Tokyo-Nagoya Algebra Seminar

Seminar information archive ~12/02Next seminarFuture seminars 12/03~

Organizer(s) Noriyuki Abe, Aaron Chan, Erik Darpoe, Osamu Iyama, Tsutomu Nakamura, Hiroyuki Nakaoka, Ryo Takahashi

Seminar information archive

2022/10/20

16:40-18:10   Online
Please see the reference URL for details on the online seminar.
Martin Kalck (Freiburg University)
A surface and a threefold with equivalent singularity categories (English)
[ Abstract ]
We discuss a triangle equivalence between singularity categories of an
affine surface and an affine threefold.
Both are isolated cyclic quotient singularities.
This seems to be the first (non-trivial) example of a singular
equivalence involving varieties of even and odd Krull dimension.

The same approach recovers a result of Dong Yang showing a singular
equivalence between certain cyclic quotient singularities in dimension
2 and certain finite dimensional commutative algebras.

This talk is based on https://arxiv.org/pdf/2103.06584.pdf
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2022/07/20

10:30-12:00   Online
Please see the reference URL for details on the online seminar.
Yuki Imamura (Osaka University)
Grothendieck enriched categories (Japanese)
[ Abstract ]
Grothendieck圏は、入射的余生成子の存在や随伴関手定理の成立など、アーベル圏の中でも特に良い性質を持つことで知られる。通常Grothendieck圏は、生成子を持つ余完備なアーベル圏であって、フィルター余極限を取る関手が完全関手になるような圏として内在的な性質で以て定義されるが、加群圏の"良い部分圏"として実現できるという外在的な特徴づけ(Gabriel-Popescuの定理)も存在する。アーベル圏が自然なプレ加法圏(アーベル群の圏Ab上の豊穣圏)の構造を持つことから、Gabriel-Popescuの定理はAb-豊穣圏に対する定理だと思うことができる。本講演では、より一般のGrothendieckモノイダル圏V上の豊穣圏に対してGabriel-Popescuの定理の一般化を定式化し証明する。特にVとしてアーベル群の複体の圏Chを取ることによりGrothendieck圏のdg圏類似とそのGabriel-Popescuの定理が得られることも確認する。
[ Reference URL ]
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2022/07/06

10:30-12:00   Online
Please see the reference URL for details on the online seminar.
Nicholas Williams (The University of Tokyo)
Cyclic polytopes and higher Auslander--Reiten theory 3 (English)
[ Abstract ]
This continues part 2. In the third talk, we consider the relationship between the objects from the first two talks. We explain how triangulations of even-dimensional cyclic polytopes may be interpreted in terms of tilting modules, cluster-tilting objects, or d-silting complexes. We then proceed in the d-silting framework, and show how the higher Stasheff--Tamari orders may be interpreted algebraically for even dimensions. We explain how this allows one to interpret odd-dimensional triangulations algebraically, namely, as equivalence classes of d-maximal green sequences. We briefly digress to consider the issue of equivalence of maximal green sequences itself. We then show how one can interpret the higher Stasheff--Tamari orders on equivalence classes of d-maximal green sequences. We finish by drawing out some consequences of this algebraic interpretation of the higher Stasheff--Tamari orders.
[ Reference URL ]
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2022/06/29

10:30-12:00   Online
Please see the reference URL for details on the online seminar.
Nicholas Williams (The University of Tokyo)
Cyclic polytopes and higher Auslander--Reiten theory 2 (English)
[ Abstract ]
This continues part 1. In the second talk, we focus on higher Auslander--Reiten theory. We survey the basic setting of this theory, starting with d-cluster-tilting subcategories of module categories. We then move on to d-cluster-tilting subcategories of derived categories in the case of d-representation-finite d-hereditary algebras. We explain how one can construct (d + 2)-angulated cluster categories for such algebras, generalising classical cluster categories. We finally consider the d-almost positive category, which is the higher generalisation of the category of two-term complexes. Throughout, we illustrate the results using the higher Auslander algebras of type A, and explain how the different categories can be interpreted combinatorially for these algebras.
[ Reference URL ]
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2022/06/22

17:00-18:30   Online
Please see the reference URL for details on the online seminar.
Martin Kalck (Freiburg University)
Update on singular equivalences between commutative rings (English)
[ Abstract ]
We will start with an introduction to singularity categories, which were first studied by Buchweitz and later rediscovered by Orlov. Then we will explain what is known about triangle equivalences between singularity categories of commutative rings, recalling results of Knörrer, D. Yang (based on our joint works on relative singularity categories. This result also follows from work of Kawamata and was generalized in a joint work with Karmazyn) and a new equivalence obtained in arXiv:2103.06584.

