论文标题
有限尺寸的倾斜理论$ 1 $ -IWANAGA-GORENSTEIN代数
Tilting theory for finite dimensional $1$-Iwanaga-Gorenstein algebras
论文作者
论文摘要
在分级iwanaga-gorenstein代数的表示理论中，稳定类别的倾斜理论$ \下划线{\ mathsf {cm}}}^{\ mathbb {z}} a $分级的cohen-macaulay模块扮演着重要角色。在本文中，我们研究了以下两个$ \下划线倾斜理论的核心问题{\ MathSf {cm}}}^{\ Mathbb {z}} a $在$ a $是有限维度的情况下：（1）$ a $ a $ a $ $ \ suestline {\ suestline {\ mathsf {cm Mathsf {cm}}^\ a} （2）$ \ usevinline {\ mathsf {cm}}}}^{\ mathbb {z}} a $的倾斜对象的内态代数是否具有有限的全局维度？对于问题（2），我们给出完整的答案。我们表明，$ \ useverline {\ mathsf {cm}}}^{\ mathbb {z}}中的任何倾斜对象的内态代数。对于问题（1），我们给出了部分答案。为此，首先，我们为有限维度分级的代数$ a $ a $ a $ a $ a $ a引入了不变的$ g（a）$。然后，我们证明，在$ a $为1- iwanaga-gorenstein的情况下，$ g（a）$的不平等给出了足够的条件，即特定的cohen-macaulay模块$ v $成为稳定类别中的倾斜对象。作为一个应用程序，我们研究了$ \下列中的倾斜对象的存在{\ mathsf {cm}}}}^{\ mathbb {z}}π（q）_w $其中$π（q）_w $是the truncated profotivative prefotive profotrogrojective $ q $ quiver $ q $ a Quiver $ q $ fo $ w__我们证明，如果$ q $的下属图是树，则$ \ upessline {\ mathsf {cm}}}}^{\ mathbb {z}}}π（q）_w $具有倾斜对象。
In representation theory of graded Iwanaga-Gorenstein algebras, tilting theory of the stable category $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ of graded Cohen-Macaulay modules plays a prominent role. In this paper we study the following two central problems of tilting theory of $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ in the case where $A$ is finite dimensional: (1) Does $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ have a tilting object? (2) Does the endomorphism algebras of tilting objects in $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ have finite global dimension? To the problem (2) we give the complete answer. We show that the endomorphism algebra of any tilting object in $\underline{\mathsf{CM}}^{\mathbb{Z}}A$ has finite global dimension. To the problem (1) we give a partial answer. For this purpose, first we introduce an invariant $g(A)$ for a finite dimensional graded algebra $A$. Then, we prove that in the case where $A$ is 1-Iwanaga-Gorenstein, an inequality for $g(A)$ gives a sufficient condition that a specific Cohen-Macaulay module $V$ becomes a tilting object in the stable category. As an application, we study the existence of tilting objects in $\underline{\mathsf{CM}}^{\mathbb{Z}}Π(Q)_w$ where $Π(Q)_w$ is the truncated preprojective algebra of a quiver $Q$ associated to $w\in W_Q$. We prove that if the underling graph of $Q$ is tree, then $\underline{\mathsf{CM}}^{\mathbb{Z}}Π(Q)_w$ has a tilting object.
