Ring Theory and Representation Theory Seminar
2015
 Since April 2015
 January 23 (Friday), Room 309 in Math. Bldg.
10:3012:00 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.
16:3018:00 Kazuhiko Kurano (Meiji University)
Title: DPD (DolgachevPinkhamDemazure) construction for normal Z^ngraded 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 (DolgachevPinkhamDemazure) construction. The purpose of this talk is to give the DPD construction for Noetherian Z^ngraded integrally closed domains whose homogeneous component of degree (0,...,0) is a field.
 January 9 (Friday), Room 309 in Math. Bldg.
15:0016:30 Martin Herschend (Uppsala University)
Title: Survey on 2representation finite algebras and selfinjective quivers with potential
Abstract: In higher dimensional AuslanderReiten theory one studies algebras whose module categories have subcategories with suitable homological properties. The first case to consider are so called drepresentation finite algebras. These are algebras of global dimension d having dcluster tilting subcategories with finitely many indecomposables. Thus 1representation finite algebras are hereditary representation finite algebras, which by Gabriel's Theorem, are given by Dynkin quivers. For 2representation finite algebras no similar classification is known. Nevertheless, every 2representation 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.
16:3018:00 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
14:0015:30 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 Rmodule 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 ntorsionfree has been introduced by Auslander and Bridger as generalization of reflexive. In this talk, we will characterize ntorsionfreeness 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 CohenMacaulay rings which is Gorenstein on the punctured spectrum.
 September 16 (Tuesday), Room 428 in Sci. Bldg. A
14:0015:30 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 localglobal principle for triangulated categories which is an analogue of the wellknown 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.
16:0017:30 Alexander Zimmermann (Universite de Picardie)
Title: Stable equivalences of Morita type and tensor products
Abstract: Rickard showed that if A, B, C, D are kalgebras 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 KönigLiu 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 2groups, 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.
16:3018:00 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 quasihereditary algebras.
 August 8 (Friday), Room 428 in Sci. Bldg. A
13:0014:30 Erik Darpö (Mälardalen University)
Title: The representation rings of the dihedral 2groups
Abstract: In this talk, I will review some results concerning tensor products of modules of dihedral 2groups in characteristic 2. This includes results by Archer on odddimensional string modules, and some recent work by Gill, Hida and myself.
14:5016:20 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 wellknown) examples.
16:4018:10 RagnarOlaf 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 CohenMacaulay 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
13:0014:30 Boris Lerner (Nagoya University)
Title: GeigleLenzing spaces and GeigleLenzing orders
Abstract: In this talk I will discuss 3 recent, related, papers:
[1] "Representation theory of GeigleLenzing 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 GeigleLenzing weighted projective varieties" by Iyama, Oppermann and myself.
I will introduce the category of coherent sheaves on GeigleLenzing (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 dcanonical 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
13:0014:30 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 kCat^b whose objects are the small kcategories C, 1morphisms are the bimodules over them and 2morphisms are the bimodule morphisms (more precisely the 1morphisms C > C' are the C'Cbimodules and the composite C > C' > C'' is given by the tensor over C'). For a lax functor X: I > kCat^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 kcategories by gluing kcategories together along bimodules, in particular, triangular matrix algebras and the tensor algebras of kspecies. We also define a notion of derived equivalences between lax functors I > kCat^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 > kCat^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.
15:0016:30 Hiroyuki Minamoto (Osaka Prefecture University)
Title: Derived bicommutator ring and DGcompletion
Abstract: We discuss a derived version of bicommutator ring (also known double centralizer ring or biendomorphism ring), which is a basic operation in ring theory. In classical setting, it is wellknown that the first commutator ring and third commutator ring are isomorphic via the canonical morphism. Hence, for n > 1 the nth commutator ring and n + 2th commutator ring are canonically isomorphic. Because of this property, we may regard taking bicommutator 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 DwyerGreenleesIyengar) of the center DGmodule of a commutator ring ensures that the property holds.