In the remainder of the talk, we will focus on the case of Gorenstein isolated singularities and especially hypersurfaces, where we give a complete description of quasi-equivalence classes of dg enhancements of singularity categories, answering a question of Keller & Shinder. This is based on arXiv:2108.03292.
[ Reference URL ]
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2022/06/15

10:30-12:00   Online
Please see the reference URL for details on the online seminar.
Nicholas Williams (The University of Tokyo)
Cyclic polytopes and higher Auslander--Reiten theory 1 (English)
[ Abstract ]
In this series of three talks, we expand upon the previous talk given at the seminar and study the relationship between cyclic polytopes and higher Auslander--Reiten theory in more detail.
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNA/2021/Williams-Cyclic_polytopes_and_higher_AR.pdf

In the first talk, we focus on cyclic polytopes. We survey important properties of cyclic polytopes, such as different ways to construct them, the Upper Bound Theorem, and their Ramsey-theoretic properties. We then move on to triangulations of cyclic polytopes. We give efficient combinatorial descriptions of triangulations of even-dimensional and odd-dimensional cyclic polytopes, which we will use in subsequent talks. We finally define the higher Stasheff--Tamari orders on triangulations of cyclic polytopes. We give important results on the orders, including Rambau's Theorem, and the equality of the two orders.
[ Reference URL ]
https://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2022/06/08

10:30-12:00   Online
Please see the reference URL for details on the online seminar.
Masahiko Yoshinaga (Osaka University)
超平面配置の特性準多項式 II (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2022/06/01

10:30-12:00   Online
Please see the reference URL for details on the online seminar.
Masahiko Yoshinaga (Osaka University)
超平面配置の特性準多項式 I (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2022/04/13

10:30-12:00   Online
Please see the reference URL for details on the online seminar.
Yuta Kimura (Osaka Metropolitan University)
Tilting ideals of deformed preprojective algebras
[ Abstract ]
Let $K$ be a field and $Q$ a finite quiver. For a weight $\lambda \in K^{|Q_0|}$, the deformed preprojective algebra $\Pi^{\lambda}$ was introduced by Crawley-Boevey and Holland to study deformations of Kleinian singularities. If $\lambda = 0$, then $\Pi^{0}$ is the preprojective algebra introduced by Gelfand-Ponomarev, and appears many areas of mathematics. Among interesting properties of $\Pi^{0}$, the classification of tilting ideals of $\Pi^{0}$, shown by Buan-Iyama-Reiten-Scott, is fundamental and important. They constructed a bijection between the set of tilting ideals of $\Pi^{0}$ and the Coxeter group $W_Q$ of $Q$.

In this talk, when $Q$ is non-Dynkin, we see that $\Pi^{\lambda}$ is a $2$-Calabi-Yau algebra, and show that there exists a bijection between tilting ideals and a Coxeter group. However $W_Q$ does not appear, since $\Pi^{\lambda}$ is not necessary basic. Instead of $W_Q$, we consider the Ext-quiver of rigid simple modules, and use its Coxeter group. When $Q$ is an extended Dynkin quiver, we see that the Ext-quiver is finite and this has an information of singularities of a representation space of semisimple modules.
This is joint work with William Crawley-Boevey.
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2022/03/11

13:00-14:30   Online
Please see the URL below for details on the online seminar.
Shigeo Koshitani (Chiba University)
Modular representation theory of finite groups – local versus global II (English)
[ Abstract ]
We are going to talk about representation theory of finite groups. In the 1st part it will be on "Equivalences of categories ” showing up for block theory in modular representation theory, and it should be kind of introductory lecture/talk. So the audience is supposed to have knowledge only of definitions of groups, rings, fields, modules, and so on. In the 2nd part we will discuss kind of local—global conjectures in modular representation theory of finite groups, that originally and essentially are due to Richard Brauer (1901–77).
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2022/03/09

