代数K-理论导论(影印版)

通俗地讲，K-理论是一 种探究数学对象（如环或拓 扑空间）结构的工具，它利 用适当参数化的向量空间并 生成重要的内在不变量，这 些不变量在代数和几何问题 的研究中非常有用。代数K- 理论是本书的主角，主要研 究环的结构。然而，事实证 明，即使在纯代数语境下工 作，人们也需要使用同伦理 论等技术来构造高阶K-群并 进行计算。由此产生的代数 、几何和拓扑在K-理论中的 相互作用提供了数学统一性 的迷人一瞥。 本书是代数K-理论的综 合介绍。它将K0和K1的经 典代数技术与更新的用于高 等K-理论的拓扑技术（如同 伦理论、谱和上同调下降） 相融合。内容涵盖从基础知 识到最前沿的技术，包括数 域的高等K-理论的计算以及 与Riemann ζ函数的关系。

Preface Acknowledgements Chapter Ⅰ.Projective modules and vector bundles 1.Free modules, GLn, and stably free modules 2.Projective modules 3.The Picard group of a commutative ring 4.Topological vector bundles and Chern classes 5.Algebraic vector bundles Chapter Ⅱ.The Grothendieck group K0 1.The group completion of a monoid 2.K0 of a ring 3.K(X), K0(X), and KU(X) of a topological space 4.Lambda and Adams operations 5.K0 of a symmetric monoidal category 6.K0 of an abelian category 7.K0 of an exact category 8.K0 of schemes and varieties 9.K0 of a Waldhausen category Appendix.Localizing by calculus of fractions Chapter Ⅲ.K1 and K2 of a ring 1.The Whitehead group K1 of a ring 2.Relative K1 3.The Fundamental Theorems for K1 and K0 4.Negative K-theory 5.K2 of a ring 6.K2 of felds 7.Milnor K-theory of fields Chapter Ⅳ.Definitions of higher K-theory 1.The BGL+ defnition for rings 2.K-theory with finite coefficients 3.Geometric realization of a small category 4.Symmetric monoidal categories 5.х-operations in higher K-theory 6.Quillen's Q-construction for exact categories 7.The “+=Q” Theorem 8.Waldhausen's ws.construction 9.The Gillet-Grayson construction 10.Nonconnective spectra in K-theory 11.Karoubi-Villamayor K-theory 12.Homotopy K-theory Chapter Ⅴ.The Fundamental Theorems of higher K-theory 1.The Additivity Theorem 2.Waldhausen localization and approximation 3.The Resolution Theorems and transfer maps 4.Devissage 5.The Localization Theorem for abelian categoriesee 6.Applications of the Localization Theorem 7.Localization for K*(R) and K*(X) 8.The Fundamental Theorem for K*(R) and K*(X) 1oal 9.The coniveau spectral sequence of Gersten and Quillenx