## Ring Theory and Representation Theory Seminar in 2015, 2016

The seminar is coorganized by Osamu Iyama and Ryo Takahashi.
Previous seminars 2014, 2013--.
19 December (Monday), 14:45-16:15, 409 math.
Aaron Chan (Nagoya)
Examples of gendo-symmetric algebras

Abstarct: An algebra is gendo-symmetric if it is isomorphic to the endomorphism ring of a generator over some symmetric algebra. Classical examples include various versions of Auslander algebras and certain families of Schur algebras associated to the symmetric groups. I will present a way to choose a generator for a symmetric algebra so that the resulting gendo-symmetric algebra has extremely simple formulae for their dominant dimension and left/right Gorenstein dimensions. Consequently, we show that all such gendo-symmetric algebras are Iwanaga-Gorenstein. A very computable class of examples is given by the representation-finite biserial gendo-symmetric algebras. The classification of these algebras via Brauer tree combinatorics will be discussed. I will then present a gendo-symmetric version of Rickard's derived equivalence classification of symmetric Nakayama algebras. If time permits , I will also discuss the structure of the stable category of the Gorenstein projective (a.k.a. maximal Cohen-Macaulay) modules. All results are obtained in joint works with René Marczinzik.
5 December (Monday), 14:45-16:15, 552 math.
Sota Asai (Nagoya)
Semibricks

Abstarct: This talk is based on arxiv:1610.05860. In the representaion theory of finite-dimensional algebras, the notion of semibricks, generalizing semisimple modules, is classical. I deal with semibricks from a view of $\tau$-tilting theory. My main result is a one-to-one correspondence between the support $\tau$-tilting modules and the semibricks satisfying some condition called left finitess. I would like to explain a new perspective of $\tau$-tilting theory given by this bijection.
29 November (Tuesday), 13:00-14:30, 14:45-16:15, 452 math.
Hiroyuki Minamoto (Osaka Prefecture), Kota Yamaura (Yamanashi)
On finite dimensional graded Iwanaga-Gorenstein algebras and their stable (graded) CM categories

Abstarct: We discuss finite dimensional graded Iwanaga-Gorenstein algebras A. An important consequence of our main results is that the Grothendieck group of the graded stable CM category of A is free of finite rank provided that the subalgebra of 0-th degree is of finite global dimension. By quasi-Veronese algebra construction, in principle, we may reduce our study to the case where A is a trivial extension algebra A = R + M with the grading deg R = 0 and deg M = 1. Thus our talk focus on this case and obtain the following three results. First, we give a necessary and sufficient condition that A is IG in terms of R and M by using derived tensor products and derived Homs. Second, in the sequel, we assume that R is of finite global dimension.Then, the condition that A is IG, has a triangulated categorical interpretation.More precisely, we give a condition for A to be IG in terms of the action of M on the derived category of R by the derived tensor product. Third, we show that if A is IG, then the graded stable CM category of A is realized as an admissible subcategory T of the derived category of R. We point that the admissible subcategory T plays an important role in the above categorical characterization. Finally, we give several applications. Among other things, for a path algebra R = KQ of an A_{3} quiver Q, we give a classification of R-R-bimodule M such that A = R + M is IG.
14 November (Monday),14:30–15:30, 15:45–16:45, 409 math.
Kawata, Shigeto (Nagoya City)
On Auslander-Reiten quivers of integral group rings

Abstarct: Let RG be the group ring of a finite group G over R, where R is either a complete discrete valuation ring O of characteristic zero or a field k of characteristic p. Assume that a block B of RG is of infinite representation type. Let C be a stable component of the Auslander-Reiten quiver of B. Webb showed the tree class of C is either a Euclidean diagram or one of the infinite Dynkin diagrams. Moreover, in the case R = k, Erdmann proved that the tree class of C is A_{¥infty} if B is a wild block and k is algebraically closed. In this talk, we consider Auslander-Reiten components containing certain Knorr lattices and Heller lattices for OG.
5 September (Monday), 10:30-12:00, 409 math.
Ryo Kanda (Osaka)
Non-exactness of direct products in the category of quasi-coherent sheaves
Abstarct: For a noetherian scheme which has an ample family of line bundles, we prove that the exactness of direct products in the category of quasi-coherent sheaves implies that the scheme is affine. The main tool of the proof is the Gabriel-Popescu embedding and its generalization, which realizes a Grothendieck category as a certain quotient category of a module/functor category.

