/ 12月 18, 2015/ 静岡代数学セミナー

(+ := @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

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.