EQUIVARIANT MOTIVIC HOMOTOPY THEORY

GUNNAR CARLSSON AND ROY JOSHUA

Abstract. In this paper, we develop the theory of equivariant motivic homotopy theory, both unstable and stable. While our main interest is the case when the group is pro-finite, we discuss our results in a more general setting so as to be applicable to other contexts, for example when the group is in fact a smooth group scheme. We also discuss 1 how A -localization behaves with respect to ring and module spectra and also with respect to mod-l-completion, where l is a fixed prime. In forthcoming papers, we apply the theory developed here to produce a theory of Spanier-Whitehead duality and Becker-Gottlieb transfer in this framework, and explore various applications of the transfer.

Table of contents

1. Introduction

2. The basic framework of equivariant motivic homotopy theory: the unstable theory

3. The basic framework of equivariant motivic homotopy theory: the stable theory

4. Effect of A1-localization on ring and module structures and on mod-l completions 5. References.

The second author was supported by the IHES, the MPI (Bonn) and a grant from the NSA. . 1 2 Gunnar Carlsson and Roy Joshua

1. Introduction

The purpose of this paper is to set up a suitable framework for a series of papers exploring the notions of Spanier-Whitehead duality and Becker-Gottlieb transfer in the setting of stable motivic homotopy theory and their applications. Some of the notable applications we have in mind are the conjectures due to the first author on equivariant algebraic K-theory, equivariant with respect to the action of a Galois group, see [Carl1]. Due to the rather versatile nature of the classical Becker-Gottlieb transfer, other potential applications also exist which will be explored in the sequel. It has become clear that a framework suitable for such applications would be that of equivariant motivic stable homotopy, equivariant with respect to the action of a profinite group, which typically will be the Galois group of a field extension.

The study of algebraic cycles and motives using techniques from algebraic topology has been now around for about 15 years, originating with the work of Voevodsky on the Milnor conjecture and the paper of Morel and Voevodsky on A1-homotopy theory: see [Voev] and [MV]. The paper [MV] only dealt with the unstable theory, and the stable theory was subsequently developed by several authors in somewhat different contexts, for example, [Hov-3] and [Dund2]. An equivariant version of the unstable theory already appears in [MV] and also in [Guill], but both are for the action of a finite group. Equivariant stable homotopy theory, even in the classical setting, needs a fair amount of technical machinery. As a result, and possibly because concrete applications of the equivariant stable homotopy framework in the motivic setting have been lacking till now, equivariant stable motivic homotopy theory has not been worked out or even considered in any detail up until now. The present paper hopes to change all this, by working out equivariant stable motivic homotopy theory in detail.

One of the problems in handling the equivariant situation in stable homotopy is that the spectra are no longer indexed by the non-negative integers, but by representations of the group. One way of circumventing this problem that is commonly adopted is to use only multiples of the regular representation. This works when the representations are in characteristic 0, because then the regular representation breaks up as a sum of all irreducible representations. However, when considering the motivic situation, unless one wants to restrict to schemes in characteristic 0, the above decomposition of the regular representation is no longer true in general. As a result there seem to be serious technical issues in adapting the setting of symmetric spectra to study equivariant stable motivic homotopy theory. On the other hand the technique of enriched functors as worked out in [Dund1] and [Dund2] only requires as indexing objects, certain finitely presented objects in the unstable category. As result this approach requires the least amount of extra work to cover the equivariant setting, and we have chosen to adopt this as the appropriate framework for our work.

The following is an outline of the paper. Section 2 is devoted entirely to the unstable theory. One of the main results in this section is the following.

Throughout the paper G will denote one of the following: (i) either a discrete group, (ii) a profinite group or (iii) a smooth group scheme, viewed as a presheaf of groups. The profinite group will be a Galois group defined as ¯ follows. We will fix a field k0 and let Gal denote the Galois group of an algebraic closure k over k0.

Then PSh(Sm/k0, G) will denote the category of simplicial presheaves on the category Sm/k0 of smooth schemes of finite type over k0 provided with an action by G. Next we will fix a family, W, of subgroups of G, so that if H ε W and H0 ⊇ H is a subgroup of G, then H0 ε W. In the case G is a finite group, W will denote all subgroups of G and when G is profinite, it will denote all subgroups of finite index in G. When G is a smooth group-scheme, we will leave W unspecified, for now.

PShc(Sm/k0, G) will denote the full subcategory of such simplicial sheaves where the action of G is continuous with respect to W, i.e. if s ε Γ(U, P ) and P ε PSh(Sm/k0, G), then the stabilizer of s belongs to W. Then we define the following structure of a cofibrantly generated simplicial model category on PSh(Sm/k0, G) and on PShc(Sm/k0, G) starting with the projective model structure on PSh(Sm/k0). (The projective model structure is where the fibrations and weak-equivalences are defined object-wise and the cofibrations are defined using the left-lifting property with respect to trivial fibrations.)

(i) The generating cofibrations are of the form

IG = {G/H × i | i a cofibration of PSh(Sm/k0), H ε W}, (ii) the generating trivial cofibrations are of the form Equivariant motivic homotopy theory 3

JG = {G/H × j | j a trivial cofibration of PSh(Sm/k0), H ⊆ G, H ε W} and

0 H 0H H (iii) where weak-equivalences (fibrations) are maps f : P → P in PSh(Sm/k0, G) so that f : P → P is a weak-equivalence (fibration, respectively) in PSh(Sm/k0) for all H ε W.

Proposition 1.1. The above structure defines a cofibrantly generated simplicial model structure on PSh(Sm/k0, G) and on PShc(Sm/k0, G) that is proper. In addition, the smash product of pointed simplicial presheaves makes these symmetric monoidal model categories.

When G denotes a finite group or profinite group, the category PShc(Sm/k0, G) is locally presentable, i.e. there exists a set of objects, so that every simplicial presheaf P ε PShc(Sm/k0, G) is filtered colimit of these objects. In particular, it also a combinatorial model category.

One also considers the following alternate framework for equivariant unstable motivic homotopy theory. First one considers the orbit category OG = {G/H | H ε W}. A morphism G/H → G/K corresponds to γ ε G, so that o −1 OG o γ.Hγ ⊆ K. One may next consider the category PSh of OG -diagrams with values in PSh(Sm/k0). The two Oo categories PShc(Sm/k0, G) and PSh G are related by the functors:

Oo H Φ : PShc(Sm/k0, G) → PSh G ,F 7→ Φ(F ) = {Φ(F )(G/H) = F } and

Oo Θ : PSh G → PShc(Sm/k0, G),M 7→ Θ(M) = lim M(G/H). → {H|H ε W} A key result then is the following. Proposition 1.2. When G denotes a discrete group or a profinite group, the functors Φ and Θ are Quillen- equivalences.

Section 3 is entirely devoted to the stable theory and the following is one of the main results in this section. We Oo let C denote one of the categories PShc(Sm/k0), PShc(Sm/k0, G), or PSh G provided with the above structure of cofibrantly generated model categories. The above categories are symmetric monoidal with the smash-product, ∧, of simplicial presheaves as the monoidal product. The unit for the monoidal product will be denoted S0.

Since not every object in Sm/k0 has an action by G, it is not possible to consider localization of PShc(Sm/k0, G, ?) or PSh(Sm/k0, G, ?) by inverting maps associated to an elementary distinguished square in the Nisnevich topology or maps of the form U• → U where U ε Sm/k0 and U• → U is a hypercovering in the given topology. Therefore, o we instead localize PSh by inverting such maps and consider the category of diagrams of type OG with values in o o OG OG o this category. PShmot ( PShdes ) will denote the category of all diagrams of type OG with values in the localized category PShmot (PShdes, respectively). Let C0 denote a C-enriched full-subcategory of C consisting of objects closed under the monoidal product ∧, all 0 0 0 of which are assumed to be cofibrant and containing the unit S . Let C0 denote a C-enriched sub-category of C , which may or may not be full, but closed under the monoidal product ∧ and containing the unit S0. Then the 0 basic model of equivariant motivic stable homotopy category will be the category [C0, C]. This is the category whose 0 objects are C-enriched covariant functors from C0 to C: see [Dund1, 2.2]. There are several possible choices for the 0 category C0, which are discussed in detail in Examples 3.3.

0 0 We let Sph(C0) denote the C-category defined by taking the objects to be the same as the objects of C0 and where Hom 0 (T ,T ) = T if T = T ∧ T and ∗ otherwise. Since T is a sub-object of Hom (T ,T ), Sph(C0) U V W V W U W C U V 0 0 0 it follows that Sph(C0) is a sub-category of C0. Now an enriched functor in [Sph(C0), C] is simply given by a 0 collection {X(TV )|TV ε Sph(C0)} provided with a compatible collection of maps TW ∧ X(TV ) → X(TW ∧ TV ). We 0 let Spectra(C) = [Sph(C0), C]. We provide two model structures on Spectra(C): (i) the level model structure and (ii) the stable model structure. A map f : X 0 → X in Spectra(C) is a level equivalence (level fibration, level trivial fibration, level cofibration, level trivial cofibration) if each EvTV (f) is a weak-equivalence (fibration, trivial fibration, cofibration, trivial cofibration, respectively) in C. Such a map f is a projective cofibration if it has the left-lifting property with respect to every level trivial fibration. Then we obtain: 4 Gunnar Carlsson and Roy Joshua

Proposition 1.3. The projective cofibrations, the level fibrations and level equivalences define a cofibrantly gener- ated model category structure on Spectra(C) with the generating cofibrations (generating trivial cofibrations) being ISp (JSp, respectively). This model structure (called the projective model structure on Spectra(C)) is left-proper (right proper) if the corresponding model structure on C is left proper (right proper, respectively). It is cellular if the corresponding model structure on C is cellular.

See Corollary 3.8 for further details. Next we localize the above model structure suitably to define the stable model structure. A spectrum X ε Spectra(C) is an Ω-spectrum if if it is level-fibrant and each of the natural maps 0 X(TV ) → HomC(TW ,X(TV ∧ TW )), TV ,TW ε C0 is a weak-equivalence in C. Proposition 1.4. (See Proposition 3.10 for more details.) (i) The corresponding stable model structure on Spectra(C) is cofibrantly generated, left proper and cellular when C is assumed to have these properties.

(ii) The fibrant objects in the stable model structure on Spectra(C) are the Ω-spectra defined above.

The above results then enable us to define the various equivariant stable motivic homotopy categories that we consider: these are discussed in Definition 3.11.

In the last section we study how A1-localization behaves with respect to ring and module spectra and also with respect to Z/l-completion, for a fixed prime l. We make use of ring and module spectra in an essential way in this paper and the sequels: therefore it is important to show that A1 -localization preserves ring and module structures on spectra. Since comparisons are often made with the ´etalehomotopy types, completions at a prime also play a major role in our work. Unfortunately the Bousfield-Kan completion as such may not commute with A1-localization. This makes it necessary for us to redefine a Bousfield-Kan completion at a prime l that behaves well in the A1-local setting: this is a combination of the usual Bousfield-Kan completion and A1-localization. Since the resulting completion is in general different from the traditional Bousfield-Kan completion, we denote our completion functor by Z/lg∞. The main properties of the resulting completion functor are discussed in 4.0.7.

2. The basic framework of equivariant motivic homotopy theory: the unstable theory

In this section we define a framework of equivariant motivic stable homotopy theory, where the group G will be either finite, profinite or a smooth group scheme The main motivation for this formulation is so as to provide a framework for work on the conjectures in [Carl1], where the group involved will be the profinite Galois group associated to a field extension. One main point to observe in this context is that there are two different kinds of equivariant cohomology and homology theories: one in the sense of Borel and one in the sense of Bredon with the latter being a finer variant than the former. Classical equivariant homotopy theory is set in the framework of Bredon-style theories and this is also the framework for work on the above mentioned conjectures. These observations should show the necessity for developing the following framework.

We let k0 denote a fixed field of arbitrary characteristic, with k denoting a fixed algebraic closure of k0. While we allow the possibility that k0 be algebraically or separably closed, for the most part this will not be case. Sm/k0 will denote the category of separated smooth schemes of finite type over k0. If X denotes a separated smooth scheme of finite type over k0, (Sm/X) will denote the corresponding subcategory of objects defined over X. (Sm/k0)? will denote the category Sm/k0 provided with a Grothendieck topology with ? denoting one of the big Zariski, Nisnevich or ´etaletopologies. If X denotes a separated smooth scheme of finite type over k0, (Sm/X)? will denote the corresponding big site over X. If p : X → Spec k0 denotes the structure map of X, one has the obvious induced functors:

−1 −1 (2.0.1) p : (Sm/k0) → (Sm/X), p (Z) = Z × X, and Spec k

p! : (Sm/X) → (Sm/k0), p!(Y ) = Y

We will let S denote any one of the above categories or sites: since the schemes we consider are of finite type over k0, it follows that these categories are all skeletally small. PSh(S) (SSh(S)) will denote the category of all simplicial presheaves (sheaves, respectively))on S : observe that to any scheme X one may associate a (simplicial) presheaf represented by X. Equivariant motivic homotopy theory 5

2.1. Model structures and localization. Though the main objects considered in this paper are simplicial presheaves, we will often need to provide them with different model structures model structures (in the sense of [Qu]). The same holds for the stable case, where different variants of spectra will be considered. In this paper we will work with several different kinds of objects: in addition to simplicial presheaves and simplicial sheaves, we will need to consider pro-objects of simplicial presheaves and sheaves. It will also become necessary to consider what may be called spectral objects (spectra, for short) in these categories. Therefore, the basic framework will be that of model categories, often simplicial (symmetric) monoidal model categories, i.e. simplicial model categories that have the structure of monoidal model categories as in [Hov-2]. We will further assume that these categories have other nice properties, for example, are closed under all small limits and colimits, are proper and cellular. (See [Hov-1] and [Hirsh] for basic material on model categories.)

2.1.1. The category of simplicial presheaves. We fix a scheme S ε Sm/k0 and consider the category PSh(Sm/S) of pointed simplicial presheaves on Sm/S. We will need to consider different Grothendieck topologies on Sm/S: this is one reason that we prefer to work for the most part with pointed simplicial presheaves rather than with pointed simplicial sheaves. We will consider pointed simplicial sheaves only when it becomes absolutely essential to do so. If ? denotes any one of the Grothendieck topologies,et ´ , Nis or Zar, and X ε Sm/S, Sh(Sm/X?) will denote the category of pointed simplicial sheaves on the corresponding big site.

There are at least three commonly used model structures on PSh(Sm/S): (i) the object-wise model structure, (ii) the injective model structure and (iii) the projective model structure. We will mostly consider the last structure in the paper but will provide a few remarks on the other two here for comparison purposes.

By viewing any simplicial presheaf as a diagram of type Sm/S with values in simplicial sets, we obtain the model-structure where weak-equivalences and cofibrations are defined object-wise and fibrations are defined using the right-lifting property with respect to trivial cofibrations. Observe as a consequence, that all objects are cofibrant and cofibrations are simply monomorphisms. This is the object-wise model structure.

All the above sites have enough points: for the ´etalesite these are the geometric points of all the schemes considered, and for the Nisnevich and Zariski sites these are the usual points (i.e. spectra of residue fields ) of all the schemes considered. Therefore, it is often convenient to provide the simplicial topoi above with the injective model structure where the cofibrations are defined as before, but the weak-equivalences are maps that induce weak-equivalences on the stalks and fibrations are defined by the lifting property with respect to trivial cofibrations. By providing the appropriate topology ? on the category Sm/S, one may define a similar injective model structure on PSh(Sm/S). Since this depends on the topology ?, we will denote this model category structure by PSh(Sm/S)?. The projective model structure on PSh(Sm/S) is defined by taking fibrations (weak-equivalences) to be maps that are fibrations (weak-equivalences, respectively) object-wise, i.e of pointed simplicial sets on taking sections over each object in the category Sm/S. The cofibrations are defined by the left-lifting property with respect to trivial fibrations. This model structure is cofibrantly generated, with the generating cofibrations I (generating trivial cofibrations J) defined as follows.