5 September (Monday), 14:00-15:30, 409 math.
Jan Stovicek (Charles University in Prague)
Title: Tilting-cotilting correspondence
Abstract: In representation theory of finite dimensional algebras, a finite dimensional tilting module induces a derived equivalence which sends an injective cogenerator to a cotilting module on the other side. In this talk I will present a joint work with Leonid Positselski which vastly generalizes this correspondence. There is a completely analogous correspondence between Grothendieck categories with a (possibly large) tilting object on one hand, and categories of contramodules with a cotilting contramodule on the other hand. I will also discuss the notion of a contramodule, which is for many purposes a suitable generalization of a complete module over a topological ring.

5 September (Monday), 16:00-17:30, 409 math.
Exact categories with enough projectives
Abstarct: After reviewing the concept of an exact category (Quillen's sense) I will focus on exact categories with enough (relative) projectives. Under mild conditions such categories turn out to be full, extension-closed subcategories of module categories. Moreover, their exact structure is object-determined, that is, it depends only on the class of projectives. Our main focus is on categories of vector bundles (and related subcategories of coherent sheaves) and the case where the exact structure is determined by a class of line bundles. Interestingly, such an exact structure has enough projectives if and only if the chosen system of line bundles is wide. (Roughly speaking, wideness is a kind of ampleness.) Such wide systems of line bundles, in turn, play a prominent role in singularity theory. For instance, in the case of weighted projective lines, restriction of the Picard group of line bundles to suitable infinite cyclic subgroups yields the Kleinian (resp. Fuchsian) singularities.
5 August (Friday), 16:30-18:00, 452 math.
Silting objects and t-structures in silting-discrete triangulated categories

Abstarct: In this talk, we study a connection between silting objects and t-structures. For a silting-discrete triangulated category with a right adjacent t-structure, we obtain a bijection between the set of isoclasses of basic silting objects and the set of bounded t-structures. This is a generalization of Keller-Vossieck's result [KeV] and an analog of Koenig-Yang's result [KoY].
[KeV] B. Keller, D. Vossieck, Aisles in derived categories, Bull. Soc. Math. Belg. Ser. A 40 (1988), no. 2, 239--253.
[KoY] S.Koenig, D.Yang, Silting objects, simple-minded collections, t-structures and co-t-structures for finite-dimensional algebras, Doc. Math. 19 (2014), 403--438.
22 July (Friday), 13:00-14:30, 409 math.
Michio Yoshiwaki (Shizuoka University)
Decomposition theory of modules: the case of Kronecker algebra

Abstarct: This is joint work with H. Asashiba and K. Nakashima. Let $A$ be a finite-dimensional algebra over an algebraically closed field $k$. For any finite-dimensional $A$-module $M$ we give a general formula that computes the indecomposable decomposition of $M$ without decomposing it. As an example we apply this formula to the Kronecker algebra $A$ and give an explicit formula to compute the indecomposable decomposition of $M$, which enables us to make a computer program.
4 July (Monday), 16:30-18:00, 409 math.
Ryoichi Kase (Nara Women's University)
Weak orders on symmetric groups and support \tau-tilting posets

Abstarct: Mizuno showed that the poset of support \tau-tilting modules of a preprojective algebra of Dynkin type is isomorphic to the corresponding Weyl group with weak order. In particular, the support \tau-tilting poset of preprojective algebra of type A is given by the weak order on symmetric group. Recently, Iyama-Zhang proved that the Auslander algebra of truncated polynomial ring has same property. In this talk, we will characterize such algebras and give some examples.
28 June (Tuesday), 13:00-14:30, A358.
Colin Ingalls (New Brunswick)
Abstract: This is joint work with R. Buchweitz and E. Faber. Let W be subgroup of \rm{GL}(V) generated by reflections. Let S = k[V] be the polynomial ring and let z \in S cut out the hyperplane arrangement of mirrors in V. The discriminant is the image of the hyperplane arrangement in the quotient V/W which is cut out by z^2. Let A be the skew group algebra W \rtimes k[V]. Let e be the idempotent of kG corresponding to the trivial representation. Our main result is that End_{S^W}(S/zS) = A/AeA forms a noncommutative resolution of the discriminant since it is Koszul, has global dimension \rm{dim} V -1, and its centre S^W/(z^2) is polynomial functions on the discriminant.
20 June (Monday), 16:30-18:00, 409 math.
Hailong Dao (Kansas)
Cohomological support and the geometric join.

