If $f : S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor $f_\otimes : \mathcal{H}_{\bullet}(S')\to \mathcal{H}_{\bullet}(S)$, where $\mathcal{H}_\bullet(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite étale, we show that it stabilizes to a functor $f_\otimes : \mathcal{S}\mathcal{H}(S') \to \mathcal{S}\mathcal{H}(S)$, where $\mathcal{S}\mathcal{H}(S)$ is the $\mathbb{P}^1$-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 $E_\infty$-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 $H\mathbb{Z}$, the homotopy $K$-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $H\mathbb{Z}$ is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology.

If f : S → S is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor f ⊗ : H•(S ) → H•(S), where H•(S) is the pointed unstable motivic homotopy category over S. If f is finite étale, we show that it stabilizes to a functor f ⊗ : SH(S ) → SH(S), where SH(S) is the P 1 -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 E∞-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.

We compute the generalized slices (as defined by Spitzweck-Østvaer) of the motivic spectrum KO (representing Hermitian K-theory) in terms of motivic cohomology and (a version of) generalized motivic cohomology, obtaining good agreement with the situation in classical topology and the results predicted by Markett-Schlichting. As an application, we compute the homotopy sheaves of (this version of) generalized motivic cohomology, which establishes a version of a conjecture of Morel. 1125which is (at best) considered an analog of S 1 , and yet the layers are already given at double speed, something seems amiss. (2) In classical topology there is another version of K-theory, namely the K-theory of real (not complex) vector bundles, denoted KO. There is also Bott periodicity, this time resulting in the computation that the homotopy groups of KO are given by Z, Z/2, Z/2, 0, Z, 0, 0, 0 and then repeating periodically. There is an analog of topological KO in algebraic geometry, namely Hermitian K-theory [6] and (also) denoted KO ∈ SH(k). It satisfies an appropriate form of Bott periodicity, but this is not captured accurately by its slices, which are also very different from the topological analog [21]. (3) The slice filtration does not always converge. Thus just considering slices is not enough, for example, to determine if a morphism of spectra is an isomorphism.Problem (3) has lead Spitzweck-Østvaer [25] to define a refined version of the effectivity condition yielding the slice filtration which they call being 'very effective'. In this article we will argue that their filtration also solves issues (1) and (2).To explain the ideas, recall that the category SH(k) eff is the localizing (so triangulated!) subcategory generated by objects of the form Σ ∞ X + for X ∈ Sm(k) (that is, no desuspension by G m ). Then one defines SH(k) eff (n) = SH(k) ∧ T ∧n and for E ∈ SH(k) the n-effective cover f n E ∈ SH(k) eff (n) is the universal object mapping to E. (Note that since SH(k) eff is triangulated, we have SH(k) eff ∧ T ∧n = SH(k) eff ∧ G ∧n m .) In contrast, Spitzweck-Østvaer define the subcategory of very effective spectra SH(k) veff to be the subcategory generated under homotopy colimits and extensions by Σ ∞ X + ∧ S n where X ∈ Sm(k) and n 0. This subcategory is not triangulated! As before we put SH(k) veff (n) = SH(k) veff ∧ T ∧n . (Note that now, crucially, SH(k) veff ∧ T ∧n = SH(k) veff ∧ G ∧n m .) Then as before the very n-effective cover f n E ∈ SH(k) veff (n) is the universal object mapping to E. The cofibresf n+1 E →f n E →s n E are called the generalized slices of E.As pointed out by Spitzweck-Østvaer, the connectivity off n E in the homotopy t-structure increases with n, so the generalized slice filtration automatically converges. Moreover, it is easy to see thatf n KGL = f n KGL (that is, the n-effective cover of KGL is 'accidentally' already very n-effective) and thuss n KGL = s n (KGL). This explains how the generalized slice filtration solves problem (1): we see that the 'G m -slices' (that is, ordinary slices)...

Let S be a Noetherian scheme of finite dimension and denote by ρ ∈ [½, Gm] SH(S) the (additive inverse of the) morphism corresponding to −1 ∈ O × (S). Here SH(S) denotes the motivic stable homotopy category. We show that the category obtained by inverting ρ in SH(S) is canonically equivalent to the (simplicial) local stable homotopy category of the site S rét , by which we mean the small realétale site of S, comprised ofétale schemes over S with the realétale topology.One immediate application is that SH(R)[ρ −1 ] is equivalent to the classical stable homotopy category. In particular this computes all the stable homotopy sheaves of the ρ-local sphere (over R). As further applications we show that D A 1 (k, Z[1/2]) − ≃ DM W (k)[1/2] (improving a result of Ananyevskiy-Levine-Panin), reprove Röndigs' result that π i (½[1/η, 1/2]) = 0 for i = 1, 2 and establish some new rigidity results.