13:00-14:30   Online
Please see the URL below for details on the online seminar.
Shigeo Koshitani (Chiba University)
Modular representation theory of finite groups – local versus global I (English)
[ Abstract ]
We are going to talk about representation theory of finite groups. In the 1st part it will be on "Equivalences of categories ” showing up for block theory in modular representation theory, and it should be kind of introductory lecture/talk. So the audience is supposed to have knowledge only of definitions of groups, rings, fields, modules, and so on. In the 2nd part we will discuss kind of local—global conjectures in modular representation theory of finite groups, that originally and essentially are due to Richard Brauer (1901–77).
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2022/01/21

16:45-18:15   Online
Please see the URL below for details on the online seminar.
Haruhisa Enomoto (Osaka Prefecture University)
Exact-categorical properties of subcategories of abelian categories 2 (Japanese)
[ Abstract ]
Quillen's exact category is a powerful framework for studying extension-closed subcategories of abelian categories, and provides many interesting questions on such subcategories.
In the first talk, I will explain the basics of some properties and invariants of exact categories (e.g. the Jordan-Holder property, simple objects, and Grothendieck monoid).
In the second talk, I will give some results and questions about particular classes of exact categories arising in the representation theory of algebras (e.g. torsion(-free) classes over path algebras and preprojective algebras).
If time permits, I will discuss questions of whether these results can be generalized to extriangulated categories (extension-closed subcategories of triangulated categories).
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2022/01/18

15:00-16:30   Online
Please see the URL below for details on the online seminar.
Haruhisa Enomoto (Osaka Prefecture University)
Exact-categorical properties of subcategories of abelian categories 1 (Japanese)
[ Abstract ]
Quillen's exact category is a powerful framework for studying extension-closed subcategories of abelian categories, and provides many interesting questions on such subcategories.
In the first talk, I will explain the basics of some properties and invariants of exact categories (e.g. the Jordan-Holder property, simple objects, and Grothendieck monoid).
In the second talk, I will give some results and questions about particular classes of exact categories arising in the representation theory of algebras (e.g. torsion(-free) classes over path algebras and preprojective algebras).
If time permits, I will discuss questions of whether these results can be generalized to extriangulated categories (extension-closed subcategories of triangulated categories).
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2021/12/16

16:45-18:15   Online
Please see the URL below for details on the online seminar.
Nicholas Williams (University of Cologne)
Cyclic polytopes and higher Auslander-Reiten theory (English)
[ Abstract ]
Oppermann and Thomas show that tilting modules over Iyama’s higher Auslander algebras of type A are in bijection with triangulations of even-dimensional cyclic polytopes. Triangulations of cyclic polytopes are partially ordered in two natural ways known as the higher Stasheff-Tamari orders, which were introduced in the 1990s by Kapranov, Voevodsky, Edelman, and Reiner as higher-dimensional generalisations of the Tamari lattice. These two partial orders were conjectured to be equal in 1996 by Edelman and Reiner, and we prove that this conjecture is true. We further show how the higher Stasheff-Tamari orders correspond in even dimensions to natural orders on tilting modules which were studied by Riedtmann, Schofield, Happel, and Unger. This then allows us to complete the picture of Oppermann and Thomas by showing that triangulations of odd-dimensional cyclic polytopes correspond to equivalence classes of d-maximal green sequences, which we introduce as higher-dimensional analogues of Keller’s maximal green sequences. We show that the higher Stasheff-Tamari orders correspond to natural orders on equivalence classes of d-maximal green sequences, which relate to the no-gap conjecture of Brustle, Dupont, and Perotin. The equality of the higher Stasheff-Tamari orders then implies that these algebraic orders on tilting modules and d-maximal green sequences are equal. If time permits, we will also discuss some results on mutation of cluster-tilting objects and triangulations.
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2021/11/19

17:00-18:30   Online
Please see the URL below for details on the online seminar.
Yuta Kozakai (Tokyo University of Science)
有限群のブロック上の$\tau$-傾理論 (Japanese) (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2021/07/08

16:00-17:30   Online
Please see the URL below for details on the online seminar.
Tsukasa Ishibashi (RIMS, Kyoto University)
Sign-stable mutation loops and pseudo-Anosov mapping classes (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2021/06/24

