NORMS IN MOTIVIC HOMOTOPY THEORY TOM BACHMANN AND MARC HOYOIS Abstract. If f : S0 ! S is a finite locally free morphism of schemes, we construct a symmetric monoidal 0 \norm" functor f⊗ : H•(S ) ! H•(S), where H•(S) is the pointed unstable motivic homotopy category 0 over S. If f is finite ´etale,we show that it stabilizes to a functor f⊗ : SH(S ) ! SH(S), where SH(S) is 1 the P -stable motivic homotopy category over S. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic E1-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; we prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck{Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum HZ, the homotopy K-theory spectrum KGL, and the algebraic cobordism spectrum MGL. The normed spectrum structure on HZ is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology. Contents 1. Introduction 3 1.1. Norm functors 3 1.2. Normed motivic spectra 4 1.3. Examples of normed spectra 5 1.4. Norms in other contexts 7 1.5. Norms vs. framed transfers 8 1.6. Summary of the construction 8 1.7. Summary of results 9 1.8. Guide for the reader 10 1.9. Remarks on 1-categories 10 1.10. Standing assumptions 11 1.11. Acknowledgments 11 2. Preliminaries 11 2.1. Nonabelian derived 1-categories 11 2.2. Unstable motivic homotopy theory 13 2.3. Weil restriction 14 3. Norms of pointed motivic spaces 15 3.1. The unstable norm functors 15 3.2. Norms of quotients 17 4. Norms of motivic spectra 19 4.1. Stable motivic homotopy theory 19 4.2. The stable norm functors 20 5. Properties of norms 21 5.1. Composition and base change 21 5.2. The distributivity laws 22 5.3. The purity equivalence 25 5.4. The ambidexterity equivalence 26 5.5. Polynomial functors 29 6. Coherence of norms 30 6.1. Functoriality on the category of spans 30 Date: May 28, 2020. 1 2 TOM BACHMANN AND MARC HOYOIS 6.2. Normed 1-categories 32 7. Normed motivic spectra 35 7.1. Categories of normed spectra 35 7.2. Cohomology theories represented by normed spectra 39 8. The norm{pullback{pushforward adjunctions 43 8.1. The norm{pullback adjunction 44 8.2. The pullback{pushforward adjunction 47 9. Spectra over profinite groupoids 48 9.1. Profinite groupoids 49 9.2. Norms in stable equivariant homotopy theory 51 10. Norms and Grothendieck's Galois theory 53 10.1. The profinite ´etalefundamental groupoid 54 10.2. Galois-equivariant spectra and motivic spectra 55 10.3. The Rost norm on Grothendieck{Witt groups 57 11. Norms and Betti realization 58 11.1. A topological model for equivariant homotopy theory 58 11.2. The real Betti realization functor 60 12. Norms and localization 61 12.1. Inverting Picard-graded elements 62 12.2. Inverting elements in normed spectra 63 12.3. Completion of normed spectra 66 13. Norms and the slice filtration 67 13.1. The zeroth slice of a normed spectrum 67 13.2. Applications to motivic cohomology 69 13.3. Graded normed spectra 71 13.4. The graded slices of a normed spectrum 74 14. Norms of cycles 75 14.1. Norms of presheaves with transfers 75 14.2. The Fulton{MacPherson norm on Chow groups 77 14.3. Comparison of norms 78 15. Norms of linear 1-categories 82 15.1. Linear 1-categories 83 15.2. Noncommutative motivic spectra and homotopy K-theory 85 15.3. Nonconnective K-theory 90 16. Motivic Thom spectra 92 16.1. The motivic Thom spectrum functor 92 16.2. Algebraic cobordism and the motivic J-homomorphism 95 16.3. Multiplicative properties 96 16.4. Free normed spectra 100 16.5. Thom isomorphisms 101 Appendix A. The Nisnevich topology 103 Appendix B. Detecting effectivity 104 Appendix C. Categories of spans 106 C.1. Spans in extensive 1-categories 106 C.2. Spans and descent 110 C.3. Functoriality of spans 113 Appendix D. Relative adjunctions 114 Table of notation 116 References 117 NORMS IN MOTIVIC HOMOTOPY THEORY 3 1. Introduction The goal of this paper is to develop the formalism of norm functors in motivic homotopy theory, to study the associated notion of normed motivic spectrum, and to construct many examples. 