In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities a... more In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with the representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological methods, using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite dimensional symmetric and satisfy $\tau^2=\id$. In particular, we compute 2-CY tilted algebras for simple and minimally elliptic curve singularities.
The aim of this paper is to introduce tau-tilting theory, which completes (classical) tilting the... more The aim of this paper is to introduce tau-tilting theory, which completes (classical) tilting theory from the viewpoint of mutation. It is well-known in tilting theory that an almost complete tilting module for any finite dimensional algebra over a field k is a direct summand of exactly 1 or 2 tilting modules. An important property in cluster tilting theory is that an almost complete cluster-tilting object in a 2-CY triangulated category is a direct summand of exactly 2 cluster-tilting objects. Reformulated for path algebras kQ, this says that an almost complete support tilting modules has exactly two complements. We generalize (support) tilting modules to what we call (support) tau-tilting modules, and show that an almost support tau-tilting module has exactly two complements for any finite dimensional algebra. For a finite dimensional k-algebra A, we establish bijections between functorially finite torsion classes in mod A, support tau-tilting modules and two-term silting complexes in Kb(proj A). Moreover these objects correspond bijectively to cluster-tilting objects in C if A is a 2-CY tilted algebra associated with a 2-CY triangulated category C. As an application, we show that the property of having two complements holds also for two-term silting complexes in Kb(proj A).
We prove the periodicities of the restricted T and Y-systems associated with the quantum affine a... more We prove the periodicities of the restricted T and Y-systems associated with the quantum affine algebra of type C_r, F_4, and G_2 at any level. We also prove the dilogarithm identities for these Y-systems at any level. Our proof is based on the tropical Y-systems and the categorification of the cluster algebra associated with any skew-symmetric matrix by Plamondon.
Weighted projective lines, introduced by Geigle and Lenzing in 1987, are one of the basic objects... more Weighted projective lines, introduced by Geigle and Lenzing in 1987, are one of the basic objects in representation theory. One key property is that 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 rings $R$ graded by abelian groups $L$ of rank $1$, which we call Geigle-Lenzing complete intersections. We study their Cohen-Macaulay representations, and show that there always exists a tilting object in the stable category of ${\mathsf CM}^LR$. As an application we study when $(R,L)$ is $d$-Cohen-Macaulay finite in the sense of higher dimensional Auslander-Reiten theory. Secondly, by applying the Serre construction to $(R,L)$, we introduce the category ${\mathsf coh} X$ of coherent sheaves on a Geigle-Lenzing projective space $X$. We show that there always exists a tilting bundle $T$ on $X$, and study the endomorphism algebra ${\rm End}_X(T)$ which we call a $d$-canonical algebra. Further we study when ${\mathsf coh} X$ is derived equivalent to a $d$-representation infinite algebra in the sense of higher dimensional Auslander-Reiten theory. Also we show that $d$-canonical algebras provide a rich source of $d$-Fano and $d$-anti-Fano algebras from non-commutative algebraic geometry. Moreover we observe Orlov-type semiorthogonal decompositions between the stable category of ${\mathsf CM}^LR$ and the derived category $D^b({\mathsf coh} X)$.
Given a representation-finite algebra B and a subalgebra A of B such that the Jacobson radicals o... more Given a representation-finite algebra B and a subalgebra A of B such that the Jacobson radicals of A and B coincide, we prove that the representation dimension of A is at most three. By a result of Igusa and Todorov, this implies that the finitistic dimension of A is finite.
We first generalize classical Auslander-Reiten duality for isolated singularities to cover singul... more We first generalize classical Auslander-Reiten duality for isolated singularities to cover singularities with a one-dimensional singular locus. We then define the notion of CT modules for non-isolated singularities and we show that these are intimately related to noncommutative crepant resolutions (NCCRs). When R has isolated singularities, CT modules recover the classical notion of cluster tilting modules but in general the two concepts differ. Then, wanting to generalize the notion of NCCRs to cover partial resolutions of Spec R, in the main body of this paper we introduce a theory of modifying and maximal modifying modules. Under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules for three-dimensional Gorenstein rings are shown to be derived equivalent. We then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory. Our mutation works in arbitrary dimension, and in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian.
By Auslander's algebraic McKay correspondence, the stable category of Cohen-Macaulay modules over... more By Auslander's algebraic McKay correspondence, the stable category of Cohen-Macaulay modules over a simple singularity is equivalent to the $1$-cluster category of the path algebra of a Dynkin quiver (i.e. the orbit category of the derived category by the action of the Auslander-Reiten translation). In this paper we give a systematic method to construct a similar type of triangle equivalence between the stable category of Cohen-Macaulay modules over a Gorenstein isolated singularity $R$ and the generalized (higher) cluster category of a finite dimensional algebra $\Lambda$. The key role is played by a bimodule Calabi-Yau algebra, which is the higher Auslander algebra of $R$ as well as the higher preprojective algebra of an extension of $\Lambda$. As a byproduct, we give a triangle equivalence between the stable category of graded Cohen-Macaulay $R$-modules and the derived category of $\Lambda$. Our main results apply in particular to a class of cyclic quotient singularities and to certain toric affine threefolds associated with dimer models.
