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Osamu Iyama

Publications and source records attributed to Osamu Iyama.

At least 37 records · Page 2Linked to original sources
arXiv2024

Fans and polytopes in tilting theory I: Foundations

For a finite dimensional algebra $A$ over a field $k$, the 2-term silting complexes of $A$ gives a simplicial complex $Δ(A)$ called the $g$-simplicial complex. We give tilting theoretic interpretations of the $h$-vectors and Dehn-Sommerville equations of $Δ(A)$. Using $g$-vectors of 2-term silting complexes, $Δ(A)$ gives a nonsingular fan $Σ(A)$ in the real Grothendieck group $K_0(\mathsf{proj} A)_{\mathbb{R}}$ called the $g$-fan. We give several basic properties of $Σ(A)$ including sign-coherence, sign decomposition, idempotent reductions, Jasso reductions, pairwise positivity and a connection with Newton polytopes of $A$-modules. Moreover, $Σ(A)$ gives a (possibly infinite and non-convex) polytope $P(A)$ in $K_0(\mathsf{proj} A)_{\mathbb{R}}$ called the $g$-polytope of $A$. We call $A$ $g$-convex if $P(A)$ is convex. In this case, we show that it is a reflexive polytope, and that the dual polytope is given by the 2-term simple minded collections of $A$. There are precisely 7 convex $g$-polyogons up to isomorphism. We give a classification of algebras whose $g$-polytopes are smooth Fano. We study $g$-fans and $g$-polytopes of two important classes of algebras. We show that the $g$-fan of a classical or generalized preprojective algebra is given by the Coxeter fan. It is $g$-convex if and only if it is of type $A$ or $B$, and in this case, its $g$-polytope is the dual polytope of the short root polytope. Moreover we classify Brauer graph algebras which are $g$-convex, and describe their $g$-polytopes as the root polytopes of type $A$ or $C$.

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arXiv2024

Periodic trivial extension algebras and fractionally Calabi-Yau algebras

We study periodicity and twisted periodicity of the trivial extension algebra $T(A)$ of a finite-dimensional algebra $A$. Our main results show that (twisted) periodicity of $T(A)$ is equivalent to $A$ being (twisted) fractionally Calabi-Yau of finite global dimension. We also extend this result to a large class of self-injective orbit algebras. As a significant consequence, these results give a partial answer to the periodicity conjecture of Erdmann-Skowroński, which expects the classes of periodic and twisted periodic algebras to coincide. On the practical side, it allows us to construct a large number of new examples of periodic algebras and fractionally Calabi-Yau algebras. We also establish a connection between periodicity and cluster tilting theory, by showing that twisted periodicity of $T(A)$ is equivalent the $d$-representation-finiteness of the $r$-fold trivial extension algebra $T_r(A)$ for some $r,d\ge 1$. This answers a question by Darpö and Iyama. As applications of our results, we give answers to some other open questions. We construct periodic symmetric algebras of wild representation type with arbitrary large minimal period, answering a question by Skowroński. We also show that the class of twisted fractionally Calabi-Yau algebras is closed under derived equivalence, answering a question by Herschend and Iyama.

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arXiv2024

Dominant Auslander-Gorenstein algebras and mixed cluster tilting

We introduce the class of dominant Auslander-Gorenstein algebras as a generalisation of higher Auslander algebras and minimal Auslander-Gorenstein algebras, and give their basic properties. We also introduce mixed (pre)cluster tilting modules as a generalisation of (pre)cluster tilting modules, and establish an Auslander type correspondence by showing that dominant Auslander-Gorenstein (respectively, Auslander-regular) algebras correspond bijectively with mixed precluster (respectively, cluster) tilting modules. We show that every trivial extension algebra $T(A)$ of a $d$-representation-finite algebra A admits a mixed cluster tilting module and show that this can be seen as a generalisation of the well known result that $d$-representation-finite algebras are fractionally Calabi-Yau. We show that iterated SGC-extensions of a gendo-symmetric dominant Auslander-Gorenstein algebra admit mixed precluster tilting modules.

