# Ryo Takahashi

Nagoya University
Furocho, Chikusaku, Nagoya 464-8602, Japan

Department of Mathematics
University of Kansas
Lawrence, KS 66045-7523, USA

## Ring Theory and Representation Theory Seminar

### 2015

• Since April 2015

• January 23 (Friday), Room 309 in Math. Bldg.

10:30--12:00 Boris Lerner (University of New South Wales)
Title: Moduli stack of Serre stable representations

Abstract: In my recent work with Daniel Chan we introduce a new stack X^S that parameterises objects of an abelian category which are fixed, up to a shift in cohomological degree, by the Serre functor. Of particular interest to us, is when X^S is derived equivalent to the original abelian category. We do not have a answer for this yet, except in certain special cases. In particular, if the abelian category is the category of representations of a canonical algebra, then we show that X^S is a weighted protective line and it is known that the two are derived equivalent in this case. In my talk I will give a brief review on stacks, explain our construction and show how we able to recover the weighted projective lines from the canonical algebras.

16:30--18:00 Kazuhiko Kurano (Meiji University)
Title: DPD (Dolgachev-Pinkham-Demazure) construction for normal Z^n-graded domains

Abstract: Positively graded commutative Noetherian integrally closed domain with R_0 a field is described as R = \sum_n H^0(X,O(nD)), where X is a normal projective variety X and D is a Weil divisor with rational coefficient. This is called the DPD (Dolgachev-Pinkham-Demazure) construction. The purpose of this talk is to give the DPD construction for Noetherian Z^n-graded integrally closed domains whose homogeneous component of degree (0,...,0) is a field.

• January 9 (Friday), Room 309 in Math. Bldg.

15:00--16:30 Martin Herschend (Uppsala University)
Title: Survey on 2-representation finite algebras and selfinjective quivers with potential

Abstract: In higher dimensional Auslander-Reiten theory one studies algebras whose module categories have subcategories with suitable homological properties. The first case to consider are so called d-representation finite algebras. These are algebras of global dimension d having d-cluster tilting subcategories with finitely many indecomposables. Thus 1-representation finite algebras are hereditary representation finite algebras, which by Gabriel's Theorem, are given by Dynkin quivers. For 2-representation finite algebras no similar classification is known. Nevertheless, every 2-representation finite algebra is given by certain datum called a selfinjective quiver with potential and cut. I will give a survey of methods for constructing selfinjective quiver with potential developed in joint work with Osamu Iyama and more recently with Osamu Iyama, Ryo Takahashi and Kota Yamaura.

### 2014

• December 12 (Friday), Room 309 in Math. Bldg.

16:30--18:00 Ayako Itaba (Tokyo University of Science)
Title: On the decomposition of the Hochschild cohomology group of a monomial algebra satisfying a separability condition

Abstract: This is a joint work with T. Furuya and K. Sanada. In this talk, we consider the finite connected quiver Q having two subquivers Q^{(1)} and Q^{(2)} with Q=Q^{(1)}\cup Q^{(2)}=(Q_{0}^{(1)}\cup Q_{0}^{(2)},Q_{1}^{(1)}\cup Q_{1}^{(2)}). Suppose that Q^{(i)} is not a subquiver of Q^{(j)} where \{i,j\}=\{1,2\}. For a monomial algebra \Lambda=kQ/I obtained by the quiver Q, when the associated sequence of paths given by the set AP(n) (n\geq 0) of overlaps satisfies a certain separability condition, we propose the method so that we easily construct a minimal projective resolution of \Lambda as a right \Lambda^{e}-module and calculate the Hochschild cohomology group of \Lambda. In addition, we describe the conjecture about the Hochschild cohomology ring of \Lambda under the same assumpution.

• September 30 (Tuesday), Room 428 in Sci. Bldg. A

14:00--15:30 Tokuji Araya (Okayama University of Science)
Title: Gorensteinness on the punctured spectrum

Abstract: This is joint work with K. Iima. Let R be a commutative noetherian ring. A finitely generated R-module C is called semidualizing if the homothety map R\to Hom_R(C,C) is an isomorphism and if Ext^{>0}_R(C,C)=0. A free module of rank one and a dualizing module are semidualizing modules. A notion of n-torsionfree has been introduced by Auslander and Bridger as generalization of reflexive. In this talk, we will characterize n-torsionfreeness of modules with respect to a semidualizing module in terms of the Serre's condition (S_n). As an application we will give a characterization of Cohen-Macaulay rings which is Gorenstein on the punctured spectrum.