(2.1.2) J = {hX ∧ Λn,+ → hX ∧ ∆[n]+|n ≥ 0} and

(2.1.3) I = {hX ∧ δ∆[n]+ → hX ∧ ∆[n]+|n ≥ 0}

Here X denotes an object of the category Sm/S, hX denotes the corresponding pointed sheaf of sets represented by X+ and ∧ is provided by the simplicial structure on this category. Observe that every map that is a cofibration in the projective model structure is a cofibration in the other two model structures. Weak-equivalences in the object-wise model structure and the projective model structure are the same and they are both weak-equivalences in the injective model structure. It follows that any fibration in the object-wise model structure is a fibration in the projective model structure and that a fibration in the injective model structure is a fibration in the other two model structures.. Next we summarize a couple of useful observations in the following proposition.

Proposition 2.1. (i) If P ε PSh(Sm/X) is fibrant in the object-wise model structure or projective model structure, Γ(U, GP ) = holim Γ(U, G•P ) is a fibrant pointed simplicial set, where G•P is the cosimplicial object defined by the ∆ Godement resolution and U is any object in the site.

(ii) The functor a : PSh(Sm/k0) → Sh(Sm/k0?) that sends a presheaf to its associated sheaf sends fibrations to maps that are fibrations stalk-wise and weak-equivalences to maps that are weak-equivalences stalk-wise, when 6 Gunnar Carlsson and Roy Joshua

PSh(Sm/k0) is provided with the projective model structure. The same functor sends cofibrations to injective maps when PSh(Sm/k0) is provided with the injective model structure. Proof. The proof of (i) amounts to the observation that if F is a pointed simplicial presheaf that is a fibration object- wise, then the stalks of F are fibrant pointed simplicial sets. It follows that, therefore, Γ(U, GnF ) are all fibrant pointed simplicial sets. (See [B-K, Chapter XI, 5.5 Fibration lemma].) The first statement in (ii) follows from the observation that fibrations and weak-equivalences are defined object-wise and these are preserved by the filtered colimits involved in taking stalks and that the stalks of a presheaf and associated sheaf are isomorphic. The second statement in (ii) follows from the observation that a map of presheaves is a cofibration (i.e. a monomorphism) if and only if the induced maps on all the stalks are cofibrations (i.e. monomorphisms). 

Let X, Y denote two smooth schemes over Spec k0 and let f : X → Y denote a map. Then f induces functors

∗ f∗ : PSh(Sm/X)) → PSh(Sm/Y)) and f : PSh(Sm/Y)) → PSh(Sm/X)) with the former right-adjoint to the latter. (One also obtains similar functors at the level of sheaves.) In case ∗ Y = Spec k0 with f = p the structure map of X, then f is simply the restriction of a presheaf (sheaf) to Sm/X. Therefore if F ε PSh(Sm/Spec( k0))), we will denote this restriction of F to Sm/X by F|X . In general, neither of these functors preserve any of the model category structures so that one needs to consider the appropriate derived functors associated to these functors. The left-derived functor of f ∗, Lf ∗, may be defined using representables as in [MV, Chapter 2] and the right derived functor of f∗, Rf∗, may often be defined using the Godement resolution in view of the above result. In case f itself is a smooth morphism, one can readily show that f ∗ preserves cofibrations and weak-equivalences in all three of the above model structures. In the case f is again assumed to be a smooth morphism, Rf∗ preserves fibrations and weak-equivalences in the projective model structures. (To see this one first observes that filtered colimits of fibrations of simplicial sets are fibrations, so that each of the maps Gnf will be a fibration object-wise. Now homotopy inverse limits preserve fibrations.)

One may observe that the above model categories are all simplicial model categories in the sense of [Qu]. If A, B are objects in the above categories, Map(A, B) will denote the corresponding simplicial mapping space: W Map(A, B)n = Hom(A ∧ ∆[n]+,B) and one may observe that (A ∧ ∆[n]+)s = As. α ε ∆[n]s One defines a symmetric monoidal structure on any of the above categories by sending (P,Q) to P ∧ Q. One may verify readily that, with this monoidal structure, the above categories of pointed simplicial presheaves and sheaves are monoidal simplicial model categories which are proper and cellular. The projective model category structure is also weakly finitely generated in the sense of [Dund1, Definition 3.4].

2.1.4. Convention. By default we will assume the projective model structure with respect to one of the given Grothendieck topologies. Objects of P Sh(Sm/k0)? will be referred to as spaces.

1 2.2. A -localization and Simplicial sheaves up to homotopy. Model structures on the category P Sh(Sm/k0)? are discussed in detail in the G-equivariant context below: the discussion there applies to the present setting by taking the group-action to be trivial.

One may apply A1-localization to each of the above model structures as in [MV]. The resulting model categories will be denoted with the subscript A1. In view the basic results on localization as in [Hirsh, Chapter 4], they inherit the basic properties of the above model categories. The fibrant objects F in these categories are characterized by the property that they are fibrant in the underlying model structure and that Map(P × A1,F ) ' Map(P,F ), for 1 1 1 all P ε PSh(S). These will be called A -local. P ShA (Sm/k0)? will denote the A -localization of P Sh(Sm/k0)?. 1 1 Objects of P ShA (Sm/k0)? will be referred to as A -spaces. Next, there are two means of passing to sheaves or rather sheaves up to homotopy. When the topology is the Nisnevich topology, one defines a presheaf P ε PSh(Sm/k0)Nis to be motivically fibrant if (i) P is fibrant in PSh(Sm/k0)Nis, (ii) Γ(φ, P ) is contractible (where φ denotes the empty scheme), (iii) sends an elementary distinguished squares as in [MV] to a homotopy cartesian square and (iv) the obvious pull-back Γ(U, P ) → Γ(U × 1 A ,P ) is a weak-equivalence. Then a map f : A → B in P Sh(Sm/k0)Nis is a motivic weak-equivalence if the induced map Map(f, P ) is a weak-equivalence for every motivically fibrant object P , with Map denoting the simplicial mapping space. One then localizes such weak-equivalences. As shown in [Dund2, Section 2], this localization produces a model structure which is also weakly finitely generated and also which is symmetric monoidal model. The resulting model structure will be denoted PShmot(Sm/k0). Equivariant motivic homotopy theory 7

An alternate approach that applies in general is to localize by inverting hypercovers as in [DHI]. Following [DHI], a simplicial presheaf has the descent property for all hypercovers if for U in (Sm/k0)?, and all hypercoverings U• → U, the induced map P (U) → holim{Γ(Un,P )|n} is a weak-equivalence. By localizing with respect to maps ∆ 1 of the form U• → U where U• is a hypercovering of U and also maps of the form U × A → U, it is proven in [Dug, Theorem 8.1] and [DHI, Example A. 10] that we obtain a model category which is Quillen equivalent to the Voevodsky-Morel model category of simplicial sheaves on (Sm/k0)? as in [MV]. Though the resulting localized category is cellular and left-proper (see [Hirsh, ]) it is unlikely to be weakly finitely generated: the main issue is that the hypercoverings, being simplicial objects, need not be small. Nevertheless this seems to be the only alternative available in the ´etalesetting. This localized category of simplicial presheaves will be denoted by PShdes(S). Often the subscript des will be omitted if there is no chance for confusion.

Remark 2.2. A1-localization is better discussed in detail in the stable setting, which we consider below. Moreover, since the etale homotopy type of affine spaces is trivial after completion away from the characteristic of the base field, A1-localization in the ´etalesetting is often simpler than the corresponding motivic version as shown below.

2.3. Equivariant presheaves and sheaves. As we pointed out earlier we allow the group G to be one of the following:

(i) a finite group

(ii) a profinite group or

(iii) a smooth group scheme viewed as a presheaf of groups on Sm/k0.

In all these cases, one defines a G-equivariant presheaf of pointed sets on Sm/k0 to be a presheaf of sets P on Sm/k0 taking values in the category of pointed sets provided with an action by G. One defines G-equivariant presheaves of simplicial sets, pointed simplicial sets and G-equivariant sheaves of simplicial sets and pointed simplicial sets similarly. Henceforth P Sh(Sm/k0, G)(Sh(Sm/k0, G, ?)) will denote the category of G-equivariant pointed simplicial presheaves (sheaves on the ?-topology, respectively). Clearly this category is closed under all small limits and colimits.

Definition 2.3. The category (Sm/k0, Gal) will denote the subcategory of Sm/k0 consisting of schemes of finite 0 0 type over k , where k is some finite Galois extension of k0 and provided with an action of Gal through the quotient 0 Galk0 (k ). The morphisms between such schemes will be morphisms over k0 compatible with the action of Gal.

The first case, when G is a finite group, is already discussed in some detail in the literature, at least in the unstable case: see [MV] and [Guill]. Therefore, presently we will only consider the last two cases.

One obtains the Galois-equivariant framework as follows. Let k¯ denote a fixed algebraically (or separably) closed ¯ field containing the given field k0 with Gal = Gal(k/k0). We will view the Galois group Gal as an inverse system of constant presheaves (of finite groups) on Sm/k0. The main examples of Galois-equivariant pointed simplicial sheaves will be the representable ones and filtered 0 colimits of such representable ones. Here a representable sheaf is one of the form hX0 i.e. represented by X+ where 0 0 0 X = X0 × Spec k . Here X0 is a smooth scheme of finite type over k0 and k which is a finite Galois extension Spec k0 of the base field k0 and contained in the fixed algebraic closure k.

Remark 2.4. Our discussion through 2.8 will only consider the category P Sh(Sm/k0, G) explicitly though every- 1 1 thing that is said here applies equally well to the A -localized category P ShA (Sm/k0, G) as well. 2.3.1. Pointed simplicial sets with action by G. A related context that comes up, especially in the con- text of ´etalerealization is the following: let (simpl.sets)∗ denote the category of pointed simplicial sets and let (simpl.sets, G)∗ denote the category of all pointed simplicial sets with actions by the group G with the morphisms in this case being equivariant maps. By considering the constant presheaves associated to a pointed simplicial set, we may imbed this category into P Sh(Sm/k0, G).

2.3.2. Simplicial presheaves with continuous action by the group G. We let W denote a family of sub- groups of G so that if H ε W and H0 is a subgroup of G containing H, the H0 ε W. As pointed out earlier, when 8 Gunnar Carlsson and Roy Joshua the group G is finite, W will denote the family of all subgroups and when G is profinite, W will denote the family of all subgroups of finite index in G. When G is a smooth group scheme, we will leave the family W unspecified for now.

For the most part, we will only consider G-equivariant presheaves of pointed simplicial sets P on which the action of G is continuous, i.e. for each object U ε S and each section s ε Γ(U, F )n, the stabilizer Z(s) of s belongs to W. This category will be denoted PShc(Sm/k0, G). Clearly the above terminology is taken from the case where the group G is profinite. Observe that, in this case T −1 the intersection of the conjugates g g Z(s)g of this stabilizer as g varies over a set of coset representatives of Z(s) in G is a normal subgroup of G, contained in Z(s) and of finite index in G. Therefore G acts continuously on P if and only if Γ(U, P ) = lim Γ(U, P )H . → {H||G/H|<∞,HCG} For each subgroup H ε W, let P H denote the sub-presheaf of P of sections fixed by H, i.e. Γ(U, P H ) = Γ(U, P )H . If H¯ = G/H, then Γ(U, P )H = {s ε Γ(U, P )| the action of G on s factors through H¯ }.

Lemma 2.5. (i) Let φ : P 0 → P denote a map of simplicial presheaves. Then φ induces a map φH : P 0H → P H for each subgroup H ε W. The association φ 7→ φH is functorial in φ in the sense that if ψ : P 00 → P 0 is another map, then the composition (φ ◦ ψ)H = φH ◦ ψH .

(ii) The full subcategory PShc(Sm/k0, G) of simplicial presheaves with continuous action by G is closed under all small colimits, with the small colimits the same as those computed in PSh(Sm/k0, G). For the remaining statements, we will assume the group G is profinite.

(iii) The full subcategory PShc(Sm/k0, G) is also closed under all small limits, where the limit of a small diagram {Pα|α} is

H (2.3.3) lim lim Pα → ← {H||G/H|<∞} α

When the inverse limit above is finite, it commutes with the filtered colimit over H, so that in this case, the inverse limit agrees with the inverse limit computed in PSh(Sm/k0, G).

(iv) Let {Qα|α} denote a diagram of simplicial sub-presheaves of a simplicial presheaf Q ε PSh(Sm/k0, G) indexed K K by a small filtered category. If K is any subgroup of G, then (lim Qα) = lim Qα . → → α α

(v) Let {Pα|α} denote a diagram of objects in PShc(Sm/k0, G) indexed by a small filtered category. Let K denote K K a subgroup of G with finite index. Then (lim Pα) = lim (Pα) . → → α α

Proof. (i) and (ii) are clear. In order to prove (iii), we show that giving a compatible collection of maps φα : P → Pα from a simplicial presheaf P with a continuous action by G to the given inverse system {Pα|α} corresponds to H giving a map φ : P → lim lim Pα . Observe that the maps φα induce a compatible collection of maps → ← α {H||G/H|<∞,HCG} H H H {φα : P → Pα } for each fixed H. Therefore, one may now take the limit of these maps over {α} to obtain a H H H compatible collection of maps {φ : P → lim Pα |H}. One may next take the colimit over H of this collection to ← α H H obtain a map φ : P = lim P → lim lim Pα . → → ← α {H||G/H|<∞,HCG} {H||G/H|<∞,HCG}

H Clearly the latter maps naturally to lim lim Pα . The latter then projects to ← → α {H||G/H|<∞,HCG}

H Pα = lim Pα . Moreover, the composition of the above maps P → Pα identifies with the given → {H||G/H|<∞,HCG} H map φα since the corresponding maps identify after applying ( ) . Equivariant motivic homotopy theory 9

H The action of G on lim lim Pα is continuous. The above arguments show that the latter is in fact → ← α {H||G/H|<∞,HCG} the inverse limit of {Pα|α} in the category PShc(Sm/k0, G). This completes the proof of (iii). The last statement in (iii) is clear since filtered colimits commute with finite limits.

0 0 (iv) follows readily in view of the observation that for a simplicial sub-presheaf Q of Q, with Q ε PSh(Sm/k0, G), (Q0)K = Q0 ∩ QK .

H Next we consider (v). Since each Pα has a continuous action by G, we first observe that Pα = lim Pα → {H||G/H|<∞,HCG} H H H for each α. Therefore, limPα = lim lim Pα = lim limPα . Let QH = limPα ; then we → → → → → → α α α α {H||G/H|<∞,HCG} {H||G/H|<∞,HCG} see that each QH is a sub-simplicial presheaf of lim QH = limPα. Then the collection {QH |H, |G/H| < → → α {H||G/H|<∞,HCG} ∞,H C G} satisfies the hypotheses in (iv) so that we obtain: H K H K ( lim limPα ) = lim (limPα ) . → → → → α α {H||G/H|<∞,HCG} {H||G/H|<∞,HCG} Next observe that, for any simplicial presheaf Q and any fixed normal subgroup H of finite index in G, the action of G on QH is through the finite group G/H. Therefore, (QH )K = (QH )K¯ , where K is the image of K in G/H. Therefore we obtain the isomorphism

H K ∼ H K¯ ∼ H K¯ ∼ H K (lim Pα ) = (lim Pα ) = lim (Pα ) = lim (Pα ) . → → → → α α α α Making use of this identification and commuting the two colimits, we therefore obtain the identification

H K H K K lim (lim Pα ) = lim lim (Pα ) = lim (Pα) → → → → → α α α {H||G/H|<∞,HCG} {H||G/H|<∞,HCG}

H The last identification follows from (iv) and the observation that for each fixed α, each Pα ⊆ Pα as simplicial presheaves. 