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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

Classifying subcategories of modules over Noetherian algebras

The aim of this paper is to unify classification theories of torsion classes of finite dimensional algebras and commutative Noetherian rings. For a commutative Noetherian ring $R$ and a module-finite $R$-algebra $Λ$, we study the set $\mathsf{tors} Λ$ (respectively, $\mathsf{torf}Λ$) of torsion (respectively, torsionfree) classes of the category of finitely generated $Λ$-modules. We construct a bijection from $\mathsf{torf}Λ$ to $\prod_{\mathfrak{p}} \mathsf{torf}(κ(\mathfrak{p}) \otimes_R Λ)$, and an embedding $Φ_{\rm t}$ from $\mathsf{tors} Λ$ to $\mathbb{T}_R(Λ):=\prod_{\mathfrak{p}} \mathsf{tors}(κ(\mathfrak{p}) \otimes_R Λ)$, where $\mathfrak{p}$ runs all prime ideals of $R$. When $Λ=R$, these give classifications of torsionfree classes, torsion classes and Serre subcategories of $\mathsf{mod} R$ due to Takahashi, Stanley-Wang and Gabriel. To give a description of $\mathrm{Im} Φ_{\rm t}$, we introduce the notion of compatible elements in $\mathbb{T}_R(Λ)$, and prove that all elements in $\mathrm{Im} Φ_{\rm t}$ are compatible. We give a sufficient condition on $(R, Λ)$ such that all compatible elements belong to $\mathrm{Im} Φ_{\rm t}$ (we call $(R, Λ)$ compatible in this case). For example, if $R$ is semi-local and $\dim R \leq 1$, then $(R, Λ)$ is compatible. We also give a sufficient condition in terms of silting $Λ$-modules. As an application, for a Dynkin quiver $Q$, $(R, RQ)$ is compatible and we have a poset isomorphism $\mathsf{tors} RQ \simeq \mathrm{Hom}_{\rm poset}(\mathrm{Spec} R, \mathfrak{C}_Q)$ for the Cambrian lattice $\mathfrak{C}_Q$ of $Q$.

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arXiv2023

Fans and polytopes in tilting theory II: $g$-fans of rank 2

The $g$-fan of a finite dimensional algebra is a fan in its real Grothendieck group defined by tilting theory. We give a classification of complete $g$-fans of rank 2. More explicitly, our first main result asserts that every complete sign-coherent fan of rank 2 is a $g$-fan of some finite dimensional algebra. Our proof is based on three fundamental results, Gluing Theorem, Rotation Theorem and Subdivision Theorem, which realize basic operations on fans in the level of finite dimensional algebras. For each of 16 convex sign-coherent fans $Σ$ of rank 2, our second main result gives a characterization of algebras $A$ of rank 2 satisfying $Σ(A)=Σ$. As a by-product of our method, we prove that for each positive integer $N$, there exists a finite dimensional algebra $A$ of rank 2 such that the Hasse quiver of the poset of 2-term silting complexes of $A$ has precisely $N$ connected components.

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arXiv2023

Semistable torsion classes and canonical decompositions in Grothendieck groups

We study two classes of torsion classes which generalize functorially finite torsion classes, that is, semistable torsion classes and morphism torsion classes. Semistable torsion classes are parametrized by the elements in the real Grothendieck group up to TF equivalence. We give a close connection between TF equivalence classes and the cones given by canonical decompositions of the spaces of projective presentations due to Derksen-Fei. More strongly, for $E$-tame algebras and hereditary algebras, we prove that TF equivalence classes containing lattice points are exactly the cones given by canonical decompositions. One of the key steps in our proof is a general description of semistable torsion classes in terms of morphism torsion classes. We also answer a question by Derksen-Fei negatively by giving examples of algebras which do not satisfy the ray condition. As an application of our results, we give an explicit description of TF equivalence classes of preprojective algebras of type $\widetilde{\mathbb{A}}$.

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arXiv2022

Triangulations of prisms and preprojective algebras of type $A$

We show that indecomposable two-term presilting complexes over $Π_{n}$, the preprojective algebra of $A_{n}$, are in bijection with internal $n$-simplices in the prism $Δ_{n} \times Δ_{1}$, the product of an $n$-simplex with a 1-simplex. We show further that this induces a bijection between triangulations of $Δ_{n} \times Δ_{1}$ and two-term silting complexes over $Π_{n}$ such that bistellar flips of triangulations correspond to mutations of two-term silting complexes. These bijections are shown to compatible with the known bijections involving the symmetric group.