• September 16 (Tuesday), Room 428 in Sci. Bldg. A

14:00--15:30 Henning Krause (Bielefeld University)
Title: Relative homological algebra in triangulated categories

Abstract: In a paper from 1961 Butler and Horrocks studied extensions and resolutions in abelian categories relative to a central ring of operators. Soon after the theory was abandoned, probably because of the lack of serious applications. Some fifty years later we see that Butler and Horrocks' original idea is extremely useful in the study of triangulated categories. To illustrate this I will present a local-global principle for triangulated categories which is an analogue of the well-known principle from commutative algebra. This leads to a method of stratification relative a central ring of operators; it is based on joint work with Dave Benson and Srikanth Iyengar.

16:00--17:30 Alexander Zimmermann (Universite de Picardie)
Title: Stable equivalences of Morita type and tensor products

Abstract: Rickard showed that if A, B, C, D are k-algebras over a field k, and if A and B, as well as C and D are derived equivalent, then A\otimes C and B\otimes D are derived equivalent. Likewise he showed that if A and B are derived equivalent then the trivial extensions of A and of B are derived equivalent. Rickard for the first property, and König-Liu for the second posed the question if these properties might happen to hold for stable equivalences of Morita type. In joint work with Yuming Liu and Guodong Zhou we give a negative answer to both properties. The first example studies and uses stable equivalences between upper tiangular matrix rings and uses then an old result due to Auslander and Reiten. The answer to the second question uses properties of group rings of 2-groups, and in particular endotrivial modules as studied by Carlson and Thévenaz. Then, a result due to Liu and Xi allows to conclude.

• September 5 (Friday), Room 309 in Math. Bldg.

16:30--18:00 Lutz Hille (Universität Münster)
Title: Tilting bundles on rational surfaces and toric geometry

Abstract: (This is a report on joint work with Markus Perling.) We give a short introduction to (smooth, projective) rational surfaces and to toric surfaces. Then we consider exceptional sequences and tilting bundles on such surfaces. We present counter examples for the existence of full exceptional sequences of line bundles, and to several other conjectures. In particular, we use spherical twists to construct exceptional complexes of arbitrary length. As our main result, we associate to any full exceptional sequence a toric surface, which encodes the possible Cartan matrices for the endomorphism algebra. Finally, we construct, using full exceptional sequences of line bundles, a set of tilting bundles on any rational surfaces. This construction is related to universal extensions and to quasi-hereditary algebras.

• August 8 (Friday), Room 428 in Sci. Bldg. A

13:00--14:30 Erik Darpö (Mälardalen University)
Title: The representation rings of the dihedral 2-groups

Abstract: In this talk, I will review some results concerning tensor products of modules of dihedral 2-groups in characteristic 2. This includes results by Archer on odd-dimensional string modules, and some recent work by Gill, Hida and myself.

14:50--16:20 Steffen Oppermann (Norwegian University of Science and Technology)
Title: Quivers for silting mutation

Abstract: Aihara and Iyama introduced silting mutation as a generalization of tilting mutation. The idea is similar as for tilting mutation: Replace one indecomposable summand of a silting object by a different indecomposable, which is obtained from an approximation. In this talk we ask how the (derived) endomorphsim rings of silting objects change under mutation. More precisely I will give a combinatorial rule in terms of quivers. I will illustrate this rule on some simple (and well-known) examples.

16:40--18:10 Ragnar-Olaf Buchweitz (University of Toronto)
Title: Matrix Factorizations over Plane Projective Curves

Abstract: This is joint work in progress with my student Alexander Pavlov. We exhibit methods how to use Orlov's theorem on graded maximal Cohen-Macaulay modules in the particular case of hypersurfaces, especially plane projective curves, to find actual matrix factorizations on those hypersurfaces that belong to various (complexes of) sheaves on the geometric side. These methods partly extend earlier work of Eisenbud and joint work of the presenter with Avramov.