2.3.4. Finitely presented objects. Recall an object C in a category C is finitely presented if HomC(C, ) commutes with all small filtered colimits in the second argument.

Lemma 2.6. Let G denote a profinite group. Let P ε PSh(Sm/k0, G) be such that in PSh(Sm/k0) it is finitely presented and P = P H for some normal subgroup H of finite index in G. Then P is a finitely presented object in PShc(Sm/k0, G).

Proof. Let Hom denote the external Hom in the category PSh(Sm/k0) and let HomG denote the external Hom G in the category PSh(Sm/k0, G). Then HomG(P,Q) = Hom(P,Q) , where G acts on the set Hom(P,Q) through its actions on P and Q. Next suppose P = P H for some normal subgroup H of G with finite index. Then ∼ H H G/H HomG(P,Q) = HomG/H (P,Q ) = Hom(P,Q ) .

Next let {Qα|α} denote a small collection of objects in PShc(Sm/k0, G) indexed by a small filtered category. Then,

H G/H H G/H limHomG(P,Qα) = limHom(P,Qα ) = (limHom(P,Qα )) → → → α α α where the last equality follows from the fact that taking invariants with respect to the finite group G/H is a finite inverse limit which commutes with the filtered colimit over α. Next, since P is finitely presented as an object in H G/H PSh(Sm/k0), the last term identifies with (Hom(P, limQα )) . By (v) of the last lemma, this then identifies → α H G/H with (Hom(P, (limQα) )) . This clearly identifies with HomG(P, limQα). → →  α α

Next we define the following structure of a cofibrantly generated simplicial model category on PSh(Sm/k0, G) and on PShc(Sm/k0, G) starting with the projective model structure on PSh(Sm/k0)?. 10 Gunnar Carlsson and Roy Joshua

(i) The generating cofibrations are of the form

IG = {G/H × i | i a cofibration of PSh(Sm/k0), H ε W}, (ii) the generating trivial cofibrations are of the form

JG = {G/H × j | j a trivial cofibration of PSh(Sm/k0)?, H ⊆ G, H ε W} and

0 H 0H H (iii) where weak-equivalences (fibrations) are maps f : P → P in PSh(Sm/k0, G) so that f : P → P is a weak-equivalence (fibration, respectively) in PSh(Sm/k0)? for all H ε W.

Proposition 2.7. The above structure defines a cofibrantly generated simplicial model structure on PSh(Sm/k0, G) and on PShc(Sm/k0, G) that is proper. In addition, the smash product of pointed simplicial presheaves makes these symmetric monoidal model categories.

If G denotes a profinite group, the category PShc(Sm/k0, G) is locally presentable, i.e. there exists a set of objects, so that every simplicial presheaf P ε PShc(Sm/k0, G) is filtered colimit of these objects. In particular, it also a combinatorial model category. Proof. The key observation is the following: Let H ⊆ G denote a subgroup with finite index. Then the functor H P 7→ P , PSh(Sm/k0, G) → PSh(Sm/k0) has as left-adjoint, the functor Q 7→ G/H × Q. It follows from this observation that the domains of I (J) are small with respect to the subcategory I (J, respectively) so that the small object argument (see [Hirsh, ]) is possible. It follows readily then that the categories PSh(Sm/k0, G) and PShc(Sm/k0, G) are cofibrantly generated model categories. We skip the verification that they are proper, that they are simplicial model categories and that the smash-product of two pointed simplicial presheaves makes these monoidal model categories. (See, for example, [Fausk].) In view the assumption of continuity it suffices to show that the category of simplicial presheaves on Sm/k0 with the action of finite group is locally presentable, which follows readily from the corresponding assertion when the group is trivial. 

The following is an alternative approach to providing a model structure on the category of pointed simplicial presheaves with continuous action by the G. For this one considers first the orbit category OG = {G/H | H ε W}. A morphism G/H → G/K corresponds to γ ε G, so that γ.Hγ−1 ⊆ K. One may next consider the category o OG o PSh of OG -diagrams with values in PSh(Sm/k0). This category, being a category of diagrams with values in PSh(Sm/k0) readily inherits the structure of a cofibrantly generated model category. Moreover it is left-proper (right-proper, cellular, simplicial) since the model category PSh is.

Recall that the generating cofibrations of this diagram category are defined as follows. IOGo = {(G/H) × i | i = a cofibration in Psh, | H ε W}. The corresponding generating trivial cofibrations JOGo = {(G/H) × j | j = o a trivial cofibration in Psh, | H ε W}. The diagram (G/H) × i is the OG-diagram defined by ((G/H) × i)(G/K) = K Hom o (G/H, G/K) × i = Hom (G/K, G/H) × i = (G/H) × i. OG G

Oo Next we restrict to the case where G is profinite. Then the two categories PShc(Sm/k0, G) and PSh G are related by the functors:

Oo H Φ : PShc(Sm/k0, G) → PSh G ,F 7→ Φ(F ) = {Φ(F )(G/H) = F } and

Oo Θ : PSh G → PShc(Sm/k0, G),M 7→ Θ(M) = lim M(G/H). → {H|H ε W}

For any such H, there is a largest subgroup HG ⊆ H so that HG is normal in G and of finite index in G: this is often called the core of H. The obvious quotient map G/HG → G/H induces a map M(G/H) → M(G/HG) so that {M(G/K)|K normal in G and of finite index } is cofinal in the direct system used in the above colimit. Now G/K acts on G/K by translation and this induces an action by G/K on M(G/K). Therefore, the above colimit has a natural action by G which is clearly continuous. The natural transformation Θ ◦ Φ → id may be shown to be an isomorphism readily. Moreover Θ is left-adjoint to Φ. It follows, therefore, that the functor Φ is full and Oo faithful, so that Φ is an imbedding of the category PShc(Sm/k0, G) in PSh G . Proposition 2.8. Assume the above situation. Then the following hold

Oo (i) If P ε PSh G is cofibrant, the natural map η : P → ΦΘ(P ) is an isomorphism. Equivariant motivic homotopy theory 11

(ii) The two functors Φ and Θ are Quillen-equivalences.

H (iii) Let P ε PSh(Sm/k0, G) be such that in PSh(Sm/k0) it is finitely presented and P = P for some normal Oo subgroup H of finite index in G. Then Φ(P ) is a finitely presented object in PSh G . Proof. A key observation is that the the functor Θ sends the generating cofibrations, i.e. diagrams of the form (G/H) × i to G/H × i ε IG and similarly sends the generating trivial cofibrations, i.e. diagrams of the form (G/H) × j to G/H × j ε JG. Φ, on the other hand, sends the generating cofibrations in IG (generating trivial cofibrations in JG) to the generating cofibrations in IOGo (JOGo , respectively). Since any cofibrant object in the Oo model category PSh G is a retract of an I o -cell, and both Θ and Φ preserve retractions, it suffices to prove (i) OG when P is an I o -cell. Since Θ obviously preserves colimits and Φ also does the same as proven in Lemma 2.5, OG it suffices to consider the case when P = (G/H) × i, which is already observed to be true. Therefore, the first statement follows.

Oo Observe that it suffices to prove the following in order to establish the second statement. Let X ε PSh G be cofibrant and let Y ε PShc(Sm/k0, G) be fibrant. Then a morphism f : Θ(X) → Y in PShc(Sm/k0, G) is a weak- Oo equivalence if and only if the corresponding map g : X → Φ(Y ) in PSh G is a weak-equivalence. Now the induced η Φ(f) map g : X → Φ(Y ) induced by f by adjunction factors as the composition X→Φ(Θ(X)) → Φ(Y ). The first map is an isomorphism since X is cofibrant, so that the map g : X → Φ(Y ) is a weak-equivalence if and only if the map Φ(f) is a weak-equivalence. But the map Φ(f) is a weak-equivalence if and only if each of the maps f H : XH = (Θ(X))H → Y H is a weak-equivalence, which is equivalent to f being a weak-equivalence. This proves (ii). The proof of (iii) is similar to the proof of Lemma 2.6 and is therefore skipped. (One needs to first observe H0 0 0 that P = P for all subgroups H of G for which H ⊆ H.) 

Remarks 2.9. (i) One may observe that the functor Φ evidently commutes with all small limits while the functor Θ evidently commutes with all small colimits. Lemma 2.5 (v) shows that the functor Φ also commutes with all small colimits.

(ii) The above proposition proves that, instead of PShc(Sm/k0, G), it suffices to consider the diagram category Oo PSh G which is often easier to handle. Oo Proposition 2.10. The categories PShc(Sm/k0, G) (when the G is profinite) and PSh G as well as the corre- Oo sponding A1-localized model categories PSh G are weakly finitely generated in the sense of [Dund1, Definition 3.4], i.e. the following hold: (i) The domains and co-domains of the maps in the classes of the generating cofibrations and generating trivial cofibrations are finitely presented and (ii) there exists a subset J 0 of the generating trivial Oo cofibrations in either of the two model categories, PShc(Sm/k0, G) and PSh G so that a map f : A → B with fibrant co-domain is a fibration if and only if it has the right lifting property with respect to J 0.

Moreover, the above categories are locally presentable and therefore combinatorial model categories. Proof. The first statement needs the adjunction between the functor G/H × ( ) and the fixed points functor H () for the category PShc(Sm/k0, G) and between the free functor G/H × ( ) and the functor of evaluating Oo a diagram at G/H for the category PSh G . As shown in Lemma 2.5(v), the fixed point functor ( )H commutes Oo with filtered colimits in PShc(Sm/k0, G). Since colimits in PSh G are calculated point-wise, it is clear that taking colimits commutes with evaluating at G/H. The second statement follows from the corresponding property of the category PShc(Sm/k0) making use of the same adjunctions. The last statement reduces first to the case of finite group actions and then to the case where the group is trivial. 

2.3.5. Coverings and hypercoverings in the G-equivariant setting, where G is profinite. As shown in [DHI], one way to consider simplicial presheaves that are simplicial sheaves up to homotopy is to consider a localiza- tion of the category of simplicial presheaves by inverting hypercoverings. Therefore, G-equivariant hypercoverings are important which we proceed to consider in some detail next. Since we often need to consider rigid coverings and rigid hypercoverings, we will define these first.

Definition 2.11. (Rigid coverings and hypercoverings) Let X ε Sm/k0? denote a fixed scheme. Then a rigid covering of X in the topology ? is the following data: let X¯ denote a chosen conservative family of points of X appropriate to the site. (For example, if ? =et ´ , these form a conservative family of geometric points of X and if ? = Nis or ? = Zar, these form a conservative family of points of X in the usual sense.) A rigid covering of X then 12 Gunnar Carlsson and Roy Joshua is a disjoint union of pointed separated maps Ux, ux → X, x in the chosen site with each Ux connected and indexed by x ε X¯. If the topology ? =et ´ or Nis, observe that that the maps Ux → X are ´etaleand if ? =et ´ with ux a lift of the geometric point x (if ? = Nis, with ux a point of Ux lying above x so that induced map k(x) → k(ux) is an isomorphism). If ? = Zar, each Ux → X is an open immersion containing the chosen point x, so that ux = x in this case.

A rigid hypercovering of X is then a simplicial scheme U• together with a map U• → X so that U0 is a rigid X covering of X in the given topology ? and so that for each n ≥ 0, the induced map Un → (coskn−1(U•))n is a rigid X covering. (Here we use the convention that (cosk−1(U•))0 = X. ) One may extend this definition readily to define rigid hypercoverings of simplicial schemes. We will let HRR(X) denote the category of all rigid hypercoverings of X in the given topology ?. We skip the proof that this is a directed category, i.e. there is at most one map between two objects and is filtered.

Let X denote a smooth scheme of finite type over k0 and provided with an action by a finite quotient of the group G. We will denote this finite quotient of G by G. Proposition 2.12. (i) If φ : U → X is a covering in the ´etale(Nisnevich) site, then ψ : V = U × G = U × t (Spec k0) = t Ug → X defined by ψ|U = gφ is a G-equivariant cover in the ´etale(Nisnevich site, g ε G g ε G g respectively). (Here Ug denotes a copy U indexed by g.) If φ is a rigid covering so is ψ.

(ii) If φ• : U• → X is a hypercovering of X in the ´etale(Nisnevich) site, then ψ• : V• = U• × cosk0(G) → X is a G-equivariant hypercovering of X in the ´etale(Nisnevich site, respectively). Here the map U0 × (cosk0(G))0 = U0 × G → X is the obvious map sending U0 → X and G → Spec k0. If φ• is a rigid hypercovering, so is ψ•. (iii) Assume U → X a G-equivariant ´etalemap which is a covering in the ´etalesite (Nisnevich site) which factors through the structure map Vk → X of a G-equivariant hypercovering V•, for some fixed k ≥ 0. Then there exists a G-equivariant hypercovering W• → X together with a map W• → V• so that the map Wk → Vk factors through U → Vk. In case U → X is a rigid covering, W• can be chosen to be a rigid hypercovering.

Proof. (i) is obvious. For the following discussion we use the following convention: if W• → X is a simplicial scheme over X, then (cosk−1(W•) = X. Since Vn = Un × (cosk0(G))n for all n, the map Vn+1 → coskn(V•)n+1 is given by Un+1 × (cosk0(G))n+1 → (coskn(V•))n+1 × (coskn(cosk0(G)))n+1 = (coskn(V•))n+1 × (cosk0(G))n+1 which is the product of the given map Un+1 → (coskn(U•))n+1 (forming part of the structure of the hypercovering U•) and the identity map of (cosk0(G))n+1. Therefore this is a covering map in the appropriate topology, so that V• is a hypercovering. In case U• is a rigid hypercovering, then the map Un+1 → (coskn(U•))n is also a rigid covering in the appropriate topology, so that so is the induced map Vn+1 → (coskn(V•))n+1 for all n ≥ −1. One may also verify that all the structure maps of the hypercovering V• are G-equivariant for the diagonal action of G on each Vn. These prove (ii).

The proof of the (iii) follows from the standard construction of W•: Wn is defined as the pull-back of the two maps Vn → Π Vk ← Π U where the map sending Vn to the factor of Vk indexed by α is just α:[k]→[n] in ∆ α:[k]→[n] in ∆ V (α) and the maps U → Vk are all the given map. It follows readily that W• is a hypercovering, which is rigid if all the objects involved in its definition are rigid. It is also G-equivariant since all the objects involved in its definition are G-equivariant. HRR(X,G) will denote the collection of all G-equivariant rigid hypercoverings of X. 

Let G denote a fixed finite group. Let (Sm/k0, G, ?) denote the site whose objects are all smooth schemes of finite type over k0 provided with an action by G. The morphisms are morphisms in the given site that are also G-equivariant. The coverings of an object U are given by coverings V → U in this site so that this map is also G-equivariant. The points on this site are t ∗, where ∗ denotes a point in the site Sm/k0? and H is g ε G/H any subgroup of G. Let AbSh(Sm/k0, G, ?) denote the category of G-equivariant abelian sheaves on the big site Sm/k0?. An equivariant abelian presheaf P on the site (Sm/k0, G, ?) is additive if Γ(U t V,P ) = γ(U, P ) × Γ(V,P ). The category of all such abelian presheaves that are G-equivariant will be denoted AbP sh(Sm/k0, G, ?).

