K-THEORY OF THEORIES OF MODULES AND ALGEBRAIC VARIETIES A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences 2014 Amit Kuber School of Mathematics 2 Contents Abstract . .7 Declaration . .8 Copyright Statement . .8 Acknowledgements . 11 1 Introduction 13 1.1 Historical background . 13 1.2 Contents and connections . 14 1.3 Grothendieck rings of theories of modules . 16 1.4 Canonical forms for definable sets . 17 1.5 Geometric and topological ideas . 17 1.6 The Grothendieck ring of varieties . 18 1.7 Higher K-theory with definable sets . 19 1.8 Notations and Conventions . 19 2 Background 21 2.1 Grothendieck rings of semirings . 21 2.2 Euler characteristic of simplicial complexes . 23 2.3 Products of simplicial complexes . 26 3 K-Theory of Model-theoretic Structures 29 3.1 Model-theoretic Grothendieck rings . 29 3.2 K-theory of symmetric monoidal categories . 31 3.3 Defining Kn(M) for a structure M .................... 37 3.4 Definable bijections in vector spaces . 38 3.5 K1 of vector spaces . 41 4 Definable Subsets of Modules 45 4.1 Model theory of modules . 45 4.2 The condition T = T@0 ........................... 47 4.3 Representing definable sets uniquely . 51 3 4.4 Connected definable sets . 53 5 Grothendieck Rings of Modules: Case T = T@0 57 5.1 Local characteristics . 57 5.2 Global characteristic . 64 5.3 Coloured global characteristics . 67 5.4 Monoid rings . 69 5.5 Multiplicative structure of Def(g M).................... 70 5.6 Computation of the Grothendieck ring . 73 6 Grothendieck Rings of Modules: Case T 6= T@0 77 6.1 Finite indices of pp-pairs . 77 6.2 The invariants ideal . 80 6.3 Pure embeddings and Grothendieck rings . 83 6.4 Torsion in Grothendieck rings . 86 7 The Grothendieck Ring of Varieties 89 7.1 Questions under consideration . 89 7.2 The Grothendieck semiring of varieties . 92 7.3 Question 7.1.2 ≡ Conjecture 7.1.4 . 93 + 7.4 Freeness of K0 (Vark) under Conjecture 7.1.4 . 96 7.5 The associated graded ring of K0(Vark).................. 100 7.6 Further Remarks . 101 8 Conclusions 103 8.1 Overview . 103 8.2 Model theory and freeness of groupoids . 103 8.3 Transferring freeness . 105 8.4 Two weaker forms of freeness . 106 8.5 Further questions . 109 Bibliography 111 Index of notation 114 Index of terminology 117 Word count : 27524 4 List of Figures 1.1 Section dependency chart . 15 3.1 K-theory of symmetric monoidal groupoids . 35 4.1 A block and a cell (shaded regions) . 49 4.2 Nest, cores and the characteristic function . 50 5 6 The University of Manchester Amit Kuber Doctor of Philosophy K-theory of Theories of Modules and Algebraic Varieties June 9, 2014 Let (S; t; ;) denote a small symmetric monoidal category whose objects are sets (possibly with some extra structure) and the monoidal operation, t, is disjoint union. Such categories encode cut-and-paste operations of sets. Quillen gave a functorial construction of the abelian groups (Kn(S))n≥0, known as the K-theory of S, which seek to classify its objects and morphisms. In particular, the group K0(S), known as the Grothendieck group, is the group completion of the commutative monoid of isomorphism classes of objects of S and it classifies the objects of the category up to `scissors-congruence'. On the other hand, the group K1(S) classifies the automor- phisms of objects (i.e., maps which cut an object into finitely many pieces which reassemble to give the same object) in the direct limit as the objects become large with respect to t. In this thesis we consider two classes of symmetric monoidal categories, one from model theory and the other from algebraic geometry. For any language L, the category S(M) of subsets of finite cartesian powers of a first order L-structure M definable with parameters from M together with definable bijections is symmetric monoidal and thus can be used to define the K-theory of the structure M which is functorial on elementary embeddings. On the other hand, for any field k, the category Vark of algebraic varieties and rational maps is also symmetric monoidal. In both these cases, the categories carry an additional binary operation induced by the product of objects; this equips the Grothendieck group with a multiplicative structure turning it into a commutative ring known as the Grothendieck ring. The model-theoretic Grothendieck ring K0(M) := K0(S(M)) of a first order struc- ture M was first defined by Krajiˇcekand Scanlon. We compute the ring K0(MR) for a right R-module M, where R is a unital ring, and show that it is a quotient of the monoid ring Z[X ], where X is the multiplicative monoid of isomorphism classes of fundamental definable subsets - the pp-definable subgroups - of the module, by the ideal that codes indices of pairs of pp-definable subgroups. As a corollary we prove a conjecture of Prest that K0(MR) is non-trivial, whenever M is non-zero. The main proof uses various techniques from simplicial homology and lattice theory to construct certain counting functions. The K-theory of a module is an invariant of its theory. In the special case of vector spaces we also compute the model-theoretic group K1. Let k be an algebraically closed field. Larsen and Lunts asked if two k-varieties having the same class in the Grothendieck ring K0(Vark) are piecewise isomorphic. Gromov asked if a birational self-map of a k-variety can be extended to a piecewise automorphism. We show that these two questions are equivalent over any algebraically closed field. Under the hypothesis of a positive answer to these two questions we prove that the underlying abelian group of the Grothendieck ring is a free abelian group and that the associated graded ring of the Grothendieck ring is the monoid ring Z[B] where B is the multiplicative monoid of birational equivalence classes of irreducible k-varieties. 7 Declaration No portion of the work referred to in the thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. Important results of this thesis have been written-up in the articles [26] and [27]. Copyright Statement i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the \Copyright") and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intel- lectual property (the \Intellectual Property") and any reproductions of copyright works in the thesis, for example graphs and tables (\Reproductions"), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and com- mercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in any rele- vant Thesis restriction declarations deposited in the University Library, The Univer- sity Library's regulations (see http://www.manchester.ac.uk/library/aboutus/regul- ations) and in The University's Policy on Presentation of Theses. 8 tF aAIºAºA\nA smEp t To my parents 9 10 Acknowledgements I owe a great debt of gratitude to my supervisor Prof. Mike Prest who was always ready to help in times of need. He introduced me to the combinatorial(!) world of Grothendieck rings and we had many long discussions on a variety of topics. His comments on my writing were always to the point and I have learnt many lessons in fine academic writing from our conversations. Directly and indirectly he taught me how to manage time and prioritize a list of tasks. Most importantly he gave me the freedom, whenever appropriate, to undertake other responsibilities like teaching, organizing seminars and even another research project. His catalytic role in these crucial years helped me flourish as an academic and gave me a better understanding of my capabilities. I learnt a lot from long chats with Harold Simmons on topics ranging from the quest for Ω on the campus - which, by the way, is engraved on the Holy Name Church - to posets, perfect talks from Alexandre Borovik and Nige Ray, discussions with Marcus Tressl, Mark Kambites, Alena Vencovsk´aand Alex Wilkie, and administrative advice from Colin Steele. I am sincerely grateful to all of them for enriching my knowledge and for helping me to improve my presentation skills. The non-academic staff in the school is very efficient and always ready to help. Special thanks to Tracey Smith for being patient with innumerable requests for the `Small Meeting Room' and `Frank Adams Rooms' booking.