• July 25 (Friday), Room 207 in Sci. Bldg. A

13:00--14:30 Boris Lerner (Nagoya University)
Title: Geigle-Lenzing spaces and Geigle-Lenzing orders

Abstract: In this talk I will discuss 3 recent, related, papers:
[1] "Representation theory of Geigle-Lenzing complete intersections" - by Herschend, Iyama, MInamoto and Oppermann
[2] "Tilting bundles on orders on P^d" - by Iyama and myself
[3] "A recollement approach to Geigle-Lenzing weighted projective varieties" by Iyama, Oppermann and myself.
I will introduce the category of coherent sheaves on Geigle-Lenzing (GL) projective spaces which generalise the more well known category of coherent sheaves on GL weighted projective lines. These abelian categories are geometric in nature and have tilting sheaves making them derived equivalent to d-canonical algebras - which I will also introduce. I will then discuss how these categories can also be viewed from the perspective on noncommutative algebraic geometry using sheaves of noncommutative algebras called "GL orders". This insight allows for further generalisations which I will discuss. In particular, I will introduce a generalisation of the so called "squid" algebras.

• July 18 (Friday), Room 207 in Sci. Bldg. A

13:00--14:30 Hideto Asashiba (Shizuoka University)
Title: Gluing of derived equivalences along bimodules

Abstract: We fix a commutative ring k and a small category I, and consider the bicategory k-Cat^b whose objects are the small k-categories C, 1-morphisms are the bimodules over them and 2-morphisms are the bimodule morphisms (more precisely the 1-morphisms C --> C' are the C'-C-bimodules and the composite C --> C' --> C'' is given by the tensor over C'). For a lax functor X: I --> k-Cat^b, we define its "module category" Mod X, its "derived module category" D(Mod X), and the Grothendieck construction Gr(X) which enables us to construct new k-categories by gluing k-categories together along bimodules, in particular, triangular matrix algebras and the tensor algebras of k-species. We also define a notion of derived equivalences between lax functors I --> k-Cat^b. When k is a field, we can construct a derived equivalence between the Grothendieck constructions Gr(X) and Gr(X') of lax functors X and X': I --> k-Cat^b by gluing derived equivalences together between X(i) and X'(i) (i is an object of I) along bimodules X(a) and X'(a) (a is a morphism of I) if X and X' are derived equivalent.

15:00--16:30 Hiroyuki Minamoto (Osaka Prefecture University)
Title: Derived bi-commutator ring and DG-completion

Abstract: We discuss a derived version of bi-commutator ring (also known double centralizer ring or bi-endomorphism ring), which is a basic operation in ring theory. In classical setting, it is well-known that the first commutator ring and third commutator ring are isomorphic via the canonical morphism. Hence, for n > 1 the n-th commutator ring and n + 2-th commutator ring are canonically isomorphic. Because of this property, we may regard taking bi-commutator ring as kind of a closure operation". We observe that in derived setting, this property does not hold in general. As a result for a positive direction, we prove that proxy smallness (introduced by Dwyer-Greenlees-Iyengar) of the center DG-module of a commutator ring ensures that the property holds.

• July 1 (Tuesday), Room 428 in Sci. Bldg. A

14:00--15:30 Mayu Tsukamoto (Osaka City University)
Title: Hochschild cohomology of q-Schur algebras

Abstract: Let k be a field of characteristic zero. We denote the q-Schur algebra over k by S. S can be viewed as the quotient algebra of a general linear quantum group. We recall the definition of m-th Hochschild cohomology groups of S, HH^m(S):=Ext_{S^e}^m(S,S), where S^e:=S \otimes_k S^{op} acts on the left on S by left and right multiplication. In this talk, I will present calculation results of Hochschild cohomology groups of q-Schur algebras.

16:00--17:30 Yuta Kimura (Nagoya University)
Title: Tilting objects in stable categories of preprojective algebras

Abstract: Let Q be a finite acyclic quiver and \Pi the preprojective algebra of Q. Buan-Iyama-Reiten-Scott introduced the factor algebra \Pi_w associated with a element w in the Coxeter group of Q. They show that the stable category of submodules of free \Pi_w modules has cluster tilting objects. In this talk, we regard \Pi_w as a graded algebra and show that the stable category of graded submodules of free \Pi_w modules has tilting objects if w is a Coxeter sortable element.

• June 24 (Tuesday), Room 328 in Sci. Bldg. A

10:30--12:00 Ken-ichi Yoshida (Nihon University)
Title: Ulrich ideals of simple hypersurface singularities

Abstract: The notion of Ulrich ideals in a commutative Noetherian Cohen-Macaulay ring was recently introduced by S. Goto et al. It is known that any Cohen-Macaulay local ring of finite CM representation type contains only finitely many Ulrich ideals. Note that simple hypersurface singularities are rings of finite CM representation type. In this talk, we want to classify all Ulrich ideals of any simple hypersurface singularities.

