Sheaves and K-theory for F1-schemes
Sheaves and K-theory for F1-schemes
复制标题
F1 方案的滑轮和 K 理论
DOI：
10.1016/j.aim.2011.12.023
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发表时间：
2012
影响因子：
1.7
通讯作者：
Rekha Santhanam
中科院分区：
文献类型：
--
作者：
Cheng Chu;Oliver Lorscheid;Rekha Santhanam
本文研究了F1-几何中的一个公开问题：F1-格式的K-理论的发展。我们提供了所有必要的事实，从理论的幺半群作用点集，我们介绍层的M0计划和F1计划的意义上的Connes和Consani。一个广泛的结果希望在于F1上的代数几何的进一步发展的背景。特别注意F1-几何的两个方面，即正规态射和局部投射层，当我们采用Quillen的Q-构造来定义F1-概型的G-理论和K-理论时，会出现这两个方面。通过与瓦尔德豪森的S·-构造的比较，得到了K-理论的环结构。特别地，我们推广了Deitmar的K-幺半群理论，并证明了K-幺半群（SpecF 1）实现了球面的稳定同伦作为环谱。
This paper is devoted to the open problem in F1-geometry of developing K-theory for F1-schemes. We provide all necessary facts from the theory of monoid actions on pointed sets and we introduce sheaves for M0-schemes and F1-schemes in the sense of Connes and Consani. A wide range of results hopefully lies the background for further developments of the algebraic geometry over F1. Special attention is paid to two aspects particular to F1-geometry, namely, normal morphisms and locally projective sheaves, which occur when we adopt Quillenʼs Q-construction to a definition of G-theory and K-theory for F1-schemes. A comparison with Waldhausenʼs S•-construction yields the ring structure of K-theory. In particular, we generalize Deitmarʼs K-theory of monoids and show that K⁎(SpecF1) realizes the stable homotopy of the spheres as a ring spectrum.