Lectures on Topics in Algebraic K-Theory By Hyman Bass Tata Institute of Fundamental Research, Bombay 1967 Lectures on Topics in Algebraic K-Theory By Hyman Bass Note by Amit Roy No part of this book may be reproduced in any form by print, microfilm or any other means with- out written permission from the Tata Institute of Fundamental Research, Colaba, Bombay 5 Tata Institute of Fundamental Research, Bombay 1967 Preface These notes are based upon my lectures at the Tata Institute from De- cember, 1965 through February, 1966. The fact that the volume of ma- terial treated was excessive for so brief a period manifests itself in the monotone increasing neglect of technical details in the last chapters. The notes are often a considerable improvement on my lectures, and I express my warm thanks to Amit Roy, who is responsible for them. Portions of the research on which these lectures are based was car- ried out with NSF contract support at Columbia University. Hyman Bass iii Contents Introduction 1 1 The exact sequence of algebraic K-theory 7 1 Categories with product, and their functors . 7 2 Directed categories of abelian groups . 15 3 K1C asadirectlimit . ..... ...... ..... .. 18 4 The exact sequence . 22 5 The category P ...................... 28 6 The category FP ..................... 30 7 The category Pic ..................... 35 2 Categories of modules and their equivalences 39 1 Categories of modules; faithfully projective modules . 40 2 k-categories and k-functors . 44 3 Right continuous functors . 46 4 Equivalences of categories of modules . 47 5 Faithfully projective modules . 52 6 Wedderburn structure theory . 56 7 Autoequivalence classes; the Picard group . 60 3 The Brauer group of a commutative ring 65 1 Separable Algebras . 66 2 Assortedlemmas ..................... 69 3 Local criteria for separability . 78 4 Azumaya algebras . 81 v vi Contents 5 Splittingrings....................... 86 6 The exact sequence . 87 4 The Brauer-Wall group of graded Azumaya algebras 95 1 Graded rings and modules . 95 2 Separable algebras . 97 3 The group of quadratic extensions . 99 4 Azumaya algebras . 101 5 Automorphisms ......................102 5 The structure of the Clifford Functor 105 1 Bilinear modules . 106 2 The hyperbolic functor . 111 3 The CliffordFunctor .... ..... ...... ....116 4 The orthogonal group and spinor norm . 125 Introduction In order to construct a general theory of (non-singular) quadratic forms 1 and orthogonal groups over a commutative ring k, one should first in- vestigate the possible generalizations of the basic classical tools (when k is a field). These are (I) Diagonalization (if char k , 2), and Witt’s theorem. (II) Construction of the classical invariants: dimension, discriminant, Hasse invariant. This course is mostly concerned with the algebraic apparatus which is preliminary to a generalization of II, particularly of the Hasse invari- ant. Consequently, quadratic forms will receive rather little attention, and then only at the end. It will be useful, therefore, to briefly outline now the material to be covered and to indicate its ultimate relevance to quadratic forms. We define a quadratic module over k to be a pair (P, q) with P ∈ P, the category of finitely generated projective k-modules, and with = q : P → k a map satisfying q(ax) = a2q(x) (aǫk, aǫP) and such that (x, y) 7−→ q(x + y) − q(x) − q(y) is a bilinear form. This form then in- ∗ duces a homomorphism P → P = Homk(P, k) (by fixing a variable), and we call (P, q) non-singular if P → P∗ is an isomorphism. If (P1, q1) and (P2, q2) are quadratic modules, we have the “or- thogonal sum” (P1, q1) ⊥ (P2, q2) = (P1 ⊕ P2, q), where q(x1, x2) = q1(x1) + q2(x2). Given P ∈ P, in order to find a q so that (P, q) in non-singular we 2 = 1 2 Contents must at least have P ≈ P∗. Hence for arbitrary P, we can instead take P⊕ P∗, which has an obvious isomorphism, “ 0 1P∗ ”, with its dual. In- 1P 0 deed this is induced by the bilinear form associated with the hyperbolic module = ∗ H(P) (P ⊕ P , qP ), ∗ where qp(x, f ) = f (x)(xǫP, f ǫP ). The following statement is easily proved: (P, q) is non-singular ⇔ (P, q) ⊥ (P, −q) ≈ H(P). Let Q denote the category of non-singular quadratic modules and = their isometrics. In P we take only the isomorphisms as morphisms. = Then we can view H as the hyperbolic functor H : P → Q, = = − where, for f : P → P′, H( f ) is the isometry f ⊕ f ∗ 1 : H(P) → H(P′). Moreover, there is a natural isomorphism H(P ⊕ P′) ≈ H(P) ⊥ H(P′). With this material at hand I will now begin to describe the course. In chapter 1 we establish an exact sequence of Grothendieck groups of certain categories, in an axiomatic setting. Briefly, suppose