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arXiv2021

Positive Fuss-Catalan numbers and Simple-minded systems in negative Calabi-Yau categories

We establish a bijection between $d$-simple-minded systems ($d$-SMSs) of $(-d)$-Calabi-Yau cluster category ${\cal C_{-d}}(H)$ and silting objects of ${\cal D^{\rm b}}(H)$ contained in $\cal D^{\le 0}\cap \cal D^{\ge 1-d}$ for hereditary algebra $H$ of Dynkin type and $d\ge 1$. We show that the number of $d$-SMSs in ${\cal C_{-d}}(H)$ is the positive Fuss-Catalan number $C_{d}^{+}(W)$ of the corresponding Weyl group $W$, by applying this bijection and Buan-Reiten-Thomas' and Zhu's results on Fomin-Reading's generalized cluster complexes. Our results are based on a refined version of silting-$t$-structure correspondence.

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arXiv2021

Distributive lattices and Auslander regular algebras

Let $L$ denote a finite lattice with at least two points and let $A$ denote the incidence algebra of $L$. We prove that $L$ is distributive if and only if $A$ is an Auslander regular ring, which gives a homological characterisation of distributive lattices. In this case, $A$ has an explicit minimal injective coresolution, whose $i$-th term is given by the elements of $L$ covered by precisely $i$ elements. We give a combinatorial formula of the Bass numbers of $A$. We apply our results to show that the order dimension of a distributive lattice $L$ coincides with the global dimension of the incidence algebra of $L$. Also we categorify the rowmotion bijection for distributive lattices using higher Auslander-Reiten translates of the simple modules.

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arXiv2020

Tilting modules over Auslander-Gorenstein Algebras

For a finite dimensional algebra $Λ$ and a non-negative integer $n$, we characterize when the set $\tilt_nΛ$ of additive equivalence classes of tilting modules with projective dimension at most $n$ has a minimal (or equivalently, minimum) element. This generalize results of Happel-Unger. Moreover, for an $n$-Gorenstein algebra $Λ$ with $n\geq 1$, we construct a minimal element in $\tilt_{n}Λ$. As a result, we give equivalent conditions for a $k$-Gorenstein algebra to be Iwanaga-Gorenstein. Moreover, for an $1$-Gorenstein algebra $Λ$ and its factor algebra $Γ=Λ/(e)$, we show that there is a bijection between $\tilt_1Λ$ and the set $\sttiltΓ$ of isomorphism classes of basic support $τ$-tilting $Γ$-modules, where $e$ is an idempotent such that $eΛ$ is the additive generator of projective-injective $Λ$-modules.

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arXiv2020

Classifying $τ$-tilting modules over the Auslander algebra of $K[x]/(x^n)$

We build a bijection between the set $\sttiltΛ$ of isomorphism classes of basic support $τ$-tilting modules over the Auslander algebra $Λ$ of $K[x]/(x^n)$ and the symmetric group $\mathfrak{S}_{n+1}$, which is an anti-isomorphism of partially ordered sets with respect to the generation order on $\sttiltΛ$ and the left order on $\mathfrak{S}_{n+1}$. This restricts to the bijection between the set $\tiltΛ$ of isomorphism classes of basic tilting $Λ$-modules and the symmetric group $\mathfrak{S}_n$ due to Brüstle, Hille, Ringel and Röhrle. Regarding the preprojective algebra $Γ$ of Dynkin type $A_n$ as a factor algebra of $Λ$, we show that the tensor functor $-\otimes_ΛΓ$ induces a bijection between $\sttiltΛ\to\sttiltΓ$. This recover Mizuno's bijection $\mathfrak{S}_{n+1}\to\sttiltΓ$ for type $A_n$.

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arXiv2020

Tilting theory for Gorenstein rings in dimension one

For a $Z$-graded Gorenstein ring $R$, we study the stable category $CM^ZR$ of $Z$-graded maximal Cohen-Macaulay $R$-modules, which is canonically triangle equivalent to the singularity category of Buchweitz and Orlov. Its thick subcategory given as the stable category of $CM_0^ZR$ is central in representation theory since it enjoys Auslander-Reiten-Serre duality and has almost split triangles. In the case $dim R=1$, we prove that the stable category of $CM_0^ZR$ always admits a silting object, and that it admits a tilting object if and only if either $R$ is regular or the $a$-invariant of $R$ is non-negative.