• May 26 (Monday), Room 328 in Sci. Bldg. A

13:00--14:30 Shunsuke Takagi (University of Tokyo)
Title: Globally F-regular and Frobenius split surfaces

Abstract: Frobenius split and globally F-regular varieties are classes of projective varieties over a field of positive characteristic, defined in terms of Frobenius splitting. In this talk, I will explain relationships between globally F-regular surfaces and log Fano surfaces and between Frobenius split surfaces and log Calabi-Yau surfaces. This is joint work with Yoshinori Gongyo.

• April 24 (Thursday), Room 207 in Sci. Bldg. A

13:00--14:30 Kenta Ueyama (Shizuoka University)
Title: Examples of noncommutative graded isolated singularities

Abstract: Isolated singularities are an important class of rings in commutative ring theory. Recently, I introduced a notion of graded isolated singularity for noncommutative graded algebras, which agrees with the usual notion of isolated singularity if the algebra is commutative and generated in degree 1. In this talk, I will present some concrete examples of noncommutative graded isolated singularities.

• April 22 (Tuesday), Room 428 in Sci. Bldg. A

13:00--14:30 Ryo Kanda (Nagoya University)
Title: Classifying subcategories of quasi-coherent sheaves on locally noetherian schemes

Abstract: A prelocalizing subcategory of a Grothendieck category is a full subcategory closed under subobjects, quotient objects, and arbitrary direct sums. We classify the prelocalizing subcategories of the category of quasi-coherent sheaves on a locally noetherian scheme. The classification is given by using filters of subsheaves of the structure sheaf. Moreover, we characterize in terms of filters when a prelocalizing subcategory is closed under extensions/arbitrary direct products.

• April 15 (Tuesday), Room A428

13:00--14:30 Masahide Konishi (Nagoya University)
Title: How to basicalize KLR algebras

Abstract: KLR algebras are a class of infinite dimensional algebra, defined by two data: a quiver and a weight on its vertices. It is not so pathological, therefore we know we can obtain a basic algebra Morita equivalent to a KLR algebra as a quiver with relations in principle. In this talk, I will explain an explicit algorithm for that in some special cases.

• March 12 (Wednesday), Room A328

13:00--14:30 Akiyoshi Sannai (University of Tokyo)
Title: Invariant subring of Cox rings of K3 surfaces

Abstract: Huybrechts asked whether the invariant subrings of Cox rings by automorphism groups of K3surfaces are finitely generated. We give affirmative answer to the question.

15:00--16:30 Alvaro Nolla de Celis (UNIR)
TitleFDimer models with group actions

Abstract: Refining the work of Gullota, it is proved by A. Ishii and K. Ueda that given any lattice polygon P there exists a consistent dimer model D having P as its characteristic polygon and such the moduli space of representations M_\theta of the quiver associated to D is a non-commutative crepant resolution of the toric variety V associated to the cone over P. By considering a symmetric polygon with respect to a finite group action G and its corresponding symmetric dimer model, I will talk about the G-equivariant version of this result, showing some examples of this construction. This is a joint work (in progress) with A. Ishii and K. Ueda.

### 2013

• November 26 (Tuesday), Room 317 in Science Bldg. A

13:00--14:30 Idun Reiten (NTNU)
Title: Coxeter groups, preprojective algebras and path algebras 2

Abstract: For a finite acyclic quiver $Q$ we consider the associated Coxeter group $W_Q$, path algebra $kQ$ (for an algebraically closed field $k$) and preprojective algebra $\Pi_Q$. We discuss a one-one correspondence between the elements in $W_Q$ and the cofinite quotient closed subcategories of the category of finite dimensional $kQ$-modules, from work with Oppermann and Thomas. We include background matetial from papers with Iyama, Buan-Iyama-Scott and Amiot-Iyama-Todorov.

15:00--16:30 Timothy Logvinenko (Cardiff)
Title: Spherical DG-functors

Abstract: Seidel-Thomas twists are certain autoequivalences of the derived category D(X) of an algebraic variety X. Roughly, they are the mirror symmetry analogues of Dehn twists along Lagrangian spheres on a symplectic manifold. In this talk I will explain the definition of a Seidel-Thomas twist, illustrate it with a number of geometrical examples, and then report on my recent joint work with Rina Anno (UPitt) which generalises the notion from the twist along an object of D(X) to the twist along a functor into D(X). Geometrically, this corresponds to working, instead of a single object, with a fibration over a non-trivial base.