 July 1 (Tuesday), Room 428 in Sci. Bldg. A
14:0015:30 Mayu Tsukamoto (Osaka City University)
Title: Hochschild cohomology of qSchur algebras
Abstract: Let k be a field of characteristic zero. We denote the qSchur algebra over k by S. S can be viewed as the quotient algebra of a general linear quantum group. We recall the definition of mth 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 qSchur algebras.
16:0017:30 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. BuanIyamaReitenScott 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
10:3012:00 Kenichi Yoshida (Nihon University)
Title: Ulrich ideals of simple hypersurface singularities
Abstract: The notion of Ulrich ideals in a commutative Noetherian CohenMacaulay ring was recently introduced by S. Goto et al. It is known that any CohenMacaulay 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
13:0014:30 Shunsuke Takagi (University of Tokyo)
Title: Globally Fregular and Frobenius split surfaces
Abstract: Frobenius split and globally Fregular 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 Fregular surfaces and log Fano surfaces and between Frobenius split surfaces and log CalabiYau surfaces. This is joint work with Yoshinori Gongyo.
 April 24 (Thursday), Room 207 in Sci. Bldg. A
13:0014:30 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
13:0014:30 Ryo Kanda (Nagoya University)
Title: Classifying subcategories of quasicoherent 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 quasicoherent 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
13:0014:30 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
13:0014:30 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.
15:0016:30 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 noncommutative 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 Gequivariant 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
13:0014:30 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 oneone 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, BuanIyamaScott and AmiotIyamaTodorov.
15:0016:30 Timothy Logvinenko (Cardiff)
Title: Spherical DGfunctors
Abstract: SeidelThomas 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 SeidelThomas 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 nontrivial base.
 November 7 (Thursday), Room 317 in Science Bldg. A
13:0014:30 Joseph Karmazyn (Edinburgh)
Title: Deformed Reconstruction Algebras
Abstract: The preprojective algebras arise as noncommutative resolutions of Kleinian singularities. They have a very interesting class of deformations, the deformed preprojective algebras, which were introduced and studied by CrawleyBoevey and Holland. The reconstruction algebras were introduced by Wemyss as noncommutative 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
13:0014:30 Michael Wemyss (Edinburgh)
Title: From noncommutative deformations of curves to selfinjective 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 3folds, 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 selfinjective 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.
15:0016:30 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
13:0014:30 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.
15:0016:30 Yusuke Nakajima (Nagoya)
Title: Dual $F$signature of CohenMacaulay 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 CohenMacaulay modules over twodimensional rational double points.
 August 9 (Friday), Room 317 in Science Bldg. A
13:00 Martin Herschend (Nagoya)
Title: Tilting objects for GeigleLenzing projective spaces
15:00 Erik Darpoe (Nagoya)
Title: nrepresentation finite selfinjective algebras
 August 5 (Monday), Room 317 in Science Bldg. A
13:0014:30 Aaron Chan (University of Aberdeen)
Title: Simpleminded and mutation theories of representationfinite selfinjective algebras
Abstract: In a joint work with Steffen Koenig and Yuming Liu, we classify all simpleminded systems, a notion defined by my coauthors in their previous paper, of representationfinite selfinjective algebras. For such algebras, we also exploit some interesting connections between these systems with other "simpleminded" and "projectiveminded" objects, as well as their mutation theories. If time allows, I will also motivate the study for the connection between simpleminded systems and tautilting modules.
 July 9 (Tuesday), Room 317 in Science Bldg. A
13:00 Hailong Dao (University of Kansas)
Title: CohenMacaulay cones and asymptotic behavior of system of ideals
Abstract: In this joint project with Kazuhiko Kurano, we study cones of maximal CohenMacaulay modules inside finite dimensional quotients of the Grothendieck group of a CohenMacaulay 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 CohenMacaulay modules of rank one.