Given any smooth scheme U of finite type over k0, observe that G/H × U defines a smooth scheme of finite type ∼ H over k0 with a G-action. (Here H is any subgroup of G.) Then, HomG(G/H ×U, F ) = Hom(U, F ), where HomG (Hom) denotes the external Hom in the category of G-equivariant sheaves (sheaves, respectively). Therefore, F H H H defines a sheaf on the big site Sm/k0? by the rule Γ(U, F ) = Hom(U, F ). Now the following lemma is clear. Equivariant motivic homotopy theory 13

Lemma 2.13. AbSh(Sm/k0, G, ?) is a Grothendieck category and hence has enough injectives. A sequence of abelian sheaves 0 → F 0 → F → F 00 → 0 in the above category is exact if and only if 0 → (F 0)H → F H → (F 00)H → 0 is exact as sheaves on Sm/k0? for every subgroup H of G.

Proposition 2.14. (Cohomology from equivariant hypercoverings) Let P ε AbP sh(Sm/k0, G, ?) and let aP denote the associated abelian sheaf. Given any X ε (Sm/k0, G, ?), we obtain the isomorphism ∗ ∼ ∗ ∼ ∗ H (X, aP ) = lim H (Γ(U•, aP )) = lim H (Γ(U•,P )) ? → → U• ε HRR(X,G) U• ε HRR(X,G) that is functorial in P .

Proof. We review the standard proof here. First one observes that if U• → X is a G-equivariant hypercovering, then ZU• → ZX is a (simplicial) resolution in AbP sh(Sm/k0, G, ?), where for any scheme W , ZW denotes the abelian sheaf representing W . Therefore, if aP → I• is an injective resolution by G-equivariant abelian sheaves, • ∼ • ∼ • • (2.3.6) Γ(X,I ) = HomG(ZX ,I ) = HomG(ZU• ,I ) = Γ(U•,I ).

0 ∼ 0 i In particular H (X, aP ) = H (Γ(U•, aP ) and if aP is an injective object of AbSh(Sm/k0, G, ?), H (Γ(U•, aP ) = ∗ 0 for all i > 0. Therefore, it now suffices to show that F 7→ lim H (Γ(U•,F )) is a δ-functor and that one → U• ε HRR(X,G) obtains the last isomorphism in the statement of the proposition.

Let 0 → F1 → F2 → F3 → 0 denote a short exact sequence of G-equivariant abelian sheaves on the site Sm/k0, G, ?). (Recall that the exactness of the above sequence is equivalent to the exactness of the sequences H H H 0 → F1 → F2 → F3 → 0 of abelian sheaves on Sm/k0? for all subgroups H of G.) Let G1 = coker(F1 → F2) and G3 = coker(G1 → F3) where the cokernels are taken as abelian presheaves on the site (Sm/k0, G, ?). Then, for any G-equivariant hypercovering U• → X, we obtain the two short exact sequences of complexes:

0 → Γ(U•,F1) → Γ(U•,F2) → Γ(U•,G1) → 0 and 0 → Γ(U•,G1) → Γ(U•,F3) → Γ(U•,G3) → 0.

Presently we will show that lim (Γ(U•,G3)) = 0. The key observation here is that the sheaf associated to → U• ε HRR(X,G) G3, i.e. aG3 is trivial. Therefore, given any class α ε Γ(Un,G3) for some fixed G-equivariant rigid hypercovering U• and some integer n ≥ 0, there exists some G-equivariant cover V → Un so that α 7→ 0 in Γ(V,G3). One may then find a G-equivariant rigid hypercovering V• that dominates U• and for which the map Vn → Un factors through the given map V → Un. The last isomorphism in the proposition follows essentially from the observation that the stalks of P and aP are isomorphic. 

Corollary 2.15. Let P ε AbP sh(Sm/k0, G, ?) and let Y ε Sm/k0 be fixed scheme. Then one also obtains an iso- morphism

∗ s ∼ ∗ s Ext (Z(X+ ∧ S ∧ Y ), aP ) = lim H (HomG(Z(U•,+ ∧ S ∧ Y ),P )), where HomG denotes Hom in the G → U• ε HRR(X) category AbP sh(Sm/k0, G, ?) and ExtG denotes the derived functor of the corresponding HomG for G-equivariant abelian sheaves. s s Proof. One defines a new abelian presheaf P¯ by Γ(U, P¯) = Hom(Z(U+ ∧ S ∧ Y ),P ) = Hom(Z(U+) ⊗ Z(S ) ⊗ s s Z(Y ),P ). (The identification Z(U+ ∧ S ∧ Y ) = Z(U+) ⊗ Z(S ) ⊗ Z(Y ) is clear, see [Wei, p. 6] for example.) Then one may observe that the functor P → P¯ is exact and sends an additive abelian presheaf to an additive abelian ¯ presheaf. Therefore, the conclusion follows by applying the last proposition to the abelian presheaf P . 

Remark 2.16. Since not every object in Sm/k0 has an action by G, it is not possible to consider localization of PShc(Sm/k0, G, ?) or PSh(Sm/k0, G, ?) by inverting maps associated to an elementary distinguished square in the Nisnevich topology or maps of the form U• → U where U ε Sm/k0 and U• → U is a hypercovering in the given topology. Therefore, we instead localize PSh by inverting such maps and consider the category of diagrams of type o OG with values in this category. o o OG OG o Definition 2.17. PShmot ( PShdes ) will denote the category of all diagrams of type OG with values in the localized category PShmot (PShdes, respectively). 14 Gunnar Carlsson and Roy Joshua

o OG Lemma 2.18. PShmot is a cofibrantly generated symmetric monoidal model category which is also weakly finitely o OG generated in the sense of [Dund1, Definition 3.4] whereas PShdes is a cofibrantly generated symmetric monoidal model category which is also left proper and cellular. Both categories are locally presentable and therefore also combinatorial model categories. Oo Proof. The symmetric monoidal structure is inherited from the one on PSh G . Recall from [Hirsh, Theorem 4.3.1] that the generating cofibrations in the localized model category are the same as those in the original category. Now all the statements except the one claiming the monoidal model structure follow by standard arguments. (See [Dund2, 2.14, Lemma 2.15] for a proof of these for the category PShmot.)

o OG We will sketch an argument to show this for PShdes from which the same conclusions follow for PShdes . The generating trivial cofibrations in the localized model category PShdes are obtained as pushout-products cfg where cf : U• → cU is the cofibrant approximation to the obvious structure map f : U• → U of a hypercovering and g : A → B is a generating cofibration. (Observe that the domains of all the generating cofibrations X ∧ δ∆[n] are cofibrant, so that ∧ with these preserve weak-equivalences in PSh.) Now it suffices to show that if g0 : A0 → B0 0 is also a generating cofibration, then (cfg)g is weak-equivalence. First one uses the properness and monoidal 0 0 model structure of Psh to observe that the induced maps (cf ∧ idB)g and (cf ∧ idA)g are weak-equivalences. Then one obtains the pushout:

0 0 U• ∧ A ∧ B ∨ cU ∧ A ∧ A / 0 0 U•∧A∧A cU ∧ A ∧ B

0  0 U• ∧ B ∧ B ∨ cU ∧ B ∧ A /  0 P U•∧B∧A

0 0 which maps to cU ∧ B ∧ B . The top row is (cf ∧ idA)g and so is a weak-equivalence. The left column is a cofibration and so the pushout is a homotopy-pushout. Therefore the bottom row is also a weak-equivalence. Now 0 0 one may observe that the map (cf ∧ idB)g from the left-most vertex of the bottom row to cU ∧ B ∧ B is a weak- equivalence and that it factors through the above pushout. Therefore, the induced map from the above pushout 0 0 to cU ∧ B ∧ B given by (cfg)g is also a weak-equivalence. This proves the the monoidal model structure for PShdes and the proof for PShmot is similar. 

For the remainder of this section, we will assume that k0 is a fixed field, k a fixed algebraic closure of k0,

G = Galk0 (k). o OG Example 2.19. Then one obtains the following pushout square of objects in PShmot:

/ 1 Gm A

 /  A1 P1 0 0 where all the schemes above are obtained by extension from k0 to k which is a finite Galois extension k0 of k0. o 0 1 1 ∼ 1 OG Therefore, all of them have a natural Gk0 (k )-action. This shows that P /A = A /Gm as objects in PShmot. 1 1 1 1 Moreover the obvious map → / (obtained by sending k0 to the origin in is G-equivariant and also P P A A o 1 1 1 OG an A -equivalence. Now one also obtains the identification A /Gm ' S ∧ Gm as objects in PShmot, where Gm 1 is pointed by 1 ε Gm and S denotes the simplicial 1-sphere. Here ∧ is the usual smash product of simplicial sheaves and ' denotes an 1-equivalence. One may prove similarly using ascending induction on n ≥ 1 that A o n n−1 n n n 1 OG P /P ' A /(A − {0}) ' ∧ P in PShmot.

2.4. Sheaves with transfer and correspondences. Given two smooth schemes X, Y in (Sm/k0, G), let

CorG(X,Y ) = {Z ⊆ X × Y | closed, integral so that the projection Z → X is finite}. Observe that since X and Y are assumed to have actions through some finite quotient of G, there is natural induced action of G on CorG(X,Y ): hence the presence of the subscript G. One may also define the category of correspondences on S, by letting the objects be the smooth schemes in (Sm/k0, G) and where morphisms are elements of CorG(X,Y ). This category will be denoted CorG. An abelian G-equivariant presheaf with transfers is a contravariant functor CorG → (abelian groups, G), where the category on the right consists of abelian groups with a continuous action by G, continuous in the sense that the stabilizers are all subgroups with finite quotients. Equivariant motivic homotopy theory 15

An abelian G-equivariant sheaf with transfers is an abelian G-equivariant presheaf with transfers which is a sheaf on the site (Sm/k0)Nis. This category will be denoted by AShtr(S, G).

One obtains an imbedding of the category (Sm/k0, G) into CorG by sending a scheme to itself and a G- equivariant map of schemes f : X → Y to its graph Γf . Given a scheme X in (Sm/k, G), Ztr,G(X) will denote the sheaf with transfers defined by Γ(U, Ztr,G(X)) = CorG(U, X). One extends this to define the G-equivariant motive of X as the complex associated to the simplicial abelian sheaf n 7→ CorG(− × ∆[n],X), where − takes any n object in the big Nisnevich site as argument and ∆[n] = Spec( k0[x0, ··· , xn]/Σi=0xi − 1). We will denote this by MG(X).

It is important to realize that MG(X) is a complex of sheaves on the site (Sm/k0)Nis. The global sections of this complex, i.e. sections over Spec k0 will be denoted MG(X). 2.4.1. A key property. Let F denote an abelian G-equivariant sheaf with transfers. Then ∼ Γ(X,F ) = HomAShtr (S,G)(MG(X),F ).

3. The basic framework of equivariant motivic homotopy theory: the stable theory The theory below is a variation of the theory of enriched functors and spectra as in [Dund1], [Dund2] and also [Hov-3] modified so as to handle equivariant spectra for the action of a group G where G denotes a group as before.

3.1. Enriched functors and spectra. Let C denote a symmetric monoidal cofibrantly generated model category: recall this means C is a cofibrantly generated model category, which also has the additional structure of a symmetric monoidal category, compatible with the model structure as in [Hov-2]. We will assume that C is pointed, i.e. the initial object identifies with the terminal object, which will be denoted ∗. The monoidal product will be denoted ∧; the unit of the monoidal structure will be denoted S0 and this is assumed to be cofibrant. We will let I (J) denote the generating cofibrations (generating trivial cofibrations, respectively). We will make the following additional assumptions: (i) The model structure on C is cellular: see [Hirsh, ]. In practice this means the domains and co-domains of both I and J are small with respect to I and that the cofibrations are effective monomorphisms. We will assume the model structure is also left-proper. (ii) We will further assume that the domains and co-domains of the maps in I are cofibrant. (iii) Though this assumption is not strictly necessary, we will further assume that C is simplicial or at least pseudo-simplicial so that there is a bi-functor Map : Cop × C → (simpl.sets). We will also assume that there op exists an internal Hom functor HomC : C × C → C, so that for any finitely presented object C ε C, one obtains: ∼ (3.1.1) Map(C, HomC(K,M)) = Map(C ∧ K,M)

(In fact if Map(K,N)n = HomC(K ⊗∆[n],N) where K ⊗∆[n] denotes an object in C defined using a pairing ⊗ : C × (simpl.sets) → C, then it suffices to assume that (K ⊗ ∆[n])k is finitely presented for all k ≥ 0 and that the analogue of the isomorphism in (3.1.1) holds with Map replaced by HomC.)

Let C0 denote a C-enriched full-subcategory of C consisting of objects closed under the monoidal product ∧, all 0 0 0 of which are assumed to be cofibrant and containing the unit S . Let C0 denote a C-enriched sub-category of C , which may or may not be full, but closed under the monoidal product ∧ and containing the unit S0. (In [Dund1, section 4], there are a series of additional hypotheses put on the category C0 to obtain stronger conclusions. These are not needed for the basic conclusions as in [Dund1, Theorem 4.2] and therefore, we skip them for the time 0 being.) In view of the motivating example considered below, we will denote the objects of C0 as {TV }. Observe ∼ 0 0 0 that TW = Hom(S ,TW ) and that S → HomC(TV ,TV ). Now the pairing HomC(TV ,TV ) ∧ HomC(S ,TW ) → 0 0 0 ∼ HomC(TV ,TV ∧ TW ), pre-composed with S ∧ HomC(S ,TW ) → HomC(TV ,TV ) ∧ HomC(S ,TW ), sends TW = 0 HomC(S ,TW ) to a sub-object of HomC(TV ,TV ∧ TW ). 0 0 Definitions 3.1. (i) [C0, C] will denote the category whose objects are C-enriched covariant functors from C0 to C: see [Dund1, 2.2]. An enriched functor X sends HomC(TV ,TW ) 7→ HomC(X (TV ), X (TW )) for objects TV and 0 TW ε C0 with this map being functorial in TV and TW . The adjunction between ∧ and HomC shows that in this case, one is provided with a compatible family of pairings HomC(TV ,TW ) ∧ X (TV ) → X (TW ). On replacing TW 0 0 0 with TV ∧TW and pre-composing the above pairing with S ∧HomC(S ,TW ) → HomC(TV ,TV )∧HomC(S ,TW ) → HomC(TV ,TV ∧TW ), one sees that, an enriched functor X comes provided with the structure maps TW ∧X (TV ) → 0 0 X (TV ∧ TW ) that are compatible as TW and TV vary in C0.) A morphism φ : X → X between two such enriched 16 Gunnar Carlsson and Roy Joshua functors is given by a C-natural transformation, which means that one is provided with a compatible collection 0 of maps {X (TV ) → X (TV ) | TV ε fC0}, compatible with the pairings HomC(TV ,TW ) ∧ X (TV ) → X(TW ) and 0 0 HomC(TV ,TW ) ∧ X (TV ) → X (TW ).

0 (ii) Spectra(C). We let Sph(C0) denote the C-category defined by taking the objects to be the same as the 0 objects of C and where Hom 0 (T ,T ) = T if T = T ∧ T and ∗ otherwise. Since T is a sub-object 0 Sph(C0) U V W V W U W 0 0 0 of HomC(TU ,TV ), it follows that Sph(C0) is a sub-category of C0. Now an enriched functor in [Sph(C0), C] is 0 simply given by a collection {X(TV )|TV ε Sph(C0)} provided with a compatible collection of maps TW ∧ X(TV ) → 0 X(TW ∧ TV ). We let Spectra(C) = [Sph(C0), C].

0 0 (iii) For each fixed TV ε C0, we associate the free C-functor FTV : C → [C0, C] defined by FTV (X ) = X ∧

HomC(TV , ) and the free C-functor FTV : C → Spectra(C) that sends each X ε C to the spectrum FTV (X) defined by

(3.1.2) FTV (X)(TW ) = X ∧ TU , if TW = TU ∧ TV and = ∗ otherwise

0 (iii) For each TV ε C0, we let ΩTV : C → C denote the functor defined by ΩTV (X) = HomC(TV ,X).