Abstarct: Let M,N be finitely generated modules over a local complete intersection R. Assume that all the modules Tor_i^R(M,N) are zero for i>0. We prove that the cohomological support (in the sense of Avramov-Buchweitz) of the tensor product of M, N is equal to the geometric join of the cohomological supports of M and N. This result gives a new connection between two active areas or research, and immediately produces several corollaries as well as new questions. This is joint work with William Sanders.
10 June (Friday), 13:00-14:30, 409 math.
Michel Van den Bergh (Hasselt)
Semi-orthogonal decompositions for equivariant derived categories for some reflection groups.

Abstarct: We will discuss a number of instances of semi-orthogonal decompositions of equivariant derived categories for reflection groups. We will discuss in particular the case of the symmetric group acting on its standard representation. To prove our results we use the Springer correspondence. This is joint work with Alexander Polishchuk.
20 May (Friday), 13:00-14:30, 409 math.
Hiroyuki Nakaoka (Kagoshima)
Twin cotorsion pairs on a triangulated category and related structures

Abstract: The notion of a twin cotorsion pair on a triangulated category generalizes t-structure, co-t-structure, cluster tilting subcategory and functorially finite rigid subcategory. Cotorsion pairs can be related to some homological structures on exact/triangulated categories. In this talk, we would like to introduce relations with reduction, mutation, recollement of cotorsion pairs, and model structures. This is partly based on a joint work with Yann Palu.
8 April (Friday), 13:00-14:30, 309 math.
Yuya Mizuno (Nagoya)

Abstract: 道多元環の加群圏はもっとも基本的な対象の一つである。 この講演ではコクセター群の元が道多元環の捻じれ対などの部分圏として実現できる結果を紹介し、 コクセター群における組み合わせ的な操作を圏論的に解釈できる事について述べたい。 なお本研究はHトーマス氏との研究に基づく。
10 February (Wednesday), 13:00-14:30, 409 math.
Taro Sakurai (Chiba)
On Cartan matrix and Loewy structure

Abstract: Artin 環上の射影直既約加群が組成因子に各既約加群をいくつ持つかを並べ て得られる行列をCartan 行列といい, 有限生成加群に関するradical seriesの隣り合う商 を並べた半単純加群の列をLoewy seriesという. 本講演ではCartan 行列の変形と してq-Cartan 行列を定義し(これは射影直既約加群たちのLoewy seriesと対応する), 有限p群の群多元環や有限非輪状箙の道多元環といった例で組合せ論的な解釈ができることを紹介する.
8 February (Monday), 13:00-14:30, 14:45-16:15, A317.
Orbifold Euler characteristic-The tubular case, and Large tilting sheaves.

We show Riemann-Hurwitz type formulae for the (orbifold) Euler characteristic. Over the real numbers this generalizes a well-known formula for 2-orbifolds to noncommutative 2-orbifolds. We give a full classification of noncommutative 2-orbifolds of nonnegative Euler characteristic (= real domestic, elliptic and tubular curves).

(Joint with Lidia Angeleri.) We work in the category Qcoh(X) of quasicoherent sheaves, which is a hereditary Grothendieck category. We show a general procedure to construct large (= non-coherent) tilting sheaves and present classifications in the tubular and in the elliptic cases. In particular we show that in these cases each large tilting sheaf has a slope.
29 January (Friday), 16:30-18:00, 409 math
Noncommutative real elliptic curves.

It is well-known that complex smooth projective curves correspond to compact Riemann surfaces. Similarly, real smooth projective curves correspond to the Klein surfaces. The real (=boundary) points form so-called ovals. Witt (Crelle's, 1934) studied Klein surfaces with an even number of marked points on each of its ovals. We show that this leads to noncommutative real smooth projective curves, which we call Witt curves. We then consider those curves of Euler characteristic zero, the noncommutative real elliptic curves. Prominent commutative examples are the Klein bottle, the Moebius band and the annulus, but there are also not-commutative ones. We will show that the Klein bottle has a (noncommutative) Witt curve as a so-called Fourier-Mukai partner.

