Let A be an abelian group such that tor_n(A) is finite for every n ≥ 1 and let \({\mathbb{K}}\) be a field of characteristic zero containing roots of unity of all orders equal to finite element orders in A. In this paper we prove fundamental properties of the A-fibered Burnside ring functor \(B_{\mathbb{K}}^{A}\) as an A-fibered biset functor over K. This includes a description of the composition factors of \(B^{A}_{\mathbb{K}}\) and the lattice of subfunctors of \(B_{\mathbb{K}}^{A}\) in terms of what we call B^A-pairs and a poset structure on their isomorphism classes. Unfortunately, we are not able to classify B^A-pairs. The results of the paper extend results of Coşkun and Yılmaz for the A-fibered Burnside ring functor restricted to p-groups and results of Bouc in the case that A is trivial, i.e., the case of the Burnside ring functor as a biset functor over fields of characteristic zero. In the latter case, B^A-pairs become Bouc’s B-groups which are also not known in general.

让阿是阿贝尔群，使得TOR _Ñ（甲）是有限的，每Ñ ≥1并且让\（{\ mathbb {K}} \）是所有订单的统一的特性零含有根的字段等于有限元A中的订单。在本文中，我们证明了A纤维Burnside环函子\（B _ {\ mathbb {K}} ^ {A} \）的基本特性，它是K上的A纤维双对偶函子。其中包括对K的组成因子的描述。\（B ^ {A} _ {\ mathbb {K}} \）和\（B _ {\ mathbb {K}} ^ {A} \）的子功能格（用我们称为B ^{A的形式表示）}对和同构类上的poset结构。不幸的是，我们无法对B ^A对进行分类。本文的结果扩展了Coşkun和Yılmaz对于A纤维的Burnside环函子的结果，将其限制为p-基团，以及Bouc的结果，其中A是微不足道的，即Burnside环函子作为Biset函子的情况。特征为零的字段。在后者的情况下，乙^甲-pairs成为的Bouc的乙其也未在一般已知的-基团。