代数K理论与模符号

摘要：

本文计算了秩谱序列的微分$d^1$。通过考虑广义生成无扭模范畴的Q-构造$Q^{tf}(A)$，将Quillen谱序列从戴德金整环推广到广义积分诺特环$A$.特别地，根据归结定理，如果$A$是正则的，则$Q^{tf}(A)$同伦等价于$QP(A)$，其中$P(A)$是可分解生成的投射$A$ -模的范畴。我们推导出微分$d^1$通过使用山雀建筑物，Steinberg模，并在阿什鲁道夫意义下的模符号。奎伦的范畴同伦理论的精神将在整个文件中集中使用。

英文摘要：

In this paper, we calculate the differential $d^1$ of the rank spectral sequence. We generalize Quillen's spectral sequence from Dedekind domain to general integral Noetherian ring $A$ by considering the Q-construction $Q^{tf}(A)$ of the category of finitely generated torsion-free modules. In particular, by resolution theorem, if $A$ is regular, then $Q^{tf}(A)$ is homotopy equivalent to $QP(A)$ where $P(A)$ is the category of finitely generated projective $A$-modules. We deduce the differential $d^1$ by using Tits buildings, Steinberg modules, and modular symbols in the sense of Ash-Rudolph. The spirit of Quillen's categorical homotopy theory will be used intensively throughout this paper.

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