ${\rm P}(A)$ of the simplices associated with the cones of $Σ(A)$ is convex, we call $A$ $g$-convex. In this case, the $g$-polytope ${\rm P}(A)$ of $A$ is a reflexive polytope. Thus, in each dimension, there are only finitely many isomorphism classes of fans that can be realized as $g$-fans of $g$-convex algebras. An important problem is to classify such fans for a fixed dimension $d$. In this paper, we give a complete answer for the case $d=3$: we prove that there are precisely 61 convex $g$-fans of dimension 3 up to isomorphism. Our method is based on the decomposition of fans into the $2^3$ orthants in the real Grothendieck group of $A$, together with a detailed analysis of possible sequences of $g$-vectors arising from iterated mutations.

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arXiv2025

Torsion classes of extended Dynkin quivers over commutative rings

For a Noetherian $R$-algebra $Λ$, there is a canonical inclusion $\mathsf{tors}Λ\to\prod_{\mathfrak{p}\in \mathrm{Spec} R}\mathsf{tors}(κ(\mathfrak{p})Λ)$, and each element in the image satisfies a certain compatibility condition. We call $Λ$ compatible if the image coincides with the set of all compatible elements. For example, for a Dynkin quiver $Q$ and a commutative Noetherian ring $R$ containing a field, the path algebra $RQ$ is compatible. In this paper, we prove that $RQ$ is compatible when $Q$ is an extended Dynkin quiver and $R$ is either a Dedekind domain or a Noetherian semilocal normal ring of dimension two.

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arXiv2025

Schur roots and tilting modules of acyclic quivers over commutative rings

Let $Q$ be a finite acyclic quiver and $A_Q$ the cluster algebra of $Q$. It is well-known that for each field $k$, the additive equivalence classes of support tilting $kQ$-modules correspond bijectively with the clusters of $A_Q$. The aim of this paper is to generalize this result to any ring indecomposable commutative Noetherian ring $R$, that is, the additive equivalence classes of 2-term silting complexes of $RQ$ correspond bijectively with the clusters of $A_Q$. As an application, for a Dynkin quiver $Q$, we prove that the torsion classes of $\mathrm{mod} RQ$ corresponds bijectively with the order preserving maps from $\mathrm{Spec} R$ to the set of clusters.

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arXiv2024

On $τ$-tilting theory

We give a brief introduction to $τ$-tilting theory [AIR]. In particular, we will see how our theory unifies two different branches of tilting theory, namely, silting theory and cluster tilting theory. We also introduce the history and recent developments.

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arXiv2024

Enhanced Auslander-Reiten duality and Morita theorem for singularity categories

We establish a Morita theorem to construct triangle equivalences between the singularity categories of (commutative and non-commutative) Gorenstein rings and the cluster categories of finite dimensional algebras over fields, and more strongly, quasi-equivalences between their canonical dg enhancements. More precisely, we prove that such an equivalence exists as soon as we find a quasi-equivalence between the graded dg singularity category of a Gorenstein ring and the derived category of a finite dimensional algebra which can be done by finding a single tilting object. Our result is based on two key theorems on dg enhancements of cluster categories and of singularity categories, which are of independent interest. First we give a Morita-type theorem which realizes certain $\mathbb{Z}$-graded dg categories as dg orbit categories. Secondly, we show that the canonical dg enhancements of the singularity categories of symmetric orders have the bimodule Calabi-Yau property, which lifts the classical Auslander-Reiten duality on singularity categories. We apply our results to such classes of rings as Gorenstein rings of dimension at most $1$, quotient singularities, and Geigle-Lenzing complete intersections, including finite or infinite Grassmannian cluster categories, to realize their singularity categories as cluster categories of finite dimensional algebras.

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arXiv2024

Auslander-Reiten's Cohen-Macaulay algebras and contracted preprojective algebras

Auslander and Reiten called a finite dimensional algebra $A$ over a field Cohen-Macaulay if there is an $A$-bimodule $W$ which gives an equivalence between the category of finitely generated $A$-modules of finite projective dimension and the category of finitely generated $A$-modules of finite injective dimension. For example, Iwanaga-Gorenstein algebras and algebras with finitistic dimension zero on both sides are Cohen-Macaulay, and tensor products of Cohen-Macaulay algebras are again Cohen-Macaulay. They seem to be all of the known examples of Cohen-Macaulay algebras. In this paper, we give the first non-trivial class of Cohen-Macaulay algebras by showing that all contracted preprojective algebras of Dynkin type are Cohen-Macaulay. As a consequence, for each simple singularity $R$ and a maximal Cohen-Macaulay $R$-module $M$, the stable endomorphism algebra $\underline{End}_R(M)$ is Cohen-Macaulay. We also give a negative answer to a question of Auslander-Reiten asking whether the category $CM A$ of Cohen-Macaulay $A$-modules coincides with the category of $d$-th syzygies, where $d\ge1$ is the injective dimension of $W$. In fact, if $A$ is a Cohen-Macaulay algebra that is additionally $d$-Gorenstein in the sense of Auslander, then $CM A$ always coincides with the category of $d$-th syzygies.

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arXiv2024

Lattice theory of torsion classes: Beyond $τ$-tilting theory

The aim of this paper is to establish a lattice theoretical framework to study the partially ordered set $\operatorname{\mathsf{tors}} A$ of torsion classes over a finite-dimensional algebra $A$. We show that $\operatorname{\mathsf{tors}} A$ is a complete lattice which enjoys very strong properties, as bialgebraicity and complete semidistributivity. Thus its Hasse quiver carries the important part of its structure, and we introduce the brick labelling of its Hasse quiver and use it to study lattice congruences of $\operatorname{\mathsf{tors}} A$. In particular, we give a representation-theoretical interpretation of the so-called forcing order, and we prove that $\operatorname{\mathsf{tors}} A$ is completely congruence uniform. When $I$ is a two-sided ideal of $A$, $\operatorname{\mathsf{tors}} (A/I)$ is a lattice quotient of $\operatorname{\mathsf{tors}} A$ which is called an algebraic quotient, and the corresponding lattice congruence is called an algebraic congruence. The second part of this paper consists in studying algebraic congruences. We characterize the arrows of the Hasse quiver of $\operatorname{\mathsf{tors}} A$ that are contracted by an algebraic congruence in terms of the brick labelling. In the third part, we study in detail the case of preprojective algebras $Π$, for which $\operatorname{\mathsf{tors}} Π$ is the Weyl group endowed with the weak order. In particular, we give a new, more representation theoretical proof of the isomorphism between $\operatorname{\mathsf{tors}} k Q$ and the Cambrian lattice when $Q$ is a Dynkin quiver. We also prove that, in type $A$, the algebraic quotients of $\operatorname{\mathsf{tors}} Π$ are exactly its Hasse-regular lattice quotients.

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