(+ := @ipc.shizuoka.ac.jp)<<最近アドレスが変わりました。>>

1日だけの参加も大歓迎です。
その場合，土日の1コマ目に遅刻する可能性があれば，事前に連絡しておいてもらった方が無難です。
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### プログラム

#### 12月18日（金）

13:30 – 14:30 上山健太（弘前大学教育） Cluster tilting modules in noncommutative projective geometry, I

14:45 – 15:45 上山健太（弘前大学教育） Cluster tilting modules in noncommutative projective geometry, II

16:00 – 17:00 Dirk Kussin（Paderborn, 名古屋大学多元数理） Action of the Auslander-Reiten translation on tubes

18:30 – 懇親会

#### 12月19日（土）

09:30 – 10:30 Dirk Kussin（Paderborn, 名古屋大学多元数理） Noncommutative real elliptic curves

10:45 – 11:45 吉脇理雄（大阪市立大学数学研究所） Relative derived dimensions for cotilting modules

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### アブストラクト

• #### 上山健太

Cluster tilting modules in noncommutative projective geometry, I, II
Cluster tilting modules are crucial in the study of higher-dimensional analogues of Auslander-Reiten theory,  and also attract attention in terms of Van den Bergh's noncommutative crepant resolutions. In this series of talks, we study cluster tilting modules from the viewpoint of noncommutative projective geometry. In particular, we show that if a d-dimensional  AS-Gorenstein algebra A has a (d-1)-cluster tilting module X with some additional assumptions, then the graded endomorphism algebra B of X is two-sided noetherian ASF-regular of dimension d such that the noncommutative projective schemes of B and A are equivalent. Note that the notion of ASF-regular algebra was recently introduced by Minamoto and Mori, and it is a natural generalization of AS-regular algebra for N-graded algebras.
• #### Dirk Kussin

1. Action of the Auslander-Reiten translation on tubes
We work over a perfect field. A finite-dimensional tame hereditary or canonical algebra has a tubular family. We compute the complete local rings for the associated noncommutative curve and explain how the Auslander-Reiten translation is acting, as functor, on the tubes.
2.  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 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.
• #### 吉脇理雄

Relative derived dimensions for cotilting modules
Let $A$ be a finite-dimensional algebra over a field. We denote by $\operatorname{mod} A$ the category of finitely generated right $A$-modules. For a module $T\in\operatorname{mod} A$, we also denote by ${}^{\perp} T$ the left Ext-orthogonal subcategory of $\operatorname{mod} A$ with respect to $T$. In this talk,  we will show that for a given cotilting module $T\in\operatorname{mod} A$  of injective dimension at least $1$,  the derived dimension of $A$ with respect to ${}^{\perp} T$ is just the injective dimension of $T$.  This generalizes a result of Krause and Kussin.

http://www.ipc.shizuoka.ac.jp/~shasash/

(+ := @ipc.shizuoka.ac.jp)

1日だけの参加も大歓迎です。
その場合，土日の1コマ目に遅刻する可能性があれば，事前に連絡しておいてもらった方が無難です。
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### プログラム

#### 6月5日（金）

13:45 – 14:45 板場 綾子（静岡大学理）
Finite condition (Fg) for self-injective Koszul algebras

15:00 – 16:00 足立 崇英（名古屋大学多元数理）
Brauer tree algebras and triangulations

16:15 – 17:15 柴田 和樹（立教大学理）
マトロイドに付随するトーリックイデアルとグレブナー基底について

18:30 – 懇親会

#### 6月6日（土）

09:30 – 10:30 柴田 和樹（立教大学理）
Strong Koszulness of the toric ring associated to a cut ideal

10:45 – 11:45 足立 崇英（名古屋大学多元数理）
Tilting Brauer graph algebras

13:15 – 14:15 毛利 出（静岡大学理）
Stable categories of graded maximal Cohen-Macaulay modules over noncommutative quotient singularities

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### アブストラクト

• #### 板場 綾子（静岡大学理）

Finite condition (Fg) for self-injective Koszul algebras
Let $A$ be a graded algebra finitely generated in degree 1 over a field $k$. Mori defined a co-point module over $A$ which is a dual notion of point module introduced by Artin, Tate and Van den Bergh in terms of Koszul duality. Using a co-point module, $(E,\sigma)$ is called a cogeometric pair, where $E$ is a projective scheme and $\sigma$ is an automorphism of $E$. For a finite-dimensional algebra over a filed $k$, Erdmann, Holloway, Taillefer, Snashall and Solberg introduced certain finiteness conditions, denoted by (Fg).
In this talk, we consider a finite-dimensional algebra over an algebraically closed field. For a relationship between a cogeometric pair $(E,\sigma)$ and the finite condition (Fg), we propose the following conjecture. Let $R$ be a cogeometric self-injective Koszul algebra with the complexity of $k$ is finite. Then $R$ satisfies the condition (Fg) if and only if the order of $\sigma$ is finite. If the complexity of $k$ is $2$, then we show that this conjecture holds.
• #### 足立 崇英（名古屋大学多元数理）

(1) Brauer tree algebras and triangulations
We explain connection between two-term tilting complexes for a Brauer star algebra, triangulations of a polygon with a puncture, and certain integer sequences.
(2) Tilting Brauer graph algebras
We give a combinatorial description of two-term tilting complexes for Brauer graph algebras which are finite dimensional symmetric algebras defined by ribbon graphs. This result is a generalization of the classification result of the first talk.
• #### 柴田 和樹（立教大学理）

(1) マトロイドに付随するトーリックイデアルとグレブナー基底について
マトロイドとはWhitney(1935)によって導入された概念であり、 ベクトル空間における線形独立の概念の抽象・一般化となっている。 本講演では、初めにマトロイドの基本的性質を述べた後、 マトロイドに付随するトーリックイデアルに関する予想について紹介する。
(2) Strong Koszulness of the toric ring associated to a cut ideal
cut idealとは、グラフの切断に対応するトーリックイデアルのことであり、 代数統計の分野でよく用いられている。 また、イデアルの環論的性質と組合せ論的構造を関連付ける多くの結果が知られている。 本講演では、cut idealのグレブナー基底、及び、 それに対応するトーリック環のstrongly Koszul性について紹介する。
• #### 毛利 出（静岡大学理）

Stable categories of graded maximal Cohen-Macaulay modules over noncommutative quotient singularities
This talk is a report of a joint work with Ueyama. Let S be a noetherian AS-regular Koszul algebra and G is a finite group acting on S such that SG is an AS-Gorenstein isolated singularity. In this talk, we will show that the stable category of graded maximal Cohen-Macaulay modules over SG has a tilting object. This is a noncommutative generalization of the result due to Iyama and Takahashi with a more conceptual proof. The keys to prove this result are Buchweitz equivalence, Orlov embedding, Koszul duality, and Yamaura tilting.

http://www.ipc.shizuoka.ac.jp/~shasash/

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