Given a 0-connective motivic spectrum E ∈ SH(k) over a perfect field k, we determine h 0 of the associated motive M E ∈ DM(k) in terms of π 0 (E). Using this we show that if k has finite 2-étale cohomological dimension, then the functor M : SH(k) → DM(k) is conservative when restricted to the subcategory of compact spectra, and induces an injection on Picard groups. We extend the conservativity result to fields of finite virtual 2-étale cohomological dimension by considering what we call "real motives". h 0 (E) ∼ = π 0 (E)/η.Here we use implicitly that the heart of DM(k) can be identified with a subcategory of the heart of SH(k). This theorem shows that we lose some information in passing to the motive, and so cannot expect a perfect analogy with the classical situation. Nonetheless we can prove the following.Theorem (Conservativity Theorem I; see Theorem 16). Let k be a perfect field of finite 2-étale cohomological dimension and E ∈ SH(k) be compact. If M E ≃ 0 then E ≃ 0.(Recall that an object is called compact if Hom SH(k) (E, •) commutes with arbitrary sums.) Combined with the Motivic Hurewicz Theorem, one easily obtains the following.Corollary (Pic-injectivity Theorem; see Theorem 18). In the situation of the Theorem, the natural homomorphism P ic(SH(k)) → P ic(DM(k)) is injective.This result was one of the main motivations for our investigations. See later in this introduction for more details on applications. (We do not know if P ic(SH(k)) → P ic(DM(k)) might also be surjective.)The Conservativity Theorem (and by extension the Pic-injectivity Theorem) stated here is not optimal. Let us discuss to what extent the assumptions can be weakened.In characteristic zero, the compactness assumption can be replaced by the technical assumptions of connectivity and "slice-connectivity" (see Section 4 for a definition of this term). In characteristic p > 0 we can also replace compactness by connectivity and slice-connectivity, but in this case we must additionally assume that the natural map E p − → E is an isomorphism. If k is a non-orderable field (i.e. one in which -1 is a sum of squares; we still assume k perfect), then one may present k as a colimit of perfect subfields with finite 2-étale cohomological dimension, and the theorem can then be deduced for such k, see [5]. Compactness instead of connectivity and slice-connectivity is then essential, however.The author currently does not know how to weaken the perfectness assumption on k.A further weakening is to consider orderable k, i.e. fields where -1 is not a sum of squares. (Such fields necessarily have characteristic zero.) Assume for simplicity that for any ordering of k, there is an order preserving embedding of k into R. Write Sper(k) for the set of orderings of k. (Equivalently in this case, embeddings k ֒→ R.) If σ ∈ Sper(k), then there is a so-called real realisation functor R R σ : SH(k) → SH, related to considering the real points of a smooth variety, with their strong topology. We compose this with the singular chain complex functor to obtain the real ...

Over any field of characteristic = 2, we establish a 2-term resolution of the η-periodic, 2-local motivic sphere spectrum by shifts of the connective 2-local Witt K-theory spectrum. This is curiously similar to the resolution of the K(1)-local sphere in classical stable homotopy theory. As applications we determine the η-periodized motivic stable stems and the η-periodized algebraic symplectic and SL-cobordism groups. Along the way we construct Adams operations on the motivic spectrum representing Hermitian K-theory and establish new completeness results for certain motivic spectra over fields of finite virtual 2-cohomological dimension. In an appendix, we supply a new proof of the homotopy fixed point theorem for the Hermitian K-theory of fields. Contents 1. Introduction 1.1. Inverting the motivic Hopf map 1.2. Main results 1.3. Proof sketch 1.4. Organization 1.5. Acknowledgements 1.6. Conventions 1.7. Table of notation 2. Preliminaries and Recollections 2.1. Filtered modules 2.2. Derived completion of abelian groups 2.3. Witt groups 2.4. The homotopy t-structure 2.5. Completion and localization 2.6. Very effective spectra 2.7. Inverting ρ 3. Adams operations for the motivic spectrum KO 3.1. Summary 3.2. Motivic ring spectra KO and KGL 3.3. The Gepner-Snaith and homotopy fixed point theorems 3.4. Action on the Bott element 3.5. The 2-adic fracture square 3.6. Proof of the main theorem 3.7. The motivic orthogonal image of j spectrum 3.8. Geometric operations 4. Cobordism spectra 4.1. Summary 4.2. Real realization 4.3. Homology 4.4. Adams action 5. Some completeness results 5.1. Summary 5.2. Main result 5.3. Remaining proofs 6. The HW-homology of kw 6.1. Summary 6.2. The motivic dual Steenrod algebra 6.3. Spectra employed in the proof Date: June 4, 2020. 1 2 TOM BACHMANN AND MICHAEL J. HOPKINS 6.4. Determination of π * (kw ∧ HW) ∧ 2 41 7. Main theorem 45 8. Applications 48 8.1. Homotopy groups of the η-periodic sphere 48 8.2. Homotopy groups of MSp[η −1 ] 49 8.3. Homotopy groups of MSL[η −1 ] 52 8.4. MSp, MSL and kw 52 8.5. Cellularity results 53 8.6. HW ∧ HW (2) 54 8.7. kw * kw (2) 55 Appendix A. The homotopy fixed point theorem 56 References 58

We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

We show that the reduced motive of a smooth affine quadric is invertible as an object of the triangulated category of motives DM(k, Z[1/e]) (where k is a perfect field of exponential characteristic e). We also establish a motivic version of the conjectures of Po Hu on products of certain affine Pfister quadrics. Both of these results are obtained by studying a novel conservative functor on (a subcategory of) DM(k, Z[1/e]), the construction of which constitutes the main part of this work.