0 0 (iv) For each TV ε C0, we let RTV : C → [C0, C] denote the C-enriched functor RTV (P ) defined by RTV (P )(TW ) =

HomC(HomC(TW ,TV ),P ). Similarly we let RTV : C → Spectra(C) denote the functor defined by RTV (P )(TW ) = 0 HomC(TU ,P ) if TU ∧ TW = TV and S otherwise.

0 0 Let EvalTV :[C0, C] → C denote theC-enriched functor sending X 7→ X (TV ). Similarly, let EvalTV : Spectra(C0) → C denote the C-enriched functor that sends a spectrum X ε Spectra(C) to X (TV ).

Lemma 3.2. Now RTV is right-adjoint to EvalTV while FTV is left-adjoint to EvalTV . Similarly, RTV is right- adjoint to EvalTV while FTV is left-adjoint to EvalTV .

Example 3.3. The main examples of the above are the following. Let C denote one of the categories PShc(Sm/k0, G), Oo 1 PSh(Sm/k0, G), PSh G or the corresponding A -localized categories of simplicial sheaves up to homotopy. We may also restrict to the full subcategories of constant pointed simplicial presheaves with action by G in the first o two categories and to the full subcategory of diagrams of type OG in the category of pointed simplicial sets. One 0 has the following choices for the category C0.

(i) Let V denote an affine space over k (where k is some finite Galois extension of k0) and provided with a linear H action by G so that V = V for some H ε W. We let TV = V/V − 0 = the Thom-space of the corresponding 0 0 G/K-equivariant vector bundle over k. We let C0 = C denote the collection {TV } as one varies over all such representations of G. Observe that C0 now contains the sphere S0 (by taking the V to be the 0-dimensional vector space) and is closed under smash products. The resulting category of spectra will be denoted Spectra(C) (and by Spectra(C, C0) to clarify the choice of the sub-category C0.)

A variant of the above framework is the following: let F : C → C denote a functor that is compatible with the monoidal product ∧ (i.e. there is a natural map F (A) ∧ F (B) → F (A ∧ B)) and with all small colimits. We may 0 0 0 0 now let C0 = CF = the full subcategory of C generated by {F (A)|A ε C } together with S . Examples of such a framework appear in the later sections. The resulting category of spectra will be denoted Spectra(C,F ). This seems to be the only general framework available for G denoting any one of the three choices above.

0 0 (ii) One may also adopt the following alternate definition of C0 = C , when G denotes either a finite or profinite group. Let T = 1 = 1 . Let K denote a normal subgroup of G with finite index. Let T denote the ∧ P Pk0 K of G/K copies of T with G/K acting by permuting the various factors above. We let C0 denote the full sub- category of C generated by these objects under finite applications of ∧ as K is allowed to vary subject to the above 0 0 1 constraints. This sub-category of [C , C] will be denoted [C ,C]P and the corresponding category of spectra will be denoted Spectra(C, P1). If we fix the normal subgroup K of G, the resulting category of P1-spectra will be denoted Spectra(C, G/K, P1): here the subcategory C0 will denote the full sub-category of C generated by these objects under finite applications of ∧.

More generally, given any object P ε C together with an action by some finite quotient group G/K of G, one 0 1 may define the categories [C ,C]P and the category, Spectra(C,P ) of P -spectra similarly by replacing P above by 1 P . (For example, P could be F (P ) for a functor F as in (i).) In case char(k0) = 0, the regular representation Equivariant motivic homotopy theory 17 of any finite group breaks up into the sum of irreducible representations and contains among the summands all such irreducible representations. Therefore, in case char(k0) = 0, there is no loss of generality in adopting this framework to that of (i) when the group is finite. In arbitrary characteristics, and also for general groups, the framework of (i) is clearly more general.

(iii) Under the same hypotheses as in (ii), we may also define C0 as follows. Let T = Sn (for some fixed positive integer n) denote the usual simplicial n-sphere. Let TK denote the ∧ of G/K copies of T with G/K acting by permuting the various factors. In case K = G and n = 1, we obtain a spectrum in the usual sense and indexed by the non-negative integers. Such spectra will be called ordinary spectra: when the constituent simplicial presheaves are all simplicial abelian presheaves, such spectra will be called ordinary abelian group spectra. 3.1.3. Spectra indexed by the non-negative integers. One may construct spectra indexed by the non-negative 0 integers from Spectra(C) as follows. Let TV denote a fixed element of C0. One may now consider the full subcategory 0 0 of C0 generated by TV under iterated smash products along with the unit S . Any spectrum X in Spectra(C) may n be restricted to this sub-category and provides a spectrum X in the usual sense by defining X n = X (∧ TV ). Given any motivic spectrum X , one obtains this way, the (motivic) spectrum X indexed the non-negative integers where ∧n X n = X (TV ). Examples 3.4. Suspension spectra (i) Let C denote one of the categories as in Definition 3.11 and let A ε C. 0 Then the motivic suspension spectrum associated to A is the spectrum Σmot(A) whose value on TV ε C0 is given by 1 0 TV ∧ A. The P -motivic suspension spectrum associated to A is is the spectrum whose value on TK ε C0 is given by 1 TK ∧A. This will be denoted ΣP (A). If the subgroup K of G is fixed, we will denote the corresponding suspension 0 1 1 spectrum by ΣP ,K (A). If we take A = S , we obtain the motivic sphere spectrum Σmot and the P -motivic sphere 1 spectrum ΣP . In case F : C → C is a functor commuting with ∧ and with colimits, then we obtain the motivic 1 1 F -spectrum Σmot,F and the F (P )-motivic sphere spectrum ΣF (P ). (ii) If we take A = Sn ∧ B, for some n ≥ 1 and B ε C (as above), the resulting spectra are the Sn-suspensions of n,0 s 0 the above suspension spectra associated to A. This will be denoted ΣmotB. If we take A = TV ∧ S ∧ B, TV ε C0, 1 s+2|TV |,TV the resulting P -motivic suspension spectrum associated to A will be denoted Σ B, where |TV | = dim(V ).

1 n 1 If we take A = (Pk) ∧ B for some B ε C, the resulting P -motivic suspension spectrum associated to A will be 1 n 1 2n,n the ( ) -suspension of the -motivic suspension spectrum associated to B. This will be denoted Σ 1 (B). In P P P n 1 m 1 n+2m,m case A = S ∧ ( ) ∧ B for some B ε C, the resulting -motivic suspension spectrum will be denoted Σ 1 B. Pk P P 3.1.4. Suspending and de-suspending spectra. Let E denote a spectrum in any one of the above cate- 0 2|TV |,TV gories, with TV denoting elements of C0. Then we define the suspension Σ E to be the spectrum de- 2|TV |,TV −2|TV |,−TV fined by Σ E(TU ) = E(TV ∧ TU ). The de-suspension Σ E will be the spectrum defined by −2|TV |,−TV Σ E(TV ∧ TU ) = E(TU ) and = ∗ otherwise. 3.1.5. Equivariant vs. non-equivariant spectra. The equivariant spectra will denote spectra as in 3.3(i) 1 and 3.3(ii). Σ 1 will denote the G-equivariant sphere spectrum associated to . The non-equivariant spectra Pk,G Pk will denote the G-spectra when the group G is the trivial group. Σ 1 will denote the (non-equivariant) sphere Pk 1 spectrum associated to . When we only consider equivariant spectra, Σ 1 will also be used to denote Σ 1 . Pk Pk Pk,G

It is important to observe that the equivariant spectrum Σ 1 is a module spectrum over the non-equivariant Pk,G spectrum Σ 1 . Pk 3.1.6. Mapping spectra. Next we briefly discuss the various mapping spectra (i.e. Hom-spectra) that one needs Oo 1 to invoke. For the categories PSh(Sm/k0, G), PSh G , PSh(Sm/k0) as well as the corresponding A -localized model categories one has the external hom which will be denoted Hom. One also has the internal Hom, which will be denoted Hom and in addition there is an enriched hom, enriched in simplicial sets, which will be denoted Map. For P,Q ε PSh(Sm/k0, G), Map(P,Q)n = Hom(P ∧ ∆[n]+,Q). Observe also that Γ(U, Hom(P,Q)) = Map(P|U ,Q|U ) where P|U and Q|U denote the restriction of P and Q to the subcategory Sm/U.

Using these, one obtains the corresponding Hom-functors for spectra: HomSp will denote the external hom, while HomSp will denote the internal hom and MapSp will denote the enriched hom. One may observe that

→ HomSp(E,B)(TU ) = Eq(ΠTV Hom(E(TV ),B(TV ∧ TU )) → ΠTV ,TW Hom(TW ∧ E(TV ),B(TW ∧ TV ∧ TU ))) 18 Gunnar Carlsson and Roy Joshua where Hom denotes the internal hom in the appropriate category of simplicial presheaves and the two arrows denote the obvious maps.

3.1.7. Notational convention. When the context is clear, we will often omit the subscript Sp in HomSp. Moreover, we will also use Hom or HomSp to denote Γ(Spec k0, HomSp( , )). 3.1.8. Layer filtrations on spectra. Layer filtrations on spectra provide a convenient tool for inductive argu- ments: see [Sw, 8.7(ii)]. We extend this to our framework as follows. Let K denote a spectrum in any one of the 0 above categories and let TV denote elements of C0. Let e ε K(TV )n. We will also use e to denote the sub-simplicial V presheaf of K(TV ) generated by e. Let F (e) denote the image of the spectrum FTV (e) in K (i.e. the sub-spectrum of K generated by FTV (e)), using the terminology of 3.1(iii). For such an e, we define l(e) = the number of faces of e. (Clearly l(e) ≥ 1.)

We define an ascending filtration, {Ke n|n} on K as follows: Ke n = ∗ for n ≤ 0 and Ke n = ∪{F V (e) | l(e) ≤ n n, TV ε C0}. This is clearly an ascending filtration on K and K = ∪n≥1Ke .

Observe that K1 = W F (S0) and Kn/Kn−1 = W F (S0) for some families of {T |α}, {T |β} ⊆ C0 . e TVα e e TVβ Vα Vβ 0 For module-spectra M over a ring spectrum E considered below, one may also obtain a similar filtration with M 1 = W(F (S0) ∧ E) and M n/M n−1 = W(F (S0) ∧ E) for some families of {T |α}, {T |β} ⊆ C0 . The main f TVα f f TVβ Vα Vβ 0 V V V V difference is that we will define FE (e) = F (e)∧E and use FE (e) in the place of F (e), with the filtration defined similarly. 3.2. Smash products of spectra and ring spectra. First we recall the construction of smash products of 0 0 0 enriched functors from [Dund1, 2.3]. Given F ,F ε [C0, C], their smash product F ∧ F is defined as the Kan- 0 0 0 0 0 extension along the monoidal product ◦ : C0 ×C0 → C0 of the C-enriched functor F ×F ε [C0 ×C0, C]. Given spectra 0 X and Y in [Sph(C0), C], this also defines their smash product X ∧ Y. One also defines the derived smash product L X ∧Y by C(X ) ∧ Y, where C(X ) is a cofibrant replacement of X in the stable model structure on Spectra(C). An 0 algebra in [C0, C] is an enriched functor X provided with an associative and unital pairing µ : X ∧ X → X , i.e. for 0 TV ,TW ε C0, one is given a pairing, X (TV ) ∧ X (TW ) → X (TV ∧ TW ) which is compatible as TW and TV vary and is also associative and unital.

0 0 A ring spectrum in Spectra(C) is an algebra in [Sph(C0), C] for some choice of C0 satisfying the above hypotheses. 0 0 A map of ring-spectra is defined as follows. If X is an algebra in [Sph(C0), C] and Y is an algebra in [Sph(D0), C] 0 0 for some choice of subcategories C0 and D0, then a map φ : X → Y of ring-spectra is given by the following data: 0 0 (i) an enriched covariant functor φ : C0 → D0 compatible with ∧ and (ii) a map of spectra X → φ∗(Y ) compatible 0 with the ring-structures. (Here φ∗(Y ) is the spectrum in Spectra(C, C0) defined by φ∗(Y )(TV ) = Y (φ(TV )). Given a ring spectrum A, a left module spectrum M over A is a spectrum M provided with a pairing µ : A∧M → M which is associative and unital. One defines right module spectra over A similarly. Given a left (right) module spectrum M (N, respectively) over A, one defines (3.2.1) M∧N = coequalizer(M ∧ A ∧ N→→ M ∧ N) A where the coequalizer is taken in the category of spectra and the two maps correspond to the module structures on M and N, respectively.

Let Mod(A) denote the category of left module spectra over A with morphisms being maps of left module spectra over A. Then the underlying functor U : Mod(A) → Spectra(C) has a left-adjoint given by the functor n FA(N) = A ∧ N. The composition T = FA ◦ U defines a triple and we let TM = hocolim{T M|n}. Since a map ∆ f : M 0 → M in Mod(A) is a weak-equivalence of spectra if and only if U(f) is, one may observe readily that TM is weakly-equivalent to M. Therefore, one defines L (3.2.2) M∧N = TM∧N = coequalizer(TM ∧ A ∧ N→→ M ∧ N) A A

Examples 3.5. (i) Assume the situation of 3.3 (i). Let F : C → C denote a functor commuting with ∧ and with colimits and provided with a natural augmentation  : id → F also compatible with ∧. Then Σmot,F is a ring spectrum in Spectra(C) and  induces a map of ring spectra Σmot → Σmot,F . In case F is not provided with Equivariant motivic homotopy theory 19

the augmentation, then Σmot,F is a ring spectrum in Spectra(C,F ); in fact it is the unit of this symmetric monoidal category. .

Let φ : F → G denote a natural transformation between two functors C → C that commute with the tensor 0 0 0 structure ∧ and with all small colimits. Let C denote a subcategory of C as before and let CF , CG denote the 0 0 corresponding subcategories. Then φ induces a functor CF → CG compatible with ∧. Let Σmot,F and Σmot,G denote the motivic sphere spectra defined there. Then these are ring-spectra and φ induces a map commuting with the ring structures. (ii) Assume the situation of 3.3 (ii). Let φ : F → G be as above. Then for any P ε C,ΣF (P ) is a ring-spectrum in Spectra(C,F (P )) and ΣG(P ) is a ring-spectrum in Spectra(C,G(P )) with φ inducing a map of these ring- spectra.

3.3. Model structures on Spectra(C). We will begin with the level-wise model structure on Spectra(C) which will be defined as follows. Definition 3.6. (The level model structure on Spectra(C).) A map f : X 0 → X in Spectra(C) is a level equivalence

(level fibration, level trivial fibration, level cofibration, level trivial cofibration) if each EvTV (f) is a weak-equivalence (fibration, trivial fibration, cofibration, trivial cofibration, respectively) in C. Such a map f is a projective cofibration if it has the left-lifting property with respect to every level trivial fibration.

Let I (J) denote the generating cofibrations (generating trivial cofibrations, respectively) of the model category C. We define the generating cofibrations I to be S {F (i) | i ε I} and the generating trivial cofibrations Sp TV ε fC TV J to be S {F (j)|j ε J}. Sp TV ε fC TV

Proposition 3.7. (i) If A ε C is small relative to the cofibrations (trivial cofibrations) in C, then FTV (A) is small relative ISp. (ii) A map f in Spectra(C) is level cofibration (level trivial cofibration) if and only if it has the left lifting property with respect to all maps of the form RTV (g), where g is a trivial fibration (fibration, respectively)in C.

(iii) Every map in ISp − cof is a level cofibration and every map in JSp − cof is a level trivial cofibration.

(iv) The domains of ISp (JSp) are small relative to ISp − cof (JSp − cof, respectively).

Proof. (i) The main point here is that the functor EvTV right adjoint to FTV commutes with all small colimits.

(ii) Since RTV is right adjoint to EvTV , f has the left lifting property with respect to RTV (g) if and only if

EvTV (f) has the left-lifting property with respect to g. (ii) follows readily from this observation.