16:00-17:30   Online
Please see the URL below for details on the online seminar.
Kohei Kikuta (Chuo University)
Rank 2 free subgroups in autoequivalence groups of Calabi-Yau categories
[ Abstract ]
Via homological mirror symmetry, there is a relation between autoequivalence groups of derived categories of coherent sheaves on Calabi-Yau varieties, and the symplectic mapping class groups of symplectic manifolds.
In this talk, as an analogue of mapping class groups of closed oriented surfaces, we study autoequivalence groups of Calabi-Yau triangulated categories. In particular, we consider embeddings of rank 2 (non-commutative) free groups generated by spherical twists. It is interesting that the proof of main results is almost similar to that of corresponding results in the theory of mapping class groups.
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2021/06/02

16:00-17:30   Online
Please see the URL below for details on the online seminar.
Satoshi Murai (Waseda University)
An equivariant Hochster's formula for $S_n$-invariant monomial ideals (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2021/05/20

16:00-17:30   Online
Please see the URL below for details on the online seminar.
Ryo Kanda (Osaka city University)
This talk is based on joint work with Tsutomu Nakamura. For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. As a consequence, we show that pointwise Matlis duality gives a bijective correspondence between the isoclasses of indecomposable flat cotorsion right modules and the isoclasses of indecomposable injective left modules. This correspondence is an explicit realization of Herzog's homeomorphism induced from elementary duality between Ziegler spectra.
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2021/05/06

16:00-17:30   Online
Please see the URL below for details on the online seminar.
Liran Shaul (Charles University)
Derived quotients of Cohen-Macaulay rings (English)
[ Abstract ]
It is well known that if A is a Cohen-Macaulay ring and $a_1,\dots,a_n$ is an $A$-regular sequence, then the quotient ring $A/(a_1,\dots,a_n)$ is also a Cohen-Macaulay ring. In this talk we explain that by deriving the quotient operation, if A is a Cohen-Macaulay ring and $a_1,\dots,a_n$ is any sequence of elements in $A$, the derived quotient of $A$ with respect to $(a_1,\dots,a_n)$ is Cohen-Macaulay. Several applications of this result are given, including a generalization of Hironaka's miracle flatness theorem to derived algebraic geometry.
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2021/04/22

16:00-17:30   Online
Please see the URL below for details on the online seminar.
Julian Külshammer‬ (Uppsala University)
Exact categories via A-infinity algebras (English)
[ Abstract ]
Many instances of extension closed subcategories appear throughout representation theory, e.g. filtered modules, Gorenstein projectives, as well as modules of finite projective dimension. In the first part of the talk, I will outline a general strategy to realise such subcategories as categories of induced modules from a subalgebra using A-infinity algebras. In the second part, I will describe how this strategy has been successfully applied for the exact category of filtered modules over a quasihereditary algebra. In particular I will present compatibility results of this approach with heredity ideals in a quasihereditary algebra from joint work with Teresa Conde.
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2021/04/08

16:00-17:30   Online
Please see the URL below for details on the online seminar.
Kevin Coulembier (Univeristy of Sydney)
Abelian envelopes of monoidal categories (English)
[ Abstract ]
For the purposes of this talk, a ‘tensor category’ is an abelian rigid monoidal category, linear over some field. I will try to argue that there are good reasons (inspired by classification attempts of tensor categories, by motives, by Frobenius twists on tensor categories and by the idea of universal tensor categories), to try to associate tensor categories to non-abelian rigid monoidal categories. Then I will comment on some of the recent progress made on such constructions (in work of Benson, Comes, Entova, Etingof, Heidersdof, Hinich, Ostrik, Serganova and myself).
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2021/03/11

16:00-17:30   Online
Please see the URL below for details on the online seminar.
Akihito Wachi (Hokkaido University of Education)
相対不変式で生成されるゴレンスタイン環のレフシェッツ性 (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2021/02/24

16:00-17:30   Online
Please see the URL below for details on the online seminar.
Shunya Saito (Nagoya University)
周期三角圏上の傾理論 (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

2021/02/10

16:00-17:30   Online
Please see the URL below for details on the online seminar.
Akishi Ikeda (Josai University)
Gentle代数の2重次数付きCalabi-Yau完備化と曲面の幾何学 (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

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