1.1. Norm functors. Norm functors in unstable motivic homotopy theory were first introduced by Voevod- sky in [Del09, x5.2], in order to extend the symmetric power functors to the category of motives [Voe10a, x2]. In this work, we will be mostly interested in norm functors in stable motivic homotopy theory. For every finite ´etalemorphism of schemes f : T ! S, we will construct a norm functor f⊗ : SH(T ) ! SH(S); where SH(S) is Voevodsky's 1-category of motivic spectra over S [Voe98, Definition 5.7]. Another functor associated with f is the pushforward f∗ : SH(T ) ! SH(S). For comparison, if r: S t S ! S is the fold map, then r∗ : SH(S t S) ' SH(S) × SH(S) ! SH(S) is the direct sum functor, whereas r⊗ : SH(S t S) ' SH(S) × SH(S) ! SH(S) is the smash product functor. This example already shows one of the main technical difficulties in the theory of norm functors: unlike f∗, the functor f⊗ preserves neither limits nor colimits. The norm functors are thus a new basic functoriality of stable motivic homotopy theory, to be added to ∗ ! the list f], f , f∗, f!, f . The three covariant functors f], f∗, and f⊗ should be viewed as parametrized versions of the sum, product, and tensor product in SH(S): categorical structure parametrized extension sum f] for f smooth product f∗ for f proper tensor product f⊗ for f finite ´etale Of course, the sum and the product coincide in SH(S), and much more generally the functors f] and f∗ coincide up to a \twist" when f is both smooth and proper. The basic interactions between sums, products, and tensor products extend to this parametrized setting. For example, the associativity and commutativity of these operations are encoded in the compatibility of the assignments f 7! f]; f∗; f⊗ with composition and base change. The distributivity of tensor products over sums and products has a parametrized extension as well. To express these properties in a coherent manner, we are led to consider 2-categories of spans1, which will be ubiquitous throughout the paper. The most fundamental example is the 2-category Span(Fin) of spans of finite sets: its objects are finite sets, its morphisms are spans X Y ! Z, and composition of morphisms is given by pullback. The one-point set in this 2-category happens to be the universal commutative monoid: if C is any 1-category with finite products, then a finite-product-preserving functor Span(Fin) ! C is precisely a commutative monoid in C (we refer to Appendix C for background on 1-categories of spans and for a proof of this fact). For example, the 1-category SH(S) is itself a commutative monoid in the 1-category Cat1 of 1-categories, under either the direct sum or the smash product. These commutative monoid structures correpond to functors ⊕ ⊗ I SH(S) ; SH(S) : Span(Fin) ! Cat1;I 7! SH(S) : I ` Using that SH(S) ' SH( I S), one can splice these functors together as S varies to obtain ⊕ ⊗ SH ; SH : Span(Sch; all; fold) ! Cat1;S 7! SH(S): Here, Span(Sch; all; fold) is a 2-category whose objects are schemes and whose morphisms are spans X Y ! Z where X Y is arbitrary and Y ! Z is a sum of fold maps (i.e., there is a finite coproduct ` tni decomposition Z = i Zi and isomorphisms Y ×Z Zi ' Zi over Zi). It turns out that the functors SH⊕ and SH⊗ encode exactly the same information as the presheaves of symmetric monoidal 1-categories 1All the 2-categories that appear in this introduction are (2; 1)-categories, i.e., their 2-morphisms are invertible. 4 TOM BACHMANN AND MARC HOYOIS S 7! SH(S)⊕ and S 7! SH(S)⊗. The basic properties of parametrized sums, products, and tensor products are then encoded by extensions of SH⊕ and SH⊗ to larger 2-categories of spans: t f p ∗ SH : Span(Sch; all; smooth) ! Cat1;S 7! SH(S); (X − Y −! Z) 7! p]f ; × f p ∗ SH : Span(Sch; all; proper) ! Cat1;S 7! SH(S); (X − Y −! Z) 7! p∗f ; ⊗ f p ∗ SH : Span(Sch; all; f´et) ! Cat1;S 7! SH(S); (X − Y −! Z) 7! p⊗f ; where \f´et"is the class of finite ´etalemorphisms. A crucial difference between the first two functors and the third one is that the former are completely determined by their restriction to Schop and the knowledge ∗ t × that the functors f] and f∗ are left and right adjoint to f . The existence of the extensions SH and SH is then a formal consequence of the smooth base change theorem (which holds in SH by design) and the proper base change theorem (formulated by Voevodsky [Del02] and proved by Ayoub [Ayo08]). By contrast, the functor SH⊗ must be constructed by hand.