/ 11月 25, 2016/ 静岡代数学セミナー

(+ := @shizuoka.ac.jp)

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。
———————————————————————————————

プログラム

● 11月25日(金)

13:30 – 14:30 大川新之介（大阪大学）
Noncommutative projective planes and their moduli spaces, I

14:45 – 15:45 木村雄太（名古屋大学多元数理）
Tilting objects from reduced expressions in Coxeter groups, I

16:00 – 17:00 中村力（岡山大学）
Principle of local duality and Grothendieck type vanishing theorem, I

18:30 – 懇親会

● 11月26日(土)

09:30 – 10:30 中村力（岡山大学）
Principle of local duality and Grothendieck type vanishing theorem, II

10:45 – 11:45 大川新之介（大阪大学）
Noncommutative projective planes and their moduli spaces, II

13:30 – 14:30 木村雄太（名古屋大学多元数理）
Tilting objects from reduced expressions in Coxeter groups, II

============================

アブストラクト

Noncommutative projective planes are the most basic and important class of noncommutative algebraic varieties. Although they were first defined as certain graded rings, it has been recognized that (derived) categorical points of view are essential to understand them.
In the first talk, I will give a brief introduction to the subject and explain how these different points of view are related. In the second half, based on a joint work with Tarig Abdelgadir and Kazushi Ueda, I will introduce certain construction of compactified moduli spaces of noncommutative projective planes and discuss their properties.

Let $Q$ be a finite acyclic quiver and $w$ be an element of the Coxeter group of $Q$. Buan-Iyama-Reiten-Scott constructed and studied a triangulated category $\mathcal{E}_{w}$ associated with $w$. They showed that, for each reduced expression of $w$, there exists a cluster titling object of $\mathcal{E}_{w}$. In this talk, we consider a triangulated category $\mathcal{E}_{w}^{\mathbb{Z}}$, which is a $\mathbb{Z}$-graded version of $\mathcal{E}_{w}$. We show that, for each reduced expression of $w$, there exists a silting object of $\mathcal{E}_{w}^{\mathbb{Z}}$. We give a sufficient condition on a reduced expression such that the silting object is a tilting object.

Let R be a commutative noetherian ring. For the unbounded derived category D(R), it is known that there is a natural bijection between the set of subsets W of Spec R and the set of localizing subcategories L_W of D(R). Moreover, by a classical argument of the localization theory of triangulated categories, there exists a right adjoint functor to the inclusion functor from L_W to D(R), which we call the local cohomology functor gamma_W. If W is specialization-closed, then gamma_W coincides with the ordinary local cohomology functor RGamma_W. In this talk, we prove that the local duality theorem and the Grothendieck type vanishing theorem hold for gamma_W, where W is a general subset of Spec R.
This talk is based on a joint work with Yuji Yoshino.