From the viewpoint of mutation, we will give a brief survey of tilting theory and cluster-tilting... more From the viewpoint of mutation, we will give a brief survey of tilting theory and cluster-tilting theory together with a motivation from cluster algebras. Then we will give an introdution to \tau-tilting theory which was recently developed in [AIR].
We introduce a class of orders on $P^d$ called Geigle-Lenzing orders and show that they have tilt... more We introduce a class of orders on $P^d$ called Geigle-Lenzing orders and show that they have tilting bundles. Moreover we show that their module categories are equivalent to the categories of coherent sheaves on Geigle-Lenzing spaces introduced in Herschend, Iyama, Minamoto and Oppermann.
We study quivers with potential (QPs) whose Jacobian algebras are finite dimensional selfinjectiv... more We study quivers with potential (QPs) whose Jacobian algebras are finite dimensional selfinjective. They are an analogue of the `good QPs' studied by Bocklandt whose Jacobian algebras are 3-Calabi-Yau. We show that 2-representation-finite algebras are truncated Jacobian algebras of selfinjective QPs, which are factor algebras of Jacobian algebras by certain sets of arrows called cuts. We show that selfinjectivity of QPs is preserved under successive mutation with respect to orbits of the Nakayama permutation. We give a sufficient condition for all truncated Jacobian algebras of a fixed QP to be derived equivalent. We introduce planar QPs which provide us with a rich source of selfinjective QPs.
In this short paper we introduce a new triangulated category for rational surface singularities w... more In this short paper we introduce a new triangulated category for rational surface singularities which in the non-Gorenstein case acts as a substitute for the stable category of matrix factorizations. The category is formed as a Frobenius quotient of the category of special CM modules, and we classify the relatively projective-injective objects and thus describe the AR quiver of the quotient. Connections to the corresponding reconstruction algebras are also discussed.
In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities a... more In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with the representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological methods, using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite dimensional symmetric and satisfy $\tau^2=\id$. In particular, we compute 2-CY tilted algebras for simple and minimally elliptic curve singularities.
The aim of this paper is to introduce tau-tilting theory, which completes (classical) tilting the... more The aim of this paper is to introduce tau-tilting theory, which completes (classical) tilting theory from the viewpoint of mutation. It is well-known in tilting theory that an almost complete tilting module for any finite dimensional algebra over a field k is a direct summand of exactly 1 or 2 tilting modules. An important property in cluster tilting theory is that an almost complete cluster-tilting object in a 2-CY triangulated category is a direct summand of exactly 2 cluster-tilting objects. Reformulated for path algebras kQ, this says that an almost complete support tilting modules has exactly two complements. We generalize (support) tilting modules to what we call (support) tau-tilting modules, and show that an almost support tau-tilting module has exactly two complements for any finite dimensional algebra. For a finite dimensional k-algebra A, we establish bijections between functorially finite torsion classes in mod A, support tau-tilting modules and two-term silting complexes in Kb(proj A). Moreover these objects correspond bijectively to cluster-tilting objects in C if A is a 2-CY tilted algebra associated with a 2-CY triangulated category C. As an application, we show that the property of having two complements holds also for two-term silting complexes in Kb(proj A).
We prove the periodicities of the restricted T and Y-systems associated with the quantum affine a... more We prove the periodicities of the restricted T and Y-systems associated with the quantum affine algebra of type C_r, F_4, and G_2 at any level. We also prove the dilogarithm identities for these Y-systems at any level. Our proof is based on the tropical Y-systems and the categorification of the cluster algebra associated with any skew-symmetric matrix by Plamondon.