22 January (Friday), 16:30-18:00, 409 math.
What is a tube?

We present a local-global principle involving the Auslander-Reiten translation tau. We compare tau with tubular shift functors restricted to tubes. Many examples will be discussed.
18 January(Monday), 13:30-15:00, 555 math.
Maiko Ono (Okayama)
On general principle for Auslander-Reiten duality

Abstract: This is a joint work with Yuji Yoshino. Auslander-Reiten(AR) duality is one of the most important theorem in the theory of maximal Cohen-Macaulay rings. In this talk, we discuss the generalization of AR duality in the derived category of chain complexes of $R$-modules. It means that the most general form of the theorem which naturally gives us the AR duality and its generalization found by Iyama and Wemyss. We call our conclusion the AR principle.

18 January(Monday), 15:30-17:00, 555 math.
Tokuji Araya (Okayama University of Science)
Thick subcategories over graded simple singularities of type D

Abstract: Takahashi classified the thick subcategories of the stable category of maximal Cohen-Macaulay modules over a hypersurface local ring. By his classification, we can see that if the base ring has a simple singularity, then the thick subcategories are trivial. On the other hand, if the base ring is graded, then there exist non-trivial thick subcategories. In this talk, we will classify the thick subcategories of the stable category of graded maximal Cohen-Macaulay modules over a graded hypersurface which has a simple singularity of type D.
13 January (Wednesday), 14:45-16:15 and 16:45-18:15, A317.
14:45-16:15 Axiomatic approach to weighted projective lines.
16:45-18:15 Constructing weighted projective lines from canonical algebras.

14 January (Thursday), 13:00-14:30 and 16:30-18:00, A317.
13:00-14:30 Vector bundles and Cohen-Macaulay modules.
16:30-18:00 Representation type and Euler characteristic.
16 December (Wednesday), 10:30-12:00, 309 math.
Yingying Zhang (Nanjing)
G-stable support tau-tilting modules

Abstract: Motivated by the tau-tilting theory developed by Adachi, Iyama and Reiten, for a finite-dimensional algebra with action by a finite group G, we introduce the notion of G-stable support tau-tilting modules. Then we establish bijections among G-stable support tau-tilting modules over \Lambda, G-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective \Lambda-modules, and G-stable functorially finite torsion classes in the category of finitely generated left \Lambda-modules. In the case when \Lambda is the endomorphism of a G-stable cluster-tilting object T over a Hom-finite 2-Calabi-Yau triangulated category C with a G-action, these are also in bijection with G-stable cluster-tilting objects in C. Moreover, we investigate the relationship between stable support tau-tilitng modules over \Lambda and the skew group algebra \LambdaG

14 December (Monday), 14:00-15:30, 555 math.
Kazuho Ozeki (Yamaguchi)
The structure of the Sally module of integrally closed ideals

Abstract: This is a joint work with M. E. Rossi. The first two Hilbert coefficients of a primary ideal play an important role in commutative algebra. In this talk we will give a structure theorem of the Sally module of integrally closed ideals I in a Cohen-Macaulay local ring A satisfying the equality \mathrm{e}_1(I)=\mathrm{e}_0(I)-\ell_A(A/I)+\ell_A(I^2/QI)+1, where Q is a minimal reduction of I, and \mathrm{e}_0(I) and \mathrm{e}_1(I) denote the first two Hilbert coefficients of I. This almost extremal value of \mathrm{e}_1(I) with respect classical inequalities holds a complete description of the homological and the numerical invariants of the associated graded ring.