0 0 (iii) Recall every object of C0 is assumed to be cofibrant. Therefore, smashing with any TV ε C0 preserves cofibrations of C. Therefore, every map in ISp is a level cofibration. By (ii) this means RTV (g) ε ISp − inj for all trivial fibrations g in C. Recall every map in ISp − cof has the left lifting property with respect to every map in

ISp − inj and in particular with respect to every map RTV (g), with g a trivial fibration in C. Now the adjunction between EvTV and RTV completes the proof for ISp − cof. The proof for JSp − cof is similar.

(iv) follows readily in view of the adjunction between the free functor FTV and EvTV . 

We now obtain the following corollary.

Corollary 3.8. The projective cofibrations, the level fibrations and level equivalences define a cofibrantly generated model category structure on Spectra(C) with the generating cofibrations (generating trivial cofibrations) being ISp (JSp, respectively). This model structure (called the projective model structure on Spectra(C)) has the following properties: (i) It is left-proper (right proper) if the corresponding model structure on C is left proper (right proper, respec- tively). It is cellular if the corresponding model structure on C is cellular. (ii) The objects in S {F (C0 )} are all finitely presented. The category Spectra(C) is symmetric monoidal TV ε fC TV 0 0 0 with the pairing X ∧ X(TV ) = X (TV ) ∧ X(TV ) and with the unit being the inclusion functor [ 0 i : {FTV (C0)} → Spectra(C) 0 TV ε C0 . 20 Gunnar Carlsson and Roy Joshua

(iii) The projective model structure on Spectra(C) is weakly finitely generated when the given model structure on C is weakly finitely generated. (iv) The category Spectra(C) is locally presentable if C is. Proof. The assertions in (ii) are clear. Those in (i) and (iii) may be proven making use of the last Proposition: see also [Hov-3, proof of Theorem 1.14]. 

One may characterize the projective cofibrations and projective trivial cofibrations as follows: a map f : X 0 → X is a projective cofibration (projective trivial cofibration) if and only if (i) each EvTV (f) is a cofibration (trivial 0 cofibration, respectively) and (ii) the induced map X (TV ∧ TW ) t TV ∧ (X(TW ) → X(TV ∧ TW ) is a 0 TV ∧(X (TW )) 0 cofibration (trivial cofibration, respectively) for all TV ,TW ε C0. This may be proven as in [Hov-3, Proposition 1.15].

3.3.1. The stable model structure on Spectra(C). We proceed to define the stable model structure on Spectra(C) by applying a suitable Bousfield localization to the projective model structure on Spectra(C). This follows the approach in [Hov-3, section 3]. (One may observe that the domains and co-domains of objects in I are cofibrant, so that there is no need for a cofibrant replacement functor Q as in [Hov-3, section 3].) Definition 3.9. (Ω-spectra) A spectrum X ε Spectra(C) is an Ω-spectrum if if it is level-fibrant and each of the 0 natural maps X(TV ) → HomC(TW ,X(TV ∧ TW )), TV ,TW ε C0 is a weak-equivalence in C.

Observe that giving a map FTV ∧TW (TW ∧ C) → FTV (C) corresponds by adjunction to giving a map TW ∧ C →

(FTV (C))(TV ∧ TW ) = TW ∧ C. Therefore we let 0 (3.3.2) S denote the morphisms {FTV ∧TW (TW ∧C) → FTV (C) | C ε Domains and Co-domains of I, TV, TW ε C0} corresponding to the identity maps TW ∧ C → TV ∧ C by adjunction. The stable model structure on Spectra(C) is obtained by localizing the projective model structure on Spectra(C) with respect to the maps in S. (The reason for considering such maps is as follows: let C ε C be an object as above.

Then Map(C, EvalTV (X)) ' Map(FTV (C),X) and Map(C, HomC(TW , EvalTV ∧TW (X))) ' Map(FTV ∧TW (TW ∧ C),X) provided X is fibrant in the projective model structure considered above. Therefore to convert X into an Ω -spectrum, it suffices to invert the maps in S.) The S-local weak-equivalences (S-local fibrations) will be referred to as the stable equivalences (stable fibrations, respectively). The cofibrations in the localized model structure are the level cofibrations in Spectra(C). Proposition 3.10. (i) The corresponding stable model structure on Spectra(C) is cofibrantly generated, left proper and cellular when C is assumed to have these properties.

(ii) The fibrant objects in the stable model structure on Spectra(C) are the Ω-spectra defined above.

(iii) The category Spectra(C) is symmetric monoidal with the same structure as in the projective model structure.

(iv) The stable model structure on Spectra(C) is weakly finitely generated when the given model structure on C is. In general it is cellular and left proper assuming this holds for C. It is also locally presentable, assuming C is locally presentable. Proof. The proof of (i) is entirely similar to the proof of [Hov-3, Theorem 3.4] and is therefore skipped. If C is left proper (cellular), so is the projective model structure on Spectra(C) as proved above. It is shown in [Hirsh, ] that then the localization of the projective model structure is also left proper and cellular. The last two statements are obvious from the corresponding properties of the projective model structure. 

o 1 OG 0 Definition 3.11. Motivic and ´etalespectra, P -motivic and ´etalespectra. (i) If C = PShmot and C0 chosen as in Examples 3.3(i), the resulting category of spectra, Spectra(C), with the stable model structure will o OG 0 be called motivic spectra and denoted Sptmot(k0, G). If C = PShdes with the ´etaletopology and C0 chosen as in Examples 3.3(i), the resulting category of spectra, Spectra(C) with with the stable model structure will be called ´etale spectra and denoted Sptet(k0, G). In case F : C → C is a functor as in 3.3 (i), Sptmot,F(k0, G)(Sptet,F(k0, G) will denote the corresponding category.) Observe from Examples 3.3(i), that there are several possible choices for 0 the sub-categories C0. Equivariant motivic homotopy theory 21

o OG 0 (ii) If C = PShmot and C0 chosen as in Examples 3.3(ii), the resulting category of spectra, Spectra(C), with the o 1 1 OG stable model structure will be called P -motivic spectra and denoted Sptmot(k0, G, P ). If C = PShdes with the 0 ´etaletopology and C0 chosen as in Examples 3.3(ii), the resulting category of spectra, Spectra(C) with the stable 1 1 model structure will be called P -´etalespectra and denoted Sptet(k0, G, P ). In case F : C → C is a functor as 1 1 in 3.3 (i), Sptmot(k0, G, F(P )) (Sptet(k0, G, F(P )) will denote the corresponding category.) Observe again from 0 Examples 3.3(ii), that there are several possible choices for the sub-categories C0.

If Sn for some fixed positive integer n is used in the place of P1 above, the resulting categories will be denoted n n Sptmot(k0, G, S ) and Sptet(k0, G, S ). The group G will be suppressed when we consider spectra with trivial action by G.

o OG 0 3.3.3. Diagrams of spectra vs. spectra of diagrams. If C = PShmot with C0 chosen as in Examples 3.3(i) or (ii), then the resulting categories of spectra, Spectr 0 (C) obtained above are in fact spectra with values in C which C0 ¯ ¯0 is a diagram category. One may instead start with C = PShmot and with C0 chosen as in Examples 3.3(i) or (ii), then we also obtain the resulting categories of spectra, Spectra ¯0 (C¯). The main observation we want to make here C0 Oo is that there is a natural isomorphism Spectra ¯0 (C¯) G ' Spectr 0 (C). The category on the left is a category of C0 C0 o diagrams of type O with values in the category of spectra, Spectra ¯0 (C¯). Therefore, we will routinely identify G C0 objects of Spectra 0 (C) with objects in the above diagram category of spectra. The same discussion applies also C0 o OG 0 to the case where C = PShdes with the ´etaletopology and with C0 chosen as in Examples 3.3(i) or (ii).

Oo H Notational convention: Given an object X ε Spectra ¯0 (C¯) G , we let X denote the diagram X evaluated at C0 G/H.

o OG 1 Remark 3.12. Clearly it is possible to use any of the categories C = PShc(Sm/k0, G) PShA ,c(Sm/k0, G), or PShdes 0 with C0 chosen as in 3.3(i) or as in 3.3(ii). However, the simplicial presheaves forming the terms of these spectra are not sheaves upto homotopy. Hence the reason for adopting the above definitions. Clearly, for certain applications, it may be enough to consider such spectra.

Definition 3.13. (i) HSptmot(k0, G)(HSptet(k0, G)) will denote the stable homotopy category of motivic (´etale) spectra.

1 1 1 (ii) HSptmot(k0, G, P )(HSptet(k0, G, P )) will denote the corresponding stable homotopy category of P - motivic (´etale)spectra.

The group G will be suppressed when we consider spectra with trivial action by G.

3.3.4. An alternate construction of the category of spectra. In the construction of the stable category of spectra considered above, we start with unstable category of pointed simplicial presheaves, then localize this to invert A1-equivalences, then stabilize to obtain the category of spectra with the projective model structure which is then localized to obtain the category of spectra with the stable model structure. Instead of these steps, one may adopt the following alternate series of steps to construct the category of spectra with its stable model structure, where all the localization is carried out at the end.

Oo One starts with the category of pointed simplicial presheaves C = PSh G . Next we apply the stabilization construction as discussed above and obtain Spectra(C) with its projective model structure considered above. This is left proper, cellular and weakly finitely generated. Assume first that ? = Nis. One defines a presheaf P ε PSh(Sm/k0)Nis to be motivically fibrant if (i) P is fibrant in Spectra(C), (ii) Γ(φ, P ) is contractible (where φ denotes the empty scheme), (iii) sends an elementary distinguished squares as in [MV] to a homotopy carte- 1 sian square and (iv) the obvious pull-back Γ(U, P ) → Γ(U+ ∧ A+,P ) is a weak-equivalence. Then a map f : A → B in Spectra(C) is a motivic weak-equivalence if the induced map Map(f, P ) is a weak-equivalence for every motivically fibrant object P , with Map denoting the simplicial mapping space. Next we localize this by inverting maps that belong to any one of the following classes: (i) f is a motivic weak-equivalence and (ii) 0 {FTV ∧TW (TV ∧ C) → FTW (C) | C ε Domains and Co-domains of I, TV, TW ε C0}. (Here I is the set of generating cofibrations of C.) The cofibrations in the localized model structure will be the same as those of Spectra(C). The fibrations are defined by the lifting property. By [Hirsh, Theorem 4.1.2], it follows that this is a left-proper cellular model category. The resulting stable model structure identifies with the stable model structures on motivic spectra and P1-motivic spectra obtained above. 22 Gunnar Carlsson and Roy Joshua

One may in fact carry out this process in two stages. First one only inverts maps of the following form: 0 {FTV ∧TW (TV ∧ C) → FTW (C) | C ε Domains and Co-domains of I, TV, TW ε C0}. (Here I is the set of generating cofibrations of C. This produces a stable model structure on Spectra(C) where the fibrant objects are Ω-spectra. This stable model category will be denoted Spectrast(C). Then one inverts the maps f which are motivic weak- equivalences to obtain the A1-localized stable category of spectra. It may be important to specify the generating trivial cofibrations for the localized category, which we proceed to do now: see [Dund1, Definition 2.14]. For any elementary distinguished square:

Q = P / Y

φ  /  U ψ X we factor the induced map hP → hY as a cofibration (using the simplicial mapping cylinder) hP → C followed by a simplicial homotopy equivalence C → hY and similarly factor the induced map sq = hU t C → hX as a cofibration hP q 1 sq → tq followed by a simplicial homotopy equivalence tq → hX . Similarly we factor the obvious map hU×A → hU 1 into a cofibration u : hU×A → Cu followed by a simplicial homotopy equivalence Cu → hU . We also similarly factor the above map FTV ∧TW (TV ∧ C) → FTW (C) into a cofibration uV,W : FTV ∧TW (TV ∧ C) → DV,W followed by a simplicial homotopy equivalence DV,W → FTW (C). Let

0 ˜ 1 (3.3.5) J0 = {∗ → hφ} ∪ {u : hU×A → Cu | U ε Sm/k0} ∪ {q : sq → tq | qis an elementary distinguished square } ˜ 0 J1 = {uV,W : FTV ∧TW (TV ∧ C) → DV,W | C ε Domains and Co-domains of I, TV, TW ε C0} ˜ ˜0 0 ˜ J = {FTV (J0)|V ε C0} ∪ J1 (3.3.6)

0 ˜ The set of generating trivial cofibrations, JSp will then be the set {fg|f ε J, g : δ∆[n] → ∆[n], n ≥ 0} ∪ JSp where fg denotes the pushout-product and JSp is the set of generating trivial cofibrations in the stable model structure on Spectra(C).

˜0 ˜ In order to obtain a set of generating trivial cofibrations for Spectrast(C), one just lets J0 = {∗ → hφ} and J1 and J˜ as before.

Using the above information one may verify readily that this is a also a symmetric monoidal model category which is also simplicial and that it is weakly finitely generated.

If one uses the framework of Examples 3.3 (i), one obtains the categories Sptmot(k0, G), Sptmot,F(k0, G) and 1 1 if one uses the framework of Examples 3.3(ii), one obtains the categories Sptmot(k0, G, P ), Sptmot(k0, G, F(P )). One obtains a similar alternate construction of ´etalespectra as well.

1 1 3.4. Key properties of Sptmot,F(k0, G), Sptet,F(k0, G), Sptmot(k0, G, F(P )) and Sptet(k0, G, F(P )). We summarize the following properties which have already been established above. Similar properties also hold for the n n categories Sptmot(k0, G, S ) and Sptet(k0, G, S ) for any n ≥ 1 though we do not state them explicitly. (i) Weak-equivalences and fibration sequence A map f : A → B in any one of the above categories of G-equivariant spectra is a weak-equivalence if and only if the induced map f H : AH → BH is a weak-equivalence of spectra for all subgroups H ε W. A diagram F → E → B is a fibration sequence in any one of the above categories of Hal-equivariant spectra if and only if the induced diagrams F H → EH → BH are all fibration sequences of spectra for all subgroups H ε W. (ii) Stably fibrant objects. A spectrum E is fibrant in the above stable model structure if and only if each E(TV ) is fibrant in C= the appropriate unstable category of simplicial presheaves and the induced map 0 E(TV ) → HomC(TW , (E(TV ∧ TW ))) is a weak-equivalence of G-equivariant spaces for all TV ,TW ε C0. (iii) Cofiber sequences A diagram A → B → B/A is a cofiber sequence if and only if the the homotopy fiber of the map B → B/A is stably weakly-equivalent to A. (iv) Finite sums. Given a finite collection {Eα|α} of objects, the finite sum VαEα identifies with the product ΠαEα up to stable weak-equivalence. Equivariant motivic homotopy theory 23

(v) Additive structure. The corresponding stable homotopy categories is an additive category. 0 (vi) Shifts. Each TV ε C0 defines a shift-functor E → E[TV ], where E[TV ](TW ) = E(TV ∧ TW ). This is an automorphism of the corresponding stable homotopy category. (vii) Cellular left proper simplicial model category structure. All of the above categories of spectra have the structure 1 of cellular left proper simplicial model categories. The categories Sptmot,F(k0, G), Sptmot(k0, G, F(P )) and the corresponding categories of spectra on the ´etalesite are weakly finitely generated. The above categories are locally presentable, so that the corresponding model categories are combinatorial. (viii) Symmetric monoidal structure. There is a symmetric monoidal structure on all the above categories of spectra, 0 where the product is denoted ∧. The sphere spectrum, (i.e. the inclusion of C0 into C) is the unit in this symmetric monoidal structure. Given a fixed spectrum E, the functor F 7→ E ∧ F has a right adjoint which will be denoted Hom or often Hom. This is the internal hom in the above categories of spectra. The derived functor of this Hom denoted RHom may be defined as follows. RHom(F,E) = Hom(C(F ),Q(E)), where C(F ) is a cofibrant replacement of F and Q(E) is a fibrant replacement of E. (ix) Ring spectra. Algebras in the above categories of spectra will be called these G-equivariant ring spectra (and ring-spectra for short.) Definition 3.14. (Stable homotopy groups). Let E denote a fibrant spectrum in any of the above categories of spec- tra and let H denote any subgroup of G so that H ε W and so that H acts trivially on the affine space V on which G acts linearly. Recall that since E is fibrant, the obvious map E(TV ) → ΩTW (E(TV ∧TW )) = HomC(TW ,E(TV ∧ k T )) is a weak-equivalence. Therefore, the k-th iteration of the above map, E(T ) → Ωk (E(T ∧ T ∧ )) is also W V TW V W a weak-equivalence. Therefore, we define

n n H s+2|TV |,TV H ∧ s H ∧ π (E(U)) = [Σ U+,E ] = lim [TV ∧ T ∧ S ∧ U+,E (T )], U ε (Sm/k0, G)?. s+2|TV |,TV |U n→∞ W |U W Here [A, B] denotes Hom in the homotopy category associated to the corresponding category of G-equivariant spectra and where H also acts trivially on W . Next assume that G denotes a profinite group and let HG denote the largest normal subgroup of G contained in H. (This is the core of H we considered earlier and has finite index in G.)

s+2|TV|,TV,H s+2|TV|,TV HG H (U+, E) = HomSp,G/HG (Σ U+, E|U ) =

n n ∧ s HG ∧ = lim HomSp,,G/H (TV ∧ T ∧ S ∧ U+,E (T )), U ε (Sm/k0, G)? n→∞ G W |U W

where HomSp denotes the internal hom-functor defined in 3.1.6 for G/HG-equivariant objects and ? denotes either s+2|TV|,TV,H s+2|TV|,TV,H the Nisnevich or the ´etalesites. We will use the notation Hmot (U+, E) (Het (U+, E)) to denote the corresponding hypercohomology when E ε Sptmot(k0, G, T)(E ε Sptet(k0, G, T), respectively).

