A matrix approach to lower k theory and algebraic number theory

A MATRIX APPROACH TO LOWER K-THEORY AND ALGEBRAIC NUMBER THEORY JI FENG (B.Sc., NUS, Singapore) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2013 To my parents iv v f Acknowledgements First of all, I would like to thank my supervisor A. J. Berrick for his guidance and encouragement in this project. I would like to thank Ye Shengkui, Yuan Zihong and Zhang Wenbing. We formed a discussion group on algebraic topology. Our regular discussion enrich my knowledge; and I learn a lot from the three of them. I would like to thank some other graduate students in our department who helped me in one way and another. To mention a few of them, I am particularly grateful to Ai Xinghuan, Chen Weidong, Gao Rui, Ma Jiajun, Wang Yi and Wang Haitao. I would like to thank Ivo Dell’Ambrogio and Fabrice Castel. We had fruitful discussions when they visited NUS as research fellows. I would like to thank my friends Qiu Xun and Wang Xuancong. I would like to thank CheeWhye Chin, T. Lambre and M. Karoubi for many helpful discussions and suggestions. I would like to thank Professor Lambre and Professor Karoubi for their hospitality during my visit to France. I am also grateful to Hou Likun and Sun Xiang, who helped me solve problems in TEX-typing. I would like thank Professor V. Gebhardt for introducing the Magma (software) to me. vii viii Acknowledgements Lastly, I would like thank my parents for their continuous support. Contents Acknowledgements vii Introduction The matrix theory 2.1 The modified plus construction . . . . . . . . . . . . . . . . . . . . . 2.2 Re-interpretation of the ideal class group . . . . . . . . . . . . . . . . 13 2.3 Re-interpretation of K-theory of the extension functor . . . . . . . . 18 2.4 Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5 A geometric interpretation . . . . . . . . . . . . . . . . . . . . . . . . 33 Local-global principle of the matrix theory 37 3.1 Torsion part of K0 (ext) . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2 Group schemes with local components . . . . . . . . . . . . . . . . . 43 3.3 Torsion free part of K0 (ext) . . . . . . . . . . . . . . . . . . . . . . . 50 Matrices, nonabelian cohomology and the Chern character 4.1 59 Matrices and nonabelian cohomology in general . . . . . . . . . . . . 61 ix x Contents 4.2 Specialization to number fields with GLn (R) as the coefficient group . 68 4.3 A group in the set of nonabelian cohomology . . . . . . . . . . . . . . 78 4.4 S-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.5 Explicit calculation of the Chern character via matrices . . . . . . . . 87 4.6 The formula tr(S −1 dS) . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.7 The Chern character from the Galois cohomology groups . . . . . . . 95 Bibliography 98 4.5 Explicit calculation of the Chern character via matrices Proof. ((C1j S1k )/ det)d((C1k S1j )/ det) 1≤j,k≤n = ((C1j S1j )/ det)d((C1j S1j )/ det) 1≤j≤n + ((C1j S1k )/ det)d((C1k S1j )/ det) 1≤j=k≤n = ((C1j S1j )/ det)d((C1j S1j )/ det) 1≤j≤n d((C1k S1j )(C1j S1k )/det2 ) + 1≤j[...]... into the group K0 (R) as all the spaces considered are connected and K0 (R) of a ring R is in general nontrivial There are several ways to overcome this problem, for example, Quillen’s Q-construction (see [32] and [37]) on an exact category and Waldhausen’s 3 S-construction on a category with cofibrations (see [41]) But due to highly abstract categorical and simplicial machinery, both constructions are... the ideal class group Cl(R) of R A similar idea applies to an extension KR ⊆ KR of number fields We try to understand the capitulation kernel of the ideal class group using matrix groups We try to relate our theory to the Bass exact sequence involving K- theory of functors In the next two chapters, we explore further the matrix approach developed in the second chapter In the third chapter, we mainly focus... mentioned matrix group Our main tool is the local-global principle We try to make some estimation of the size of the capitulation kernel using the Chebotarev density theorem and Galois cohomology The last chapter is inspired by the following observation The Galois group acts on the ideal class group; and on the other hand due to our matrix interpretation, certain matrix groups act on the ideal class group as... Evidently, the centralizer × CGLn (KR ) (GLn (R)) is easily identified with KR , the nonzero elements of KR , via diagonal embedding as scalar matrices We describe the normalizers Remark 2.1.3 This is a generalization of an important well-known concept borrowed from the theory of Lie groups and the theory of algebraic groups Suppose G is a connected algebraic group and T is a maximal torus in G Then the... find applications We use the local-global principle to study the capitulation kernel of an extension of number rings We also use the results in the first part to construct elements in the nonabelian Galois cohomology to detect nontrivial elements in the ideal class group of a number ring Chapter 1 Introduction The advent of algebraic K- theory dates back to mid-1950s It was invented by Grothendieck in an... difficult to compute Inspired by [6], we try to go back and modify the plus construction The main idea is that instead of looking at the general linear group GLn (R), we study larger matrix groups More precisely, we shall look at certain matrix groups containing GLn (R) as a normal subgroup, as described below In the second chapter, we lay down the foundations of the theory For a Dedekind domain R with... R is a Dedekind domain with field of fractions KR ; and n is a positive integer We study the normalizer of GLn (R) in GLn (KR ), denoted by NGLn (KR ) (GLn (R)) We investigate this group by a local argument, namely, embedding NGLn (KR ) (GLn (R)) in various localizations of R at different prime ideals Since we assume R is a Dedekind domain, we can use valuation methods to give a local characterization...Summary This thesis focuses on lower K- theory and algebraic number theory We modify Quillen’s plus construction Our new construction gives the same higher K- groups and more information on the group K0 of a ring In such a way, we are able to get new information on the ideal class group of a number ring The basis of the theory is established in the first part of the thesis In the second and the third part... vertical arrow i is injective if and only if the left hand square is a pullback (ii) The right vertical arrow i is trivial if and only if we have f : G → N making 19 20 Chapter 2 The matrix theory the following diagram commute: - G f N  i ? N i ? - G In order to apply these to the situation we are interested in, we need a characterization of W p (R) in W p (R ) Lemma 2.3.3 Suppose p is typical Then... Grothendieck in an attempt to generalize the Riemann-Roch theorem The group K0 (R) of a ring R is thus known as the Grothendieck group If R is a Dedekind domain, the group K0 (R) is not something unfamiliar, as it is nothing but Z⊕Cl(R), the direct sum of Z and the ideal class group of R If we try to recall the history of algebraic K- theory, K1 (R) and K2 (R) of a ring R were introduced next Fixing a ring R, . A MATRIX APPROACH TO LOWER K- THEORY AND ALGEBRAIC NUMBER THEORY JI FENG (B.Sc., NUS, Singapore) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL. Qiu Xun and Wang Xuancong. I would like to thank CheeWhye Chin, T. Lambre and M. Karoubi for many helpful discussions and suggestions. I would like to thank Professor Lambre and Professor Karoubi. Jiajun, Wang Yi and Wang Haitao. I would like to thank Ivo Dell’Ambrogio and Fabrice Castel. We had fruitful discussions when they visited NUS as research fellows. I would like to thank my friends