Weighted projective lines, introduced by Geigle and Lenzing in 1987, are one of the basic objects... more Weighted projective lines, introduced by Geigle and Lenzing in 1987, are one of the basic objects in representation theory. One key property is that 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 rings $R$ graded by abelian groups $L$ of rank $1$, which we call Geigle-Lenzing complete intersections. We study their Cohen-Macaulay representations, and show that there always exists a tilting object in the stable category of ${\mathsf CM}^LR$. As an application we study when $(R,L)$ is $d$-Cohen-Macaulay finite in the sense of higher dimensional Auslander-Reiten theory. Secondly, by applying the Serre construction to $(R,L)$, we introduce the category ${\mathsf coh} X$ of coherent sheaves on a Geigle-Lenzing projective space $X$. We show that there always exists a tilting bundle $T$ on $X$, and study the endomorphism algebra ${\rm End}_X(T)$ which we call a $d$-canonical algebra. Further we study when ${\mathsf coh} X$ is derived equivalent to a $d$-representation infinite algebra in the sense of higher dimensional Auslander-Reiten theory. Also we show that $d$-canonical algebras provide a rich source of $d$-Fano and $d$-anti-Fano algebras from non-commutative algebraic geometry. Moreover we observe Orlov-type semiorthogonal decompositions between the stable category of ${\mathsf CM}^LR$ and the derived category $D^b({\mathsf coh} X)$.
Given a representation-finite algebra B and a subalgebra A of B such that the Jacobson radicals o... more Given a representation-finite algebra B and a subalgebra A of B such that the Jacobson radicals of A and B coincide, we prove that the representation dimension of A is at most three. By a result of Igusa and Todorov, this implies that the finitistic dimension of A is finite.
We first generalize classical Auslander-Reiten duality for isolated singularities to cover singul... more We first generalize classical Auslander-Reiten duality for isolated singularities to cover singularities with a one-dimensional singular locus. We then define the notion of CT modules for non-isolated singularities and we show that these are intimately related to noncommutative crepant resolutions (NCCRs). When R has isolated singularities, CT modules recover the classical notion of cluster tilting modules but in general the two concepts differ. Then, wanting to generalize the notion of NCCRs to cover partial resolutions of Spec R, in the main body of this paper we introduce a theory of modifying and maximal modifying modules. Under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules for three-dimensional Gorenstein rings are shown to be derived equivalent. We then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory. Our mutation works in arbitrary dimension, and in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian.
By Auslander's algebraic McKay correspondence, the stable category of Cohen-Macaulay modules over... more By Auslander's algebraic McKay correspondence, the stable category of Cohen-Macaulay modules over a simple singularity is equivalent to the $1$-cluster category of the path algebra of a Dynkin quiver (i.e. the orbit category of the derived category by the action of the Auslander-Reiten translation). In this paper we give a systematic method to construct a similar type of triangle equivalence between the stable category of Cohen-Macaulay modules over a Gorenstein isolated singularity $R$ and the generalized (higher) cluster category of a finite dimensional algebra $\Lambda$. The key role is played by a bimodule Calabi-Yau algebra, which is the higher Auslander algebra of $R$ as well as the higher preprojective algebra of an extension of $\Lambda$. As a byproduct, we give a triangle equivalence between the stable category of graded Cohen-Macaulay $R$-modules and the derived category of $\Lambda$. Our main results apply in particular to a class of cyclic quotient singularities and to certain toric affine threefolds associated with dimer models.
From the viewpoint of mutation, we will give a brief survey of tilting theory and cluster-tilting... more From the viewpoint of mutation, we will give a brief survey of tilting theory and cluster-tilting theory together with a motivation from cluster algebras. Then we will give an introdution to \tau-tilting theory which was recently developed in [AIR].
We introduce a class of orders on $P^d$ called Geigle-Lenzing orders and show that they have tilt... more We introduce a class of orders on $P^d$ called Geigle-Lenzing orders and show that they have tilting bundles. Moreover we show that their module categories are equivalent to the categories of coherent sheaves on Geigle-Lenzing spaces introduced in Herschend, Iyama, Minamoto and Oppermann.
We study quivers with potential (QPs) whose Jacobian algebras are finite dimensional selfinjectiv... more We study quivers with potential (QPs) whose Jacobian algebras are finite dimensional selfinjective. They are an analogue of the `good QPs' studied by Bocklandt whose Jacobian algebras are 3-Calabi-Yau. We show that 2-representation-finite algebras are truncated Jacobian algebras of selfinjective QPs, which are factor algebras of Jacobian algebras by certain sets of arrows called cuts. We show that selfinjectivity of QPs is preserved under successive mutation with respect to orbits of the Nakayama permutation. We give a sufficient condition for all truncated Jacobian algebras of a fixed QP to be derived equivalent. We introduce planar QPs which provide us with a rich source of selfinjective QPs.
In this short paper we introduce a new triangulated category for rational surface singularities w... more In this short paper we introduce a new triangulated category for rational surface singularities which in the non-Gorenstein case acts as a substitute for the stable category of matrix factorizations. The category is formed as a Frobenius quotient of the category of special CM modules, and we classify the relatively projective-injective objects and thus describe the AR quiver of the quotient. Connections to the corresponding reconstruction algebras are also discussed.
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Papers by Osamu Iyama