14 December (Monday), 16:00-17:30, 555 math.
Naoki Taniguchi (Meiji)
Almost Gorenstein homogeneous rings

Abstract: This is a joint work with Shiro Goto and Ryo Takahashi. Almost Gorenstein rings are one of the candidates for a new class of Cohen-Macaulay ring, which may not be Gorenstein but still good, hopefully next to the Gorenstein rings. Theory of the almost Gorenstein ring was originally established by V. Barucci and R. Froberg. They innovated almost Gorenstein rings in the case where the local rings are analytically unramified of dimension one. However, since their notion of almost Gorenstein ring was not flexible for the analysis of analytically ramified case, in 2013 S. Goto, N. Matsuoka and T. T. Phuong extended the notion over arbitrary Cohen-Macaulay local rings but still of dimension one. More recently, in 2015 S. Goto, R. Takahashi and N. Taniguchi finally gave the definition of almost Gorenstein graded/local rings of higher dimension. In my talk we study the almost Gorenstein property for the homogeneous rings. Examples are explored to illustrate our theorems.
27 November (Friday), 14:45-16:15, A-317.
Naoya Hiramatsu (Kure national college of technology)
On the contravariantly determinedness of a stable category of Cohen-Macaulay modules.

Abstract: The concept of a contravariantly (or covariantly) determined category was introduced by Auslander and Reiten. They use it for investigating when indecomposable modules are determined by the composition factors. In this talk, we investigate when a stable category of Cohen-Macaulay modules is contravariantly determined. We give the sufficiently condition when R is of finite representation type. We also consider the converse of Butler, Auslander and Reiten's Theorem, which is the result on the relation for the Grothendieck group.

16:30-18:00 , A-317.
Hiroki Matsui (Nagoya)
Classifying resolving subcategories of exact categories via Grothendieck groups

Abstract:Classification problems of subcategories have been deeply considered so far, e.g., Serre subcategories of module categories by Gabriel, thick subcategories of perfect complexes by Hopkins-Neeman. They classified such subcategories via the spectra of noetherian commutative rings. On the other hand, Thomason classified dense triangulated subcategories of triangulated categories via their Grothendieck groups. In this talk, we discuss classifying resolving subcategories of exact categories via their Grothendieck groups.

14 September (Monday), 13:00-14:30, 309 math.
Claus Ringel (Bielefeld)
The Catalan combinatorics of non-crossing partitions, binary trees and parking functions: an interpretation in terms of the representation theory of artin algebras.

Abstract: The lecture will outline that the representation theory of artin algebras provides a categorification of the lattice of non-crossing partitions (and generalized non-crossing partitions). We will show that in this way several combinatorial constructions can be understood well.

Remark:The schedule has changed.

4 September (Friday), 13:00-14:30, 309 math.
Philipp Lampe (Bielefeld)
Ring theoretic properties of cluster algebras.
Abstract: When we categorify a cluster algebra via representation theory, its algebraic structure plays a crucial role. In this talk, we wish to discuss the effect of cluster structures on a given commutative algebra. In particular, we wish to address the following questions: When is a cluster algebra a unique factorization domain? What are its irreducible elements? Our main tool is its attached divisor class group.

14:45-16:15, 309 math.
Xiaojin Zhang (Nanjing)
On Gorenstein Projective Conjecture.
Abstract: In this talk, we try to present some recent developments of Gorenstein projective conjecture. More precisely, we show that Gorenstein projective conjecture is left and right symmetric, the vanishing condition can not be reduced in general and Gorenstein projective conjecture holds true for CM-finite algebras. In addition, we also show that for two Artin algebras A and B, if A is standard derived equivalent to B, then A satisfies Gorenstein projective conjecture if and only if so does B. This is a joint work with S. Pan.

16:30-18:00, 309 math.
Aaron Chan (Uppsala)
On the 2-representation theory of finitary 2-categories.
Abstract: In a series of works by Mazorchuk and Miemietz, they formalised the study of 2-representation theory stemmed from the categorification of Kazhdan-Lusztig theory and related areas. The goal of this talk is to provide a reading guide for this series of works. I will introduce the definitions, notions, and results used in these papers by looking at elementary examples which are more appealing for finite dimensional algebraists. If time allows, I will briefly talk about our attempts to develop homological algebra of 2-representations.

2 September (Wednesday), 16:30-18:00, 309 math.
Philipp Lampe (Bielefeld)
Quantization spaces of cluster algebras.

Abstract: This is joint work with Florian Gellert. Berenstein and Zelevinksy have introduced quantum cluster algebras as non-commutative deformations of ordinary cluster algebras. Unfortunately, a given cluster algebra does not necessarily admit a quantization, and if a quantization exists, then it is not necessarily unique. In this talk we prove that a cluster algebra admits a quantization if and only if the initial exchange matrix has full rank. Moreover, we give an explicit and natural basis for the space of all possible quantizations.