1 0 Observe that if we restrict to P -motivic spectra, then a general TW ε C0 is a finite iterated smash product of terms of the form ∧ P1, for some normal subgroup K of G with finite index. When the group actions are ignored (or G/K 0 1 trivial), a general TW ε C0 is a finite iterated smash product of P , so that in this case the above stable homotopy groups may be just indexed by two integers. 3.5. Eilenberg-Maclane and Abelian group spectra. It is shown in [Dund2, Example 3.4], how to interpret the motivic complexes, Z = ⊕r≥0Z(r) and Z/l = ⊕r≥0Z/l(r) as ring-spectra in the above stable motivic homotopy category. While this is discussed there only in the non-equivariant framework, the constructions there extend verbatim to the G-equivariant setting.

This spectrum will be denoted HG(Z). The generalized cohomology (and homology) with respect to it in the sense defined in [CJ1, ], will define G-equivariant motivic cohomology (homology, respectively). HG(Z/l) will denote the mod−l variant for each prime l.

Oo With C denoting either PSh(Sm/k0, G) or PSh G (or one of their subcategories considered earlier), we let CAb denote the Abelian group objects in C. (Observe that these are nothing but simplicial Abelian presheaves together with a continuous action by G in the first case and diagrams of simplicial Abelian presheaves indexed by the orbit o category OG in the second case.)

0 Let C0 denote the subcategory

{TV |V = an affine space provided with an action by G, so that there exists an H ε W acting trivially on W }. 24 Gunnar Carlsson and Roy Joshua

0 0 Then we define an Abelian group spectrum A to be a functor A : C0 → CAb so that for each TV ,TW ε C0, one is pro- 0 vided with maps Ztr(TV ) ⊗ A(TW ) → A(TV ∧ TW ) and these are compatible as W and V vary in C0. The category of Abelian group spectra in C will be denoted SptAb(k0, G). If C = PShmot(Sm/k0, G)(C = PShdes(Sm/k0, G)) the corresponding category of abelian group spectra will be denoted SptAb,mot(k0, G)(SptAb,et(k0, G), respec- tively). The objects in this category will be called motivic Abelian group spectra (´etaleAbelian group 1 spectra, respectively). One may now define the P -suspension spectrum associated to the motive MG(X) as an 1 object in the above category of Abelian group spectra This will be denoted ΣP (MG(X)).

Let E ε SptAb,mot(k0, G). Then one also observes the following universal property for any X ε (Sm/k0, G):

(3.5.1) Γ(X, E) =∼ Hom (Σ 1 (M (X)), E) SptAb,mot(k0,G) P G

3.5.2. Spectra of Z/l-vector spaces.

(i) For a pointed simplicial presheaf P ε PSh(Sm/k0), Z/l(P ) will denote the presheaf of simplicial Z/l-vector spaces defined in [B-K, Chapter, 2.1]. Observe that the presheaves of homotopy groups of Z/l(P ) identify with the reduced homology presheaves of P with respect to the ring Z/l. Hence these are all l-primary torsion. The functoriality of this construction shows that if P has an action by the group G, then Z/l(P ) inherits this action. Moreover, if the action by G on P is continuous, then so is the induced action on Z/l(P ). Oo (ii) Next one extends the construction in the previous step to spectra. Let C = P sh G and let E ε Spectra(C). 0 Then first one applies the functor ( ) 7→ Z/l( ) to each E(TW ), TW ε C0. Then there exist natural maps Z/l(TV ) ⊗ Z/l(E(TW )) → Z/l(TV ∧ E(TW )). Therefore, one may compose the above maps with the obvious Z/l map Z/l(TV ∧ E(TW )) → Z/l(E(TV ∧ TW )) to define an object in SpectraZ/l(C). ∼= (iii) A pairing M ∧ N → P in PSh(Sm/k0, G) induces a pairing Z/l(M) ∧ Z/l(N)→Z/l(M) ⊗ Z/l(N) → Z/l Z/l(M ∧ N) → Z/l(P ). Similarly a pairing M ∧ N → P in Spectra(C) induces a similar pairing Z/l(M) ∧ ∼ Z/l(N)→= Z/l(M) ⊗ Z/l(N) → Z/l(P ). (To see this, one needs to recall the construction of the smash-product Z/l 0 0 of spectra from [Dund1, 2.3]: first one takes the point-wise smash-product M ∧ N : Sph(C0)xSph(C0) → C 0 0 0 and the one applies a left-Kan extension of this along the ∧ : Sph(C0) × Sph(C0) → Sph(C0). The functor Z/l commutes with left-Kan extension.) This shows that if R ε Spectra(C) is a ring spectrum so is Z/l(R) and that if M ε Spectra(C) is a module over the ring spectrum R, Z/l(M) is a module object over the ring object Z/l(R). 0 (iv) If {f : A → B} = JSp is a set of generating trivial cofibrations for Spectra(C), then, {Z/l(f): Z/l(A) → 0 Z/l(B)|f ε JSp} is a set of generating trivial cofibrations for SpectraZ/l(C). The category of spectra of Z/l-vector spaces in Cl will be denoted SpectraZ/l(C). If C = PShmot(Sm/k0, G)(C = PShdes(Sm/k0, G)) the corresponding category of spectra will be denoted SptZ/l,mot,F(k0, G)(SptZ/l,et,F(k0, G), respectively) where F is a functor C → C as in 3.3(i). These categories are also locally presentable and hence the 0 model categories are combinatorial. Applying Z/l( ) to the pairing TV ∧ E(TW ) → E(TV ∧ TW ), TV ,TW ε C0 and pre-composing with the pairing Z/l(TV ) ⊗ Z/l(E) → Z/l(TV ∧ E), one observes that the functor Z/l( ) induces a functor

Z/l(): Sptmot,F(k0, G) → SptZ/l,mot,F(k0, G) and Z/l(): Sptet,F(k0, G) → SptZ/l,et,F(k0, G).

(One also obtains a similar functor for the category of F (P1)-spectra.) 1 3.6. A -localization in the ´etalesetting. Recall that Sptet(k0, G) denotes the category of spectra on the big ´etalesite. We will let Sptet(k0) denote spectra with trivial action by the group G. Observe that the ´etalehomotopy type of affine-spaces over a separably closed field k, are trivial when completed away from char(k).

Proposition 3.15. Assume the base field k0 is separably closed and char(k0) = p. Let E ε Sptet(k0) be a constant sheaf of spectra so that all the (sheaves of) homotopy groups πn(E) are l-primary torsion, for some l 6= p. Then E is 1-local in Spt (k ), i.e. the projection P ∧ 1 → P induces a weak-equivalence: Map(P,E) ' Map(P ∧ A et 0 Ak0,+ 1 ,E), P ε Spt (k ). Ak0,+ et 0 Proof. First let P denote the suspension spectrum associated to some scheme X ε S. Then Map(P,E)(Map(P ∧ 1 ,E)) identifies with the spectrum defining the generalized ´etalecohomology of X (of X × 1 , respectively) Ak0,+ Ak0 with respect to the spectrum E. There exist Atiyah-Hirzebruch spectral sequences that converge to these generalized Equivariant motivic homotopy theory 25

´etalecohomology groups with the Es,t-terms being Hs (X, π (E)) and Hs (X × 1 , π (E)), respectively. Since 2 et −t et Ak0 −t the sheaves of homotopy groups π−t(E) are all l-torsion with l 6= p, and k0 is assumed to be separably closed, X and X × 1 have finite l-cohomological dimension. Therefore these spectral sequences converge strongly and Ak0 the conclusion of the proposition holds in this case. For a general simplicial presheaf P , one may find a simplicial resolution where each term is a disjoint union of schemes as above (indexed by a small set). Therefore, the conclusion of the proposition holds also for suspension spectra of all simplicial schemes and therefore for all spectra P . 

Corollary 3.16. Assume the base field k0 is separably closed and char(k0) = p. Let E ε Sptet(k0, G) be a constant sheaf of spectra so that all the (sheaves of) homotopy groups πn(E) are l-primary torsion, for some l 6= p. Then E is 1-local in Spt (k , G), i.e. the projection P ∧ 1 → P induces a weak-equivalence: Map (P,E) ' A et 0 Ak0,+ G Map (P ∧ 1 ,E) where Map is part of the simplicial structure on Spt (k , G). G Ak0,+ G et 0 Proof. The last proposition shows that the conclusion is true were it not for the group action. The functor Φ:¯ P 7→ {P H | H ε W} sending

Sptet(k0, G) to Π Sptet(k0) H ε W W has a left-adjoint defined by sending {Q(G/H) | H ε W} to Q(G/H) ∧ (G/H)+. This provides a triple, whereby one may find a simplicial resolution of a given P ε Sptet(k0, G) by spectra of the form Q ∧ G/H+ where ¯ Q ε Sptet(k0). (To see this is a simplicial resolution, it suffices to show this after applying Φ in view of the fact that a H map f : A → B in Sptet(k0, G) is a weak-equivalence if and only if the induced maps f are all weak-equivalences. On applying Φ,¯ the above augmented simplicial object will have an extra degeneracy.) Therefore, one reduces to considering P which are of the form Q ∧ G/H+. Then one obtains by adjunction, the following weak-equivalences: H 1 1 H MapG(Q ∧ G/H+,E) ' Map(Q, E ) and MapG(Q ∧ A+ ∧ G/H+,E) ' Map(Q ∧ A+,E ) where Map is part H 1 H of the simplicial structure on Sptet(k0). One obtains the weak-equivalence Map(Q, E ) ' Map(Q ∧ A+,E ) by the last proposition. This completes the proof of the corollary. 

4. Effect of A1-localization on ring and module structures and on mod-l completions We make strong use of ring and module structures on spectra in [CJ1] and sequels. Therefore, it is important to know that the process of A1-localization is (or can be made to be) compatible with these structures. Similarly completions, especially at a prime l, plays a key role in our work: therefore, it is again important to show that such completions may be carried out so as to be compatible with the process of A1-localization. Since the arguments needed to show both of these have several common features, we will present the common arguments in a unified manner.

4.0.1. Therefore, we will let Spt(k0, G) denote either Spectrast(C) as in 3.3.4 or SptZ/l,mot(k0, G). The first observation is that both categories are locally presentable. Therefore, there exists a set {Gα|α} so that any object P ε SptZ/l,mot(k0, G) is the filtered colimit of a diagram involving the Gα, i.e. P = limGα. We will let ⊗ (⊕) → denote the monoidal product (sum, respectively) in these categories.

0 0 0 Lemma 4.1. Let f : A → B, f : A → B denote two maps in Spt(k0, G) that are trivial cofibrations and let 0 0 0 0 g : A → Q, g : A → Q be maps in Spt(k0, G) with Q, Q be among the {Gα | α} considered above. Assume that 0 0 0 0 f ⊗idQ and f ⊗idQ0 are both trivial cofibrations. Then (i) the induced map Q⊗A ⊕A⊗A0 A⊗Q → Q⊗B ⊕A⊗A0 B⊗ 0 0 0 0 0 0 Q is a trivial cofibration and (ii) so is the induced map Q⊗B ⊕A⊗A0 B⊗Q → (Q⊗B )⊕A⊗A0 B⊗Q ⊕A⊗A0 B⊗B . Proof. The proof of (i) follows from the commutative diagram where all the squares are co-cartesian:

A ⊗ A0 / A ⊗ Q0 / B ⊗ Q0

0  0 0  0  0 / Q ⊗ A ⊕ A ⊗ Q / Q ⊗ A ⊕ B ⊗ Q Q ⊗ A A⊗A0 A⊗A0

0  0 0  0  0 / Q ⊗ B ⊕ A ⊗ Q / Q ⊗ B ⊕ B ⊗ Q Q ⊗ B A⊗A0 A⊗A0 26 Gunnar Carlsson and Roy Joshua

The left arrow and the top arrow in the bottom right-most square are trivial cofibrations and hence so are the other arrows in the same square. The required map in (i) is the composition of the left arrow and the bottom arrow in the same square, so that it is a trivial cofibration.

(ii) follows from the commutative diagram

A ⊗ A0 / B ⊗ B0 / C

0  0 Q ⊗ B ⊕ B ⊗ Q /  /  0 A⊗A0 P C

The first arrow in the top row is a cofibration, so that so is the first arrow in the bottom row. Therefore, it follows that C0 =∼ C. Since the first arrow in the top row is also a trivial cofibration, C and hence C0 is acyclic. Therefore, the first arrow in the bottom row is also a weak-equivalence, thereby proving (ii). 

0 Let J denote the set of generating trivial cofibrations chosen as in spectra.construct.2 for both Spectrast(C) and Sptmot(k0, G). 0 0 0 Lemma 4.2. Let Ai → Ai, i = 1, 2 be trivial cofibrations in Spt(k0, G) and let fi : A0 → Ai, i = 1, 2 be given. 0 0 0 Assume that g : A → A3 is also a trivial cofibration in Spt(k0, G). (i) Then the induced map A = A ⊕A → 0 1 0 2 A0 0 0 B = A1 ⊕A2 and the induced map B = A ⊕A → C = A1 ⊕A2 ⊕A3 are trivial cofibrations. 0 1 0 2 0 0 A0 A0 A0 A0

(ii) If Q ε Spt(k0, G) so that fi ⊗ idQ, for i = 1, 2 and g ⊗ idQ are trivial cofibrations, then the obvious induced maps A ⊗ Q → B ⊗ Q and B ⊗ Q → C ⊗ Q are trivial cofibrations.

0 (iii) If Spt(k0, G) denotes SptZ/l,mot(k0, G) and each fi and g belong to Z/l(JSp) then the hypothesis of (ii) holds. Proof. The proof of (i) is identical to the proof of the corresponding statement in the previous lemma. (ii) follows 0 0 from (i) by replacing the Ai and Ai in (i) with Ai ⊗ Q and Ai ⊗ Q. Then the hypothesis in (ii) ensures that the 0 hypothesis in (i) holds for Ai ⊗ Q.

