论文标题
伯恩赛环单位的功能演示
A functorial presentation of units of Burnside rings
论文作者
论文摘要
让$ b^\ times $成为$ \ mathbb {f} _2 $上的Biset函数，将有限的组〜$ g $发送给该组的$ b^\ times（g）burnside ring $ b（g）$，然后让$ \ wideHat {b^\ times} $是其双重函数。本文的主要定理给出了自然注射的cokernel的$ b^\ times $在双重伯恩赛德函数中的$ \ widehat {\ mathbb {f} _2b} $，或等效地，这是一套明确的发电机$ \ nathcal {g} _s _s $ uspuction of of $ \ mathbb {f} _2b \ to \ wideHat {b^\ times} $。这产生了两个术语的投影分辨率为$ \ wideHat {b^\ times} $，从而导致了有关扩展功能器$ \ mathrm {ext}^1（ - ，b^\ times）$的一些信息。对于有限的组$ g $，这还允许描述$ b^\ times（g）$作为$ b^\ times $ b^\ times（t/s）$的限制，$（t/s）$ axpions $（t，s）$ $ g $的$ g $，因此$ t/s $是奇数prime的循环，奇数prime，klein fourd，klein fours fours of diron，dihedral of 8，或roquette 8，或roquette 2 group dihedral of dihedral of dihedral of dihedral of Roquette 2 group 2 group。另一个结果是，Biset Founcor $ b^\ times $不是有限生成的，并且其dual $ \ wideHat {b^\ times} $是有限生成的，但没有有限地呈现。本文的最后结果还表明，$ \ Mathcal {g} _s $是$ L $的最低发电机集，因此，$ L $的亚功能的晶格是无关的。
Let $B^\times$ be the biset functor over $\mathbb{F}_2$ sending a finite group~$G$ to the group $B^\times(G)$ of units of its Burnside ring $B(G)$, and let $\widehat{B^\times}$ be its dual functor. The main theorem of this paper gives a characterization of the cokernel of the natural injection from $B^\times$ in the dual Burnside functor $\widehat{\mathbb{F}_2B}$, or equivalently, an explicit set of generators $\mathcal{G}_S$ of the kernel $L$ of the natural surjection $\mathbb{F}_2B\to \widehat{B^\times}$. This yields a two terms projective resolution of $\widehat{B^\times}$, leading to some information on the extension functors $\mathrm{Ext}^1(-,B^\times)$. For a finite group $G$, this also allows for a description of $B^\times(G)$ as a limit of groups $B^\times(T/S)$ over sections $(T,S)$ of $G$ such that $T/S$ is cyclic of odd prime order, Klein four, dihedral of order 8, or a Roquette 2-group. Another consequence is that the biset functor $B^\times$ is not finitely generated, and that its dual $\widehat{B^\times}$ is finitely generated, but not finitely presented. The last result of the paper shows in addition that $\mathcal{G}_S$ is a minimal set of generators of $L$, and it follows that the lattice of subfunctors of $L$ is uncountable.