8 August (Friday), 10:30-12:00, 555 math.
Ryo Kanda (Nagoya)
Atom-molecule correspondence in Grothendieck categories.

Abstract: For a one-sided noetherian ring, Gabriel constructed two maps between the isomorphism classes of indecomposable injective modules and the two-sided prime ideals. We generalize these maps as maps between two spectra of a Grothendieck category with some property. The two spectra are called the atom spectrum and the molecule spectrum. This generalization provides a simple way to understand the construction of Gabriel's maps, and it is shown that they induce a bijection between the minimal elements of the atom spectrum and those of the molecule spectrum.

15 July (Wednesday), 10:30-12:00, 309 math.
Yasuyoshi Yonezawa (Nagoya)
MOY diagram and related topics.

17 June (Wednesday), 10:30-12:00, 309 math.

Osamu Iyama (Nagoya)
Lattice structure of preprojective algebras and Weyl groups

Abstract: Tilting theory of the preprojective algebra A of an acyclic quiver Q categorifies the corresponding Coxeter group W. When Q is Dynkin, there exists an isomorphism of lattices between W with the opposite weak order and torsion classes of A (that is, a full subcategory of mod A closed under factor modules and extensions). This give bijections between join-irreducible elements in W and stones of A (that is, A-modules X satisfying End_A(X)=k and Ext^1_A(X,X)=0). As an application, for type A, we characterize the lattice quotients of W coming from algebra quotients of A. This is a joint work with N. Reading, I. Reiten and H. Thomas.

13:00-14:30, A-328.
Laurent Demonet (Nagoya)
Algebras of partial triangulation

Abstract: Given a partial triangulation of a surface with marked points, and certain parameters, we define an algebra by quiver and relations and investigate its general properties. Thus, we get a family of algebras which are finite dimensional with explicit dimension which turns out to contain all Jacobian algebras coming from triangulation of surfaces and all Brauer graph algebras. It permits to extend results which are known for both families to a wider context and to unify the treatment. Moreover, in the case of Jacobian algebra, it gives a new presentation which does not need completion.

18 June (Thursday), 10:30-12:00, 307 math.
Yasuyoshi Yonezawa (Nagoya)
KLR algebra type A and matrix factorizations.

20 May (Wednesday), 15:00--, A328.

Kenichi Shimizu (Nagoya)
The Serre functor for a representation of a finite tensor category

Abstract: A finite tensor category over a field k is a k-linear abelian rigid monoidal category that is equivalent to the representation category of a finite-dimensional k-algebra (as just a k-linear category). As in the case of many other algebraic objects, the representation theory is an important subject in the study of finite tensor categories. A representation M of a finite tensor category C has a natural structure of a C-enriched category (via a kind of the tensor-hom adjunction), and thus the notion of the C-enriched Serre functor on M is defined by replacing Hom's in the definition of the ordinary Serre functor with the internal Hom functor. The first important observation is that the C-enriched Serre functor on M exists if and only if M is an exact C-module category in the sense of Etingof and Ostrik. This allows us to use the enriched Serre functor in the study of exact module categories. In this talk, I will introduce basic properties of the enriched Serre functor and give their applications to finite tensor categories.

23 April (Thursday), 16:30--, 555 math.

Masahide Konishi (Nagoya)
kS_{n} structures inside some cyclotomic KLR algebras

Abstract: Chuang-Miyachi proved that a block corresponding to Rouquier-core of level 1 cyclotomic Hecke algebras is isomorphic to $L^{\otimes w} \rtimes kS_{n}$ where $L$ is Brauer tree algebra of type A. Brundan-Kleshchev proved that cyclotomic KLR algebras of type $widetilde{A}$ and cyclotomic Hecke algebras are isomorphic. From these two theorems, we have natural question: find $L^{\otimes w} \rtimes kS_{n}$ constructions in level 1 cyclotomic KLR algebras of type $widetilde{A}$. In this talk, we give an answer for $q=-1$ case.

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