Lower bounds for Auslander's representation dimension
Lower bounds for Auslander's representation dimension
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DOI：
10.1215/00127094-2009-025
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发表时间：
2009-06
影响因子：
2.5
通讯作者：
Steffen Oppermann
中科院分区：
文献类型：
--
作者：
Steffen Oppermann
The notion of representation dimension of a finite dimensional algebra has been introduced by Auslander. He has shown that an algebra is of finite representation type if and only if its representation dimension is at most two. He has expected that the representation dimension of a representation infinite algebra should measure how far this algebra is from being representation finite. In many cases, and, thanks to Iyama, also in general, upper bounds for the representation dimension are known. However there are non-trivial lower bounds only for two (similarly constructed) series of algebras. In this thesis new lower bounds for Auslander’s representation dimension are constructed, and more general methods are developed. First we consider group algebras of elementary abelian groups. We show that their representation dimension is greater than the rank of the group. This leads to a proof of a conjecture of Benson concerning the Loewy length of blocks of group algebras. Then a notion of dimension of a module category is introduced. This notion enables us to find better lower bounds for the representation dimension. In particular it allows us for the first time to determine the value of the representation dimension of non-self-injective algebras of representation dimension greater than three. Finally we give a general criterion for finding lower bounds for Auslander’s representation dimension: Suppose there is a family of modules, such that sufficiently many members of the family have certain d-extensions. Then the representation dimension of the algebra is at least d + 2. We illustrate how this criterion is applied on some interesting families of algebras. It turns out that the lower bound is often the precise value of the representation dimension.
