控制维数和整体维数是有限维代数的两个基本的同调维数. Auslander 将控制维数和整体维数结合起来定义了表示维数, 用来衡量一个代数距离表示有限有多远. Iyama 推广了表示维数的概念, 定义了高阶表示维数. 最近, Chen 和 Fang 等人同样将控制维数和整体维数结合起来提出了刚性维数的概念, 用来测量有限维代数分解的质量. 刚性维数与高阶表示维数, Schur-Weyl 对偶, Hochschild 上同调等都有密切的关系.

一个基本的问题是如何计算给定代数的刚性维数. 难点在于判断生成子- 余生成子 M 的自同态代数 End__A(M ) 的整体维数是否有限以及计算它的控制维数. 而根据 Mueller 的结论, End__A(M ) 的控制维数等千 M 的刚度加 2, 因此模范畴的 Ext-结构对刚性维数的计算有重要作用.

本文主要研究表示有限型自入射代数的刚性维数,   取得的主要结果如下.

我们完全确定了 A 型和 D 型表示有限型自入射代数不可分解模的刚度, 并给了计算公式. 

此外我们研究了 A 型表示有限型自入射代数的刚性维数. 我们主要考虑自入射 Nakayama 代数 A_n,m 的刚性维数, 其中 n 是单模个数, m 是投射模的 Loewy 长度. 当 m 不小于 n 时, 我们给出了刚性维数的精确值.

Dominant dimension and global dimension are two fundamental homological dimen- sions of finite dimensional algebras. Their interplay occurs in Auslander’s definition of representation dimension, which is to measure how far an algebra is from being representation-finite. More generally, Iyama defined the higher representation di- mension. Recently, a somewhat symmetric dimension called rigidity dimension is introduced by Chen and Fang et al., which measures the quality of resolutions of finite dimensional algebras. Rigidity dimension is related to higher representation dimension, Schur-Weyl duality, Hochschild cohomology and so on.

A basic question is how to get the precise value of the rigidity dimension of a given  algebra. The difficulties lie in the finiteness of the global dimension and the value of the dominant dimension of the endomorphism algebra End_A(M ) of a generator- cogenerator M . By Mueller’s criterion, the dominant dimension of End_A(M ) is pre- cisely the rigidity degree of M plus two, thus the Ext-structure of the module category plays an important part in rigidity dimension.

In this paper, we focus on representation-finite self-injective algebras. Our main results are stated as follows.

We present explicit formulae for rigidity degrees of all indecomposable modules for type A and D.

Moreover, we study the rigidity dimensions of representation-finite self-injective algebras of type A. We mainly concentrate on self-injective Nakayama algebra A_n,m with n simple modules and the Loewy length m>=n. The values of rigidity dimensions are shown.