A1-Algebraic topology over a field Fabien Morel Foreword This work should be considered as a natural sequel to the foundational paper [65] where the A1-homotopy category of smooth schemes over a base scheme was defined and its first properties studied. In this text the base scheme will always be the spectrum of a perfect field k. One of our first motivation is to emphasize that, contrary to the first impression, the relationship between the A1-homotopy theory over k and the category Smk of smooth k-schemes is of the same nature as the relation- ship between the classical homotopy theory and the category of differentiable manifolds. This explains the title of this work; we hope to convince the reader in this matter. This slogan was already discussed in [61], see also [5]. This text is the result of the compilation of two preprints \A1-algebraic topology over a field” and \A1-homotopy classification of vector bundles over smooth affine schemes" to which we added some recent new stuff, consist- ing of the two sections: \Geometric versus canonical transfers" (Section 3) and \The Rost-Schmid complex of a strongly A1-invariant sheaf" (Section 4). The main objective of these new sections was primarily to correctly es- tablish the equivalence between the notions of \strongly A1-invariant" and \strictly A1-invariant" for sheaves of abelian groups, see Theorems 16 below. These new Sections appear also to be interesting in their own. As the reader will notice, the introduction of the notion of strictly A1-invariant sheaves with generalized transfers in our work on the Friedlander-Milnor conjecture [62][63][64] is directly influenced from these. 1 Our treatment of transfers in Section 3, which is an adaptation of the original one of Bass and Tate for Milnor K-theory [11], is, we think, clar- ifying. Besides reaching the structure of strictly A1-invariant sheaves with generalized transfers, we obtain on the way a new proof of the fact [42] that the transfers in Milnor K-theory do not depend on choices of a sequence of generators, see Theorem 3.27 and Remark 3.32 below. Our proof is in spirit different from the one of Kato [42], as we prove directly by geometric means the independence in the case of two generators. The construction in Section 4 for a general strongly A1-invariant sheaf of abelian groups of its \Rost-Schmid" complex is directly influenced by the work of Rost [75] and its adaptation to Witt groups in [77]. Our philosophy on transfers in this work was to use them as less as possible, and to only use them when they really show up by themselves. For instance we will define below in Section 2 the sheaves of unramified Milnor K-theory (as well as Milnor-Witt K-theory) without using any transfers. The proof of Lemma 4.24 contains the geometric explanation of the formula for the differential of the \Gersten-Rost-Schmid" complex (Corollary 4.44) and the appearance of transfers in it. The results and ideas in this work have been used and extended in several different directions. In [18] some very concrete computations on A1-homotopy classes of rational fractions give a nice interpretation of our sheaf theoretic A1 P1 A1 computations of π1 ( ). The structure and property of the -fundamental group sheaves as well as the associated theory of A1-coverings has been used in [3, 4, 94]. It is also the starting point of [5]. Our result (Section 8) con- A1 cerning the Suslin-Voevodsky construction Sing• (SLn), n ≥ 3, has been generalized to a general split semi-simple group of type not SL2 in [95] and to the case of SL2 in [66]. The present work plays also a central role in our approach to the Friedlander- Milnor conjecture [62, 63, 64]. Conventions and notations. Everywhere in this work, k denotes a fixed perfect field and Smk denotes the category of smooth finite type k- schemes. We denote by Fk the category of field extensions k ⊂ F of k such that F 2 is of finite transcendence degree over k. By a discrete valuation v on F 2 Fk we will always mean one which is trivial on k. We let Ov ⊂ F denotes its valuation ring, mv ⊂ Ov its maximal ideal and κ(v) its residue field. For any scheme X and any integer i we let X(i) denote the set of points in X of codimension i. Given a point x 2 X, κ(x) will denote its residue field, and Mx will denote the maximal ideal of the local ring OX;x. The category Smk is always endowed with the Nisnevich topology [68, 65], unless otherwise explicitly