• November 7 (Thursday), Room 317 in Science Bldg. A

13:00--14:30 Joseph Karmazyn (Edinburgh)
Title: Deformed Reconstruction Algebras

Abstract: The preprojective algebras arise as non-commutative resolutions of Kleinian singularities. They have a very interesting class of deformations, the deformed preprojective algebras, which were introduced and studied by Crawley-Boevey and Holland. The reconstruction algebras were introduced by Wemyss as non-commutative resolutions of general surface quotient singularities. These algebras provide a generalisation of the preprojective algebras. It is then a natural question to ask whether there is a similar class of deformations for these algebras, generalising the deformed preprojective algebras. I will recall the case of the deformed preprojective algebras, and then discuss some of my results towards finding such a class of deformations for the reconstruction algebras.

• October 8 (Thursday), Room 317 in Science Bldg. A

13:00--14:30 Michael Wemyss (Edinburgh)
Title: From noncommutative deformations of curves to self-injective algebras

Abstract: In the first half of my talk, I will explain background about noncommutative deformations of modules and coherent sheaves. I will try to motivate why we want to study this, and why commutative deformations are in general different. In the second half of my talk, I will explain how noncommutative deformations arise in the study of flopping curves in 3-folds, and how tilting allows us to calculate this very easily. I will briefly link this to birational geometry, but I will mainly focus on the algebraic aspects. In particular, it is possible to use birational geometry to construct many examples of new self-injective finite dimensional algebras, and I will try to explain how understanding aspects of the representation theory of these algebras allows us to construct objects in geometry.

15:00--16:30 Ryo Kanda (Nagoya)
Title: Specialization orders on atom spectra of Grothendieck categories

Abstract: The atom spectrum of a Grothendieck category is a generalization of the prime spectrum of a commutative ring. The inclusion relation between prime ideals of a commutative ring is generalized as the specialization order on the atom spectrum with respect to some topology on it. We show that every partially ordered set is realized as the atom spectrum of some Grothendieck category. In order to do that, we introduce some method to construct Grothendieck categories from colored quivers.

• September 19 (Thursday), Room 317 in Science Bldg. A

13:00--14:30 Mitsuyasu Hashimoto (Nagoya)
Title: Equivariant class groups and almost principal fiber bundles

Abstract: We define the equivariant class group of a locally Krull scheme with an action of a flat group scheme, study its basic properties, and apply it to prove the finite generation of the class group of an invariant subring. We also define almost principal fiber bundles, and prove that the equivariant class groups behaves well with respect to this "quotient." We will see how almost principal fiber bundles are ubiquitous in invariant theory.

15:00--16:30 Yusuke Nakajima (Nagoya)
Title: Dual $F$-signature of Cohen-Macaulay modules over rational double points

Abstract: The dual $F$-signature is a numerical invariant defined via Frobenius morphisms in positive characteristic. It is known that the dual $F$-signature characterizes some singularities. However, the value of dual $F$-signature is not well known. In this talk, we determine the dual $F$-signature of Cohen-Macaulay modules over two-dimensional rational double points.

• August 9 (Friday), Room 317 in Science Bldg. A

13:00-- Martin Herschend (Nagoya)
Title: Tilting objects for Geigle-Lenzing projective spaces

15:00-- Erik Darpoe (Nagoya)
Title: n-representation finite self-injective algebras

• August 5 (Monday), Room 317 in Science Bldg. A

13:00--14:30 Aaron Chan (University of Aberdeen)
Title: Simple-minded and mutation theories of representation-finite self-injective algebras

Abstract: In a joint work with Steffen Koenig and Yuming Liu, we classify all simple-minded systems, a notion defined by my coauthors in their previous paper, of representation-finite self-injective algebras. For such algebras, we also exploit some interesting connections between these systems with other "simple-minded" and "projective-minded" objects, as well as their mutation theories. If time allows, I will also motivate the study for the connection between simple-minded systems and tau-tilting modules.

• July 9 (Tuesday), Room 317 in Science Bldg. A

13:00-- Hailong Dao (University of Kansas)
Title: Cohen-Macaulay cones and asymptotic behavior of system of ideals

Abstract: In this joint project with Kazuhiko Kurano, we study cones of maximal Cohen-Macaulay modules inside finite dimensional quotients of the Grothendieck group of a Cohen-Macaulay local ring R. I will describe what is known about these cones, and how their shapes are related to subtle questions about asymptotic behavior of graded families of ideals. Applications will be discussed, for example we can show that certain rings have only finitley many maximal Cohen-Macaulay modules of rank one.