we are 3 given a category C in which all morphisms are isomorphisms (i.e. a groupoid) together with a product ⊥ which has the formal properties of ⊥ and ⊕ above. We then make an abelian group out of obj C in which ⊥ corresponds to +; it is denoted by K0C . A related group K1C , is constructed using the automorphisms of objects of C . Its axioms resemble those for a determinant. If H : C → C ′ is a product preserving functor (i.e. H(A ⊥ B) = HA ⊥ HB), then it induces homomorphisms ′ KiH : KiC → KiC , i = 0, 1. We introduce a relative category ΦH, and then prove the basic theorem: There is an exact sequence ′ ′ K1C → K1C → K0ΦH → K0C → K0C , Contents 3 provided H is “cofinal”. Cofinal means: given A′ǫC ′, there exists B′ → C ′ and CǫC such that A′ ⊥ B′ ≈ HC. This theorem is a special case of results of Heller [1]. The discussion above shows that the hyperbolic functor satisfies all the necessary hypotheses, so we obtain an exact sequence [H] K1P → K1Q → K0ΦH → K0P → K0Q → Witt (k) → 0. = = = = Here we define Witt (k) = coker (k0H). It corresponds exactly to the classical “Witt ring” of quadratic forms (see Bourbaki [2]). The K P, i = i = 0, 1 will be described in chapter 1. K1Q is related to the stable = structure of the orthogonal groups over k. The classical Hasse invariant attaches to a quadratic form over a 4 field k an element of the Brauer group Br (k). It was given an intrinsic definition by Witt [1] by means of the Clifford algebra. This necessitates a slight artifice due to the fact that the Clifford algebra of a form of odd dimension is not central simple. Moreover, this complication renders the definition unavailable over a commutative. ring in general. C.T.C. Wall [1] proposed a natural and elegant alternative. Instead of modifying the Clifford algebras he enlarged the Brauer group to accommodate them, and he calculated this “Brauer-Wall” group BW(k) when k is a field. Wall’s procedure generalizes naturally to any k. In order to carry this out, we present in chapters 2, 3, and 4, an exposition of the Brauer-Wall theory. Chapter 2 contains a general theory of equivalences of categories of modules, due essentially to Morita [1] (see also Bass [2]) and Gabriel [1]. It is of general interest to algebraists, and it yields, in particular, the Wedderburn structure theory in a precise and general form. It is also a useful preliminary to chapter 2, where we deal with the Brauer group Br (k) of azumaya algebras, following the work of Auslander-Goldman [1]. In chapter 4 we study the category Az of graded azumaya algebras, and = 2 extend Wall’s calculation of BW(k), giving only statements of results, without proofs. Here we find a remarkable parallelism with the phenomenon wit- 5 nessed above for quadratic forms. Let FP denote the category of “faith- = 2 4 Contents fully projective” k-modules P (see chapter 1 for definition), which have a grading modulo 2 : P = P0 ⊕ P1. Then the full endomorphism algebra END(P) (we reserve End for morphisms of degree zero) has a natural grading modulo 2, given by maps homogeneous of degree zero and one, respectively. a b = a 0 + 0 b Matricially, c d 0 d c 0 . These are the “trivial” algebras in Az ; that is BW (k) is the group of isomorphism classes of algebras in = 2 Az , with respect to ⊗, modulo those of the form END(P). It is a group = 2 because of the isomorphism A ⊗ A∗ ≈ END(A), where A∗ is the (suitably defined) opposite algebra of A, for A ∈ Az . = 2 Morever, A is faithfully projective as a k-module. Finally we note that END : FP → Az = 2 = 2 is a functor, if in both cases we take homogeneous isomorphisms as morphisms. For, if f : P → P′ and e ∈ END(P), then END( f )(e) = f e f −1 ∈ END(P′). Moreover, there is a natural isomorphism END(P ⊗ P′) ≈ END(P) ⊗ END(P′). 6 Consequently, we again obtain an exact sequence: [END] : K FP → K Az → K ΦEND → K FP → K Az → BW(k) → 0. 1 = 1 0 0 = 0 2 = 2 2 = 2 Chapter 5 finally introduces the category Q of quadratic forms. The = Clifford algebra is studied, and the basic structure theorem for the Clif- ford algebra is proved in the following form: The diagram of (product- preserving) functors P H Q = / = ∧ Clifford FP / Az = 2 END = 2 Contents 5 commutes up to natural isomorphism. Here ∧ denotes the exterior algebra, graded modulo 2 by even and odd degrees. This result simultaneously proves that the Clifford algebras lie in Az , and shows that there is a natural homomorphism of exact sequences = 2 : K1P K1Q Φ K0P K0Q [H] = / = / K0 H / = / = / Witt(k) / 0 Grassman Clifford Hasse−Wall : K1FP K1Az K ΦEND K0FP K0Az BW(k) END = 2 / = 2 / 0 / = 2 / = 2 / / 0 This commutative diagram is the promised generalization of the Hasse invariant.