stated. Thus for us \sheaf" always means sheaf in the Nisnevich topology. We will let Set denote the category of sets, Ab that of abelian groups. A space is a simplicial object in the category of sheaves of sets on Smk [65]. We will also assume the reader is familiar with the notions and results of loc. cit.. 0 We denote by Smk the category of essentially smooth k-schemes. For us, an essentially smooth k-scheme is a noetherian k-scheme X which is the inverse limit of a left filtering system (Xα)α with each transition morphism Xβ ! Xα being an ´etaleaffine morphism between smooth k-schemes (see [32]). For any F 2 Fk the k-scheme Spec(F ) is essentially k-smooth. For each point x 2 X 2 Smk, the local scheme Xx := Spec(OX;x) of X at x, as well as h Oh its henselization Xx := Spec( X;x) are essentially smooth k-schemes. In the O h same way the complement of the closed point in Spec( X;x) or Xx are essen- tially smooth over k. We will sometime make the abuse of writing \smooth k-scheme" instead of \essentially smooth k-scheme", if no confusion can arise. op Given a presheaf of sets on Smk, that is to say a functor F :(Smk) ! Sets, and an essentially smooth k-scheme X = limαXα we set F (X) := colimitαF (Xα). From the results of [32] this is well defined. When X = Spec(A) is affine we will also simply denote this set by F (A). 3 Acknowledgements. I want to take this opportunity to warmly thank Mike Hopkins and Marc Levine for their interest in this work during the past years when I conceived it; their remarks, comments and some discussions on and around this subject helped me to improve it very much. 4 Contents 0 Introduction 6 1 Unramified sheaves and strongly A1-invariant sheaves 21 1.1 Unramified sheaves of sets . 21 1.2 Strongly A1-invariant sheaves of groups . 31 1.3 Z-graded strongly A1-invariant sheaves of abelian groups . 46 2 Unramified Milnor-Witt K-theories 60 2.1 Milnor-Witt K-theory of fields . 61 2.2 Unramified Milnor-Witt K-theories . 68 2.3 Milnor-Witt K-theory and strongly A1-invariant sheaves . 85 3 Geometric versus canonical transfers 97 3.1 The Gersten complex in codimension 2 . 98 3.2 Geometric versus cohomological transfers on M−n . 115 3.3 Proof of the main Theorem . 121 4 The Rost-Schmid complex of a strongly A1-invariant sheaf 134 4.1 Absolute transfers and the Rost-Schmid complex . 134 4.2 The Rost-Schmid complex is a complex . 147 4.3 Gersten complex versus Rost-Schmid complex . 164 5 A1-homotopy sheaves and A1-homology sheaves 174 A1 A1 ≥ 5.1 Strong -invariance of the sheaves πn , n 1 . 174 5.2 A1-derived category and Eilenberg-MacLane spaces . 184 5.3 The Hurewicz theorem and some of its consequences . 192 A1 A1 Pn A1 6 -coverings, π1 ( ) and π1 (SLn) 205 A1 A1 A1 6.1 -coverings, universal -covering and π1 . 205 A1 Pn A1 ≥ 6.2 Basic computation: π1 ( ) and π1 (SLn) for n 2 . 214 A1 P1 6.3 The computation of π1 ( ) . 219 7 A1-homotopy and algebraic vector bundles 229 7.1 A1-homotopy classification of vector bundles . 229 7.2 The theory of the Euler class for affine smooth schemes . 234 7.3 A result concerning stably free vector bundles . 237 5 8 The affine B.G. property for the linear groups and the Grass- manian 239 8.1 Preliminaries and recollections on regularity . 239 8.2 Preliminaries and recollections on patching projective modules 248 8.3 The affine B-G properties for the Grassmanian and the general linear groups . 254 A The (affine) B.G. property for simplicial sheaves 260 A.1 Some recollections on the B.G. property . 260 A.2 The affine replacement of a simplicial presheaf . 263 A.3 The affine B.G. property in the Nisnevich topology . 265 A.4 A technical result . 272 B Recollection on obstruction theory 274 B.1 The Postnikov tower of a morphism . 275 B.2 Twisted Eilenberg-MacLane objects . 277 B.3 The obstruction theory we need . 282 0 Introduction Let k be a perfect field. Our aim in this work is to address in the A1-homotopy theory of smooth k-schemes considered in [65, 89, 55] the analogues of the following classical Theorems: Theorem 1 (Brouwer degree) Let n > 0 be an integer and let Sn denote the n-sphere. Then for an integer i 0 if i < n π (Sn) = i Z if i = n Theorem 2 (Hurewicz Theorem) For any pointed connected topological space X and any integer n ≥ 1 the Hurewicz morphism πn(X) ! Hn(X) is the abelianization if n = 1, is an isomorphism if n ≥ 2 and X is (n − 1)- connected, and is an epimorphism if n ≥ 3 and X is (n − 2)-connected.