Constructing Witt–Burnside rings
Constructing Witt–Burnside rings
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构建维特-伯恩赛德环
DOI：
10.1016/j.aim.2005.04.014
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发表时间：
2006
影响因子：
1.7
通讯作者：
J. Elliott
中科院分区：
文献类型：
--
作者：
J. Elliott
交换环A上的有限群G的Witt-伯恩赛德环推广了A上的虚G-集的伯恩赛德环以及泛Witt向量环和p-典型Witt向量环.本文证明了幺半群环Z[M]上G的Witt-Burnside环同构于一个范畴的Grothendieck环，该范畴的对象是几乎有限的G-集，其映射到M且在G-轨道上为常数.特别地，如果A是一个交换环，且A×表示集合A是一个乘法幺半群，则G在Z[A×]上的Witt-Burnside环同构于Graham的“系数在A中的虚G-弦”环。这个结果构成了Witt-Burnside环的一个新构造的基础，并提供了Dress和Siebeneicher [Adv. in Math.70（1988）87-132]和Graham [Adv. in Math.99（1993）248-263]的构造之间重要的缺失环节。利用这种方法，通常的截断、弗罗贝纽斯、韦氏和泰希米勒映射很容易推广到维特-伯恩赛德环之间的映射。
The Witt–Burnside ring of a profinite group G over a commutative ring A generalizes both the Burnside ring of virtual G-sets and the rings of universal and p-typical Witt vectors over A. The Witt–Burnside ring of G over the monoid ring Z[M], where M is a commutative monoid, is proved isomorphic to the Grothendieck ring of a category whose objects are almost finite G-sets equipped with a map to M that is constant on G-orbits. In particular, if A is a commutative ring and A×denotes the set A as a monoid under multiplication, then the Witt–Burnside ring of G over Z[A×] is isomorphic to Graham's ring of “virtual G-strings with coefficients in A.” This result forms the basis for a new construction of Witt–Burnside rings and provides an important missing link between the constructions of Dress and Siebeneicher [Adv. in Math. 70 (1988) 87–132] and Graham [Adv. in Math. 99 (1993) 248–263]. With this approach the usual truncation, Frobenius, Verschiebung, and Teichmüller maps readily generalize to maps between Witt-Burnside rings.