日 時： 2009年11月06日（金）, 07日（土）
場 所： 静岡大学理学部 C 棟 309 号室
案 内： http://www.shizuoka.ac.jp/ippan/shizuoka.html
理学部 A 棟1階から入り，エレベーターで4階まで上がり，渡り廊下を渡ると，理学部 C 棟の1階に着きます．
連絡： 浅芝秀人 (shasash+)，毛利出 (simouri+)
(+ := @ipc.shizuoka.ac.jp)
注意： 土曜日，理学部棟は施錠されています．
鍵を開けるため9時すぎから9時半近くまでA棟1階の入り口に人員を配置します．
土曜日だけの参加も大歓迎です． その場合，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06日（金）
13:00 – 14:30 相原琢磨（千葉大学）
Silting mutation quiver for self-injective algebras
15:00 – 16:00 Colin Ingalls（京都大学数理解析研究所）
The Algebraic Geometry of Orders I:
The algebra of orders（学部生も参加）
16:15 – 17:15 Colin Ingalls（京都大学数理解析研究所）
The Algebraic Geometry of Orders II:
The geometry of orders
18:30 - 懇親会
11月07日（土）
09:30 – 11:00 Changchang Xi（北京師範大学）
The BB-tilting module: a bridge between AR-sequence and derived equivalence
13:00 – 14:30 玉木大（信州大学）
Homotopy colimits for enriched categories
要約
- 相原琢磨:
Silting mutation quiver for self-injective algebras
Let TR be a triangulated category, and we assume that TR is Krull-Schmidt, k-linear for a fixed field k and Hom-finite, that is, dim_k Hom_TR (X, Y ) 0. The point is that one can develop nice mutation theory for silting objects (not for tilting objects!). We introduce mutation on silting objects, and consider the following conjecture:Conjecture. Does successive mutation act transitively on silting objects in TR ?
If it is true, then we can control tilting objects in TR . In my talk, we observe mutation on silting objects in K^b(proj-A) for a finite dimensional k-algebra A. Especially, we prove that the conjecture is true for symmetric algebras of finite representation type, by using Abe-Hoshino’s result [1, Lemma 3.4, Theorem 3.7]. In this proof, we need that it is“ symmetric ”, but their result does not need it (“ self-injective ”is enough). So, we want to prove it in the case of not only“ symmetric ” but also “ self-injective ”. To do it, we have to prove the following:
Question. Let A be a self-injective algebra of finite representation type. For any silting objects T in K^b(proj-A), can we get a tilting object P in K^b(proj-A) by successive mutation of T ? In other words, is there a tilting object P lying in the component of T for silting mutation quiver of K^b(proj-A) ?
If it is true, then the conjecture is true in this case. But I cannot prove the question. If there is an aware point, please teach it to me. Finally, I will close my talk by giving some examples (and questions).
References
[1] H. Abe and M. Hoshino. On derived equivalences for selfinjective algebras. Comm.alg. 34, 4441-4452, 2006. - Colin Ingalls:
In the first talk we give an introduction to orders.
We will discuss central simple algebras and the Artin-Wedderburn Theorem. Morita equivalence and the Brauer Group. Azumaya algebras and orders. We show that every order is contained in a maximal order and we explain Auslander’s criterion for maximality. Lastly we discuss ramification theory of orders and the structure theorem for hereditary orders over a discrete valuation ring.
In the second talk we will discuss connections between orders and algebraic geometry. We will present the Coniveau spectral sequence. We will introduce etale cohomology and the Brauer group of schemes. We will discuss how orders are related to the Minimal Model Program, Deligne-Mumford stacks, tame quivers, and moduli of vector bundles. - Changchang Xi：
The BB-tilting module: a bridge between AR-sequence and derived equivalence
In this talk, we shall show that there is an intimate connection among the three old notions: AR-sequence, BB-tilting module, and derived equivalence. In particular, we can produce derived equivalences in the following way: For any nodule $X$ over a self-injective algebra $A$, the two endomorphism algebras End$_A(_AA\oplus X)$ and End$_A(_AA\oplus \Omega_A^i(X))$ are derived-equivalent via a tilting module for any $i\in {\mathbb Z}$, where $\Omega_A$ is the Heller operator of $A$. [1] C.C. Xi and W. Hu, Almost $D$-split sequences and derived equivalences, Preprint, 2007, available at: http://math.bnu.edu.cn/~ccxi/.
[2] —, Derived equivalences and stable equivalences of Morita type, I., Preprint, 2008, available at: http://math.bnu.edu.cn/~ccxi/. - 玉木大：
Homotopy colimits for enriched categories
The homotopy colimit construction for diagrams of spaces has been an indispensable tool in homotopy theory. Thomason’s famous result says that there is an analogous construction for diagrams of small categories and the classifying space functor relates these two constructions up to homotopy. The construction can be generalized to diagrams of categories enriched over a monoidal category such as $k$-linear categories, dg categories, and spectral categories. Special cases has been used explicitly or implicitly in various fields, e.g. the construction of a fibered category from a prestack, skew group rings, and the orbit category construction.In this talk we investigate basic properties of the homotopy colimit construction for enriched categories. In particular, we extend the definition of group-graded categories to category-graded enriched categories and discuss its relation to the homotopy colimit construction.