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arXiv2020

Higher preprojective algebras, Koszul algebras, and superpotentials

In this article we study higher preprojective algebras, showing that various known results for ordinary preprojective algebras generalize to the higher setting. We first show that the quiver of the higher preprojective algebra is obtained by adding arrows to the quiver of the original algebra, and these arrows can be read off from the last term of the bimodule resolution of the original algebra. In the Koszul case we are able to obtain the new relations of the higher preprojective algebra by differentiating a superpotential and we show that when our original algebra is $d$-hereditary all the relations come from the superpotential. We then construct projective resolutions of all simple modules for the higher preprojective algebra of a $d$-hereditary algebra. This allows us to recover various known homological properties of the higher preprojective algebras and to obtain a large class of almost Koszul dual pairs of algebras. We also show that when our original algebra is Koszul there is a natural map from the quadratic dual of the higher preprojective algebra to a graded trivial extension algebra.

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arXiv2020

Representation theory of Geigle-Lenzing complete intersections

Weighted projective lines, introduced by Geigle and Lenzing in 1987, are important objects in representation theory. They have tilting bundles, whose endomorphism algebras are the canonical algebras introduced by Ringel. The aim of this paper is to study their higher dimensional analogs. First, we introduce a certain class of commutative Gorenstein rings $R$ graded by abelian groups $L$ of rank $1$, which we call Geigle-Lenzing complete intersections. We study the stable category of Cohen-Macaulay representations $CM^LR$, which coincides with the singularity category $D_{sg}^L(R)$. We show that the stable category of $CM^LR$ is triangle equivalent to $D^b(mod A^{CM})$ for a finite dimensional algebra $A^{CM}$, which we call the CM-canonical algebra. As an application, we classify the $(R,L)$ that are Cohen-Macaulay finite. We also give sufficient conditions for $(R,L)$ to be $d$-Cohen-Macaulay finite in the sense of higher Auslander-Reiten theory. Secondly, we study a new class of non-commutative projective schemes in the sense of Artin-Zhang, i.e. the category $coh X=mod^LR/mod^L_0R$ of coherent sheaves on the Geigle-Lenzing projective space $X$. Geometrically this is the quotient stack $[(Spec R-{R_+})/Spec k[L]]$. We show that $D^b(coh X)$ is triangle equivalent to $D^b(mod A^{ca})$ for a finite dimensional algebra $A^{ca}$, which we call a $d$-canonical algebra. We study when $X$ is $d$-vector bundle finite, and when $X$ is derived equivalent to a $d$-representation infinite algebra in the sense of higher Auslander-Reiten theory. Our $d$-canonical algebras provide a rich source of $d$-Fano and $d$-anti-Fano algebras in non-commutative algebraic geometry. We also observe Orlov-type semiorthogonal decompositions of $D_{sg}^L(R)$ and $D^b(coh X)$.

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arXiv2020

d-Representation-finite self-injective algebras

In this paper, we initiate the study of higher-dimensional Auslander-Reiten theory of self-injective algebras. We give a systematic construction of (weakly) $d$-representation-finite self-injective algebras as orbit algebras of the repetitive categories of algebras of finite global dimension satisfying a certain finiteness condition for the Serre functor. The condition holds, in particular, for all fractionally Calabi-Yau algebras of global dimension at most $d$. This generalizes Riedtmann's classical construction of representation-finite self-injective algebras. Our method is based on an adaptation of Gabriel's covering theory for $k$-linear categories to the setting of higher-dimensional Auslander-Reiten theory. Applications include $n$-fold trivial extensions and (classical and higher) preprojective algebras, which are shown to be $d$-representation-finite in many cases. We also get a complete classification of all $d$-representation-finite self-injective Nakayama algebras for arbitrary $d$.

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arXiv2020

Weighted Projective Lines and Rational Surface Singularities

In this paper we study rational surface singularities R with star shaped dual graphs, and under very mild assumptions on the self-intersection numbers we give an explicit description of all their special Cohen-Macaulay modules. We do this by realising R as a certain Z-graded Veronese subring S^x of the homogeneous coordinate ring S of the Geigle-Lenzing weighted projective line X, and we realise the special CM modules as explicitly described summands of the canonical tilting bundle on X. We then give a second proof that these are special CM modules by comparing qgr S^x and coh X, and we also give a necessary and sufficient combinatorial criterion for these to be equivalent categories. In turn, we show that qgr S^x is equivalent to qgr of the reconstruction algebra, and that the degree zero piece of the reconstruction algebra coincides with Ringel's canonical algebra. This implies that the reconstruction algebra contains the canonical algebra, and furthermore its qgr category is derived equivalent to the canonical algebra, thus linking the reconstruction algebra of rational surface singularities to the canonical algebra of representation theory.

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