For (iii), it suffices to prove the following: if f : A → B is a trivial cofibration with A and B cofibrant, Z/l(f)⊗idQ is a weak-equivalence. Let Qc → Q denote a cofibrant replacement. Then one obtains the commutative square:

Z/l(A) ⊗ Qc / Z/l(B) ⊗ Qc

  Z/l(A) ⊗ Q / Z/l(B) ⊗ Q

The top row is a weak-equivalence, since Qc is cofibrant. The vertical maps are weak-equivalences because both A and B and hence Z/l(A) and Z/l(B) are cofibrant. It follows the bottom row is also a weak-equivalence. 

In order to show that A1-localization can be carried out to be compatible with ring and module structures on 0 spectra and also with mod-l completions, it is necessary to enlarge the set JSp as follows. We define a sequence 0 JSp(i), i ≥ 0 of increasing sets of trivial cofibrations as follows. When Spt(k0, G) = SptZ/l,mot(k0, G) we let 0 0 0 0 JSp(0) = Z/l(JSp). When Spt(k0, G) = Spectrast(C), we let JSp(0) = JSp chosen as in spectra.construct.2. 0 Having defined JSp(i), for all 0 ≤ i ≤ n, we define

00 0 0 0 0 0 0 0 0 JSp(n + 1) = {Q ⊗ B ⊕A⊗A0 B ⊗ Q → (Q ⊗ B ⊕A⊗A0 B ⊗ Q ) ⊕A⊗A0 B ⊗ B |A → B,A → B ε JSp(n), g : A → 0 0 0 0 Q, g : A → Q inSpt(k0, G), Q, Q ε {Gα|α}} and

0 0 00 0 JSp(n + 1) = JSp(n) ∪ JSp(n + 1) together with all finite sums of objects in JSp(n + 1).

0 One may now prove using ascending induction on n that the domain and co-domain of the maps in JSp(n) are 0 0 0 finitely presented. We will replace JSp with ∪n≥0JSp(n) and continue to denote this larger set by JSp. Equivariant motivic homotopy theory 27

The following lemma is straightforward to prove and is therefore left to the reader. 0 Lemma 4.3. (i) The domain and co-domain of the maps in each JSp(n) are finitely presented.

0 0 (ii) The maps in each JSp(n) are among the trivial cofibrations generated by starting with JSp defined as in 3.3.4 and by iterating the following operations finitely many times: finite coproducts, co-base-change by arbitrary maps 0 and pushout products of the form fg, where f is already a trivial cofibration generated by starting with JSp as in 3.3.4 and g = idQ, where Q is among the small generators of Spt(k0, G). 4.0.2. The basic construction.

(i) For each Y ε Spt(k0, G), we let S(Y ) denote the set of all commutative squares

A / Y

f   B / 0

0 where f ε JSp. We let P(S(Y ))f denote the set of all finite subsets of S(Y ). (ii) For each finite subset T ⊆ S, we let PT (Y ) = (⊕B) ⊕ Y where the ⊕B (and ⊕A) are over the finite set T . ⊕A 0 Then we let P (Y ) = colim PT (Y ) = (⊕B) ⊕ Y where the sum is over all elements of S(Y ). T ε P(S(Y ))f (⊕A) 0 (iii) Let Y = colim Gα where J(Y ) is some filtered set. Then we let P (Y ) = colim P (Gα) α ε J(Y ) α ε J(Y ) = colim colim PT (Gα). Observe that then, P (Y ) = (⊕B) ⊕ Y where the ⊕B (and ⊕A) are over the α ε J(Y )T ε P(S(Gα))f (⊕A) set S(Y ).

Now we start with an E ε SptZ/l,mot(k0, G) and will construct a factorization of the map Z/l(E) → ∗ into Z/l(E) → F (E) → ∗ , where the first map is a cofibration (that is an A1-equivalence) followed by an A1-local fibration all in the category SptZ/l,mot(k0, G). In case E ε Spectrast(C), then we instead construct a factorization of the map E → ∗ into E → F (E) → ∗, where the first map is a cofibration (that is an A1-equivalence) followed by an A1-local 0 fibration all in the category Spectrast(C). Let F (E) = Z/l(E) in the first case and = E in the second case. The factorization is obtained as in the diagram (4.0.3) F 0(E) / F 1(E) / F 2(E) / ··· / F β(E) / ··· kkkk UUUU LL kkk UU p0 L p1 p kk UUUU LL 2 kkkpβ UUU LL kkkk UUUU LLL  u U* ∗ =% Zl/(∗)

β 0 Assume we have constructed the above diagram upto F (E) all in Spt(k0, G) beginning with F (E). To continue 0 β the construction, we consider maps from families of maps A → B in JSp to (F (E)) → ∗ = Z/l(∗). One then β+1 β β+1 β lets F (E) = be defined by the pushout (⊕B) ⊕ F (E) in the category Spt(k0, G). i.e. F (E) = P (F (E)). ⊕A γ β For a limit ordinal γ, F (E) is defined as limF (E) with the colimit taken in the same category Spt(k0, G). We → β<γ obtain the required factorization by taking F (E) = limF β(E) and with the obvious induced maps Z/l(P ) → F (E) → β<λ 0 and E → ∗. Here λ is a sufficiently large enough ordinal: if the domains of JSp are κ-small for some ordinal κ, then λ is required to be κ-filtered. This means λ is a limit ordinal and that if A ⊆ λ and |A| ≤ κ, then supA < λ.

1 It is clear from the construction that F (E) ε Spt(k0, G). That it is A -local follows from the observation that 0 the domain of any object of JSp is finitely presented. Henceforth we will denote the above object F (E) as Z/l^(E) 1 in the first case and as RA (E) in the second case.

Proposition 4.4. The functor E → Z/lg(E): Sptmot(k0, G) → SptZ/l,mot(k0, G) is left-adjoint to the forgetful functor U : SptZ/l,mot(k0, G) → Sptmot(k0, G).

Proof. We will start with a map G → U(Ee) in Sptmot(k0, G), where Ee ε SptZ/l,mot(k0, G). This corresponds to map Z/l(G) → Ee of Z/l-module spectra. We proceed to show that this map induces a map Z/lg(G) → Ee so 28 Gunnar Carlsson and Roy Joshua that pre-composing with the map Z/l(G) → Ee is the map Z/l(G) → Ee. We will show inductively the above map Z/l(G) → Ee extends to a map F β(G) → Ee for all β < λ. Assume that β is such an ordinal for which we β have extended the given map Z/l(G) → Ee to a map F (G) → Ee. Let {Aα → Bα|α} denote a family of trivial 0 β cofibrations belonging to JZ/l,Sp indexed by a small set so that one is provided with a map ⊕αAα → F (G). Since

Ee is a fibrant object of SptZ/l,mot(k0, G), and each map Aα → Bα is a generating trivial cofibration in the same category, one obtains an extension ⊕αBα → Ee.

β+1 β Next recall that F (G) = (⊕αBα) ⊕ F (G). Therefore, this has an induced map of Z/l-module spectra (⊕αAα) to Ee. Since F γ (G) for a limit ordinal γ < λ is defined as lim F β(G), it follows that we obtain the required → β<γ extension of the given map to a map Z/lg(G) → Ee. 

Proposition 4.5. Assume the above situation. Then Z/l^(E) is Z/l-complete in the following sense. For every map ∼= φ : A → B in Spectra(C) with both A and B cofibrant and which induces an isomorphism H∗(A, Z/l)→H∗(B, Z/l) of the homology presheaves with Z/l-coefficients, then the induced map φ∗ : Map(B,U(Z/l^(E))) → Map(A, U(Z/lg(E))) is a weak-equivalence of simplicial sets. Proof. This follows readily from the following observations: (i) such a homology isomorphism induces a weak- equivalence Z/l(A) → Z/l(B) and (ii) Map(B,U(Z/l^(E))) ' Map(Z/l(B), Z/l^(E)) and Map(A, U(Z/l^(E))) '

Map(Z/l(A), Z/l^(E)) where the Map on the left-side (right-side) is taken in the category Sptmot(k0, G) (SptZ/l,motk0, G) , respectively). One may observe that the functor Z/l( ) preserves cofibrant objects and since Z/l^(E) is a fibrant object in SptZ/l,mot(k0, G), the weak-equivalence Z/l(A) → Z/l(B) induces the weak- equivalence Map(Z/l(B), Z/l^(E)) ' Map(Z/l(A), Z/l^(E)). 

4.0.4. Proposition 4.4 above shows that {Z/lgn(E)|n ≥ 1} provides a cosimplicial object in Sptmot(k0, G) (with

Z/lg(E) in degree 0) which is group-like, i.e. each Z/lgn(E) belongs to SptZ/l,mot(k0, G) and that the structure maps of the cosimplicial object except for d0 are Z/l-module maps. Such a cosimplicial object is fibrant in the Reedy model structure on the category of cosimplicial objects of SptZ/l,mot(k0, G): see for example, [B-K, Chapter 10, 4.9 proposition]. Therefore, one may take the homotopy inverse limit of this cosimplicial object just as in [B-K, Chapter X].

In view of the observation, it follows that if Z/lgi≤n(E) denotes the corresponding truncated cosimplicial object, truncated to degrees ≤ n, one obtains a compatible collection of maps {holim Z/lg•(E) → holim Z/lgi≤n(E)|n ≥ 1}. 0 n−1 Moreover the fiber of the map holim Z/lgi≤n(E) → holim Z/lgi≤n−1(E) identifies with ker(s ) ∩ · · · ∩ ker(s ), i where s : Z/lgn(E) → Z/lgn−1(E) is the i-th co-degeneracy.

0 Definition 4.6. (Z/l-completions.) First we will extend the set JSp as above. We will then apply the con- struction above to define Z/l^(P ). One repeatedly applies the functor Z/lg to an object E ε Spectra(C), to de-

fine the tower {(Z/l^)n(E)|n} in Spectra(C). We let Z/lg∞(E) = holimn{(Z/l^)n(E)|n}. In contrast, we let Z/l∞(E) = holimn{(Z/l)n(E)|n}.

0 00 Proposition 4.7. (i) Let µ : E ∧ E → E denote a pairing of spectra in Sptmot(k0, G). Then µ induces a pairing 0 00 0 00 Z/lgn(E) ∧ Z/lgn(E ) → Z/lgn(E ) for all n ≥ 1 and a pairing Z/lg∞(E) ∧ Z/lg∞(E ) → Z/lg∞(E ). 0 00 1 (ii) Let µ : E ∧ E → E denote a pairing in Spectrast(C). Then one obtains an induced pairing RA µ : 0 00 1 1 1 RA E ∧ RA E → RA E . The above pairings are coherently associative. Proof. Since the proof of the second assertion is entirely similar to that of the first, we will only prove (i). Assume that Z/l^(E), Z/l^(E0) and Z/l^(E00) are defined by Z/l^(E) = lim{F β(E) | β}, Z/l^(E0) = colim{F β(E0) | β} and → Z/l^(E00) = lim{F β(E00) | β} and that we have already shown the existence of a pairing F β(E) ⊗ F β(E0) → F β(E00) → Z/l Equivariant motivic homotopy theory 29 compatible with the given pairing Z/l(E) ⊗ Z/l(E0) → Z/l(E00) for a non-limit ordinal β. Since every object in Sptmot(k0, G) is a filtered colimit of a subset of the finitely presented objects denoted {Gα | α} as in 4.0.1, we β β 0 β 00 may replace F (E)(F (E ), F (E )) by Gα (Gβ, Gγ , respectively) together with a pairing Gα ⊗ Gβ → Gγ . Recall next that

β+1 0 F (Gα) = P (Gα) = colim PT (Gα). Observe that then, PT (Gα) = (⊕B) ⊕ Gα T ε P(S(Gα))f (⊕A) where the ⊕B (and ⊕A) are over the finite subset T ⊆ S(Gα). Similarly

β+1 0 0 F (Gβ) = P (Gβ) = colim PT 0 (Gβ). Observe that then, PT 0 (Gβ) = (⊕B ) ⊕ Gβ 0 T ε P(S(Gβ ))f (⊕A0)

0 where the ⊕B (and ⊕A) are over the finite subset T ⊆ S(Gβ) and

β+1 0 00 F (Gγ ) = P (Gγ ) = colim PT 00 (Gγ ). Observe that then, PT 00 (Gγ ) = (⊕B ) ⊕ Gγ 00 T ε P(S(Gγ ))f (⊕A00)

00 where the ⊕B (and ⊕A) are over the finite subset T ⊆ S(Gγ ). Therefore, in order to show that there is an β+1 β+1 0 β+1 00 induced pairing F (E) ⊗ F (E ) → F (E ), it suffices to show that for each finite subsets T ⊆ S(Gα) and 0 00 T ⊆ S(Gβ) there exists a finite subset T ⊆ S(Gγ ) together with a pairing PT (Gα) ⊗ PT 0 (Gβ) → PT 00 (Gγ ).

0 0 One may now identify PT (Gα) ⊗ PT 0 (Gβ) with ((Gα ⊗ B ⊕A⊗A0 B ⊗ Gβ) ⊕A⊗A0 B ⊗ B ) ⊕A⊗A0 Gα ⊗ Gβ.

0 0 0 Now Lemma 4.1 with Q = Gα, Q = Gβ shows that Gα ⊗ A ⊕A⊗A0 A ⊗ Gβ → Gα ⊗ B ⊕A⊗A0 B ⊗ Gβ is a 0 0 0 trivial cofibration and (ii) so is the induced map Gα ⊗ B ⊕A⊗A0 B ⊗ Gβ → (Gα ⊗ B ) ⊕A⊗A0 B ⊗ Gβ ⊕A⊗A0 B ⊗ B . 0 Moreover these are in JSp,Z/l. i.e. We obtain the following pushout:

0 / Gα ⊗ A ⊕A⊗A0 A ⊗ Gβ Gα ⊗ Gβ

0  0 /  (Gα ⊗ B ) ⊕A⊗A0 B ⊗ Gβ ⊕A⊗A0 B ⊗ B PT 00 (Gα ⊗ Gβ)

0 0 One may see directly that, since A ⊗ A maps naturally to Gα ⊗ A ⊕A⊗A0 A ⊗ Gβ, that there is a natural map PT (Gα) ⊗ PT 0 (Gβ) → PT 00 (Gα ⊗ Gβ) and the latter maps naturally to PT 00 (Gγ ) via the map induced by the given map Gα ⊗ Gβ → Gγ . For a limit ordinal γ, one may take the colimit of the above pairings for β < γ.

We have therefore proven the existence of a pairing Z/l^(E) ∧ Z/l^(E0) → Z/l^(E) ⊗ Z/l^(E0) → Z/l^(E00). We may now repeat the above constructions to obtain the pairing

0 00 Z/l^n(E) ∧ Z/l^n(E ) → Z/l^n(E ) for all n ≥ 0.

0 00 One may view this as a pairing Z/l^n(E) × Z/l^n(E ) → Z/l^n(E ) that is Z/l-bi-linear for all n ≥ 0. Next one takes the homotopy inverse limit in the category Sptmot(k0, G): the homotopy inverse limit commutes with products, so that one obtains an induced pairing

0 00 Z/lg∞(E) ∧ Z/lg∞(E ) → Z/lg∞(E ). 

Completions will be most often applied to pro-objects in Spectra(C). Let X = {Xi|i ε I} ε pro − Spectra(C) denote a pro-object indexed by a small category I. Let E ε Spectra(C). Then we let

(4.0.5) HomSp(X, Z/lg∞(E)) = holimncolimI HomSp(Xi, (Z/l^)n(E))

(4.0.6) Map(X, Z/lg∞(E)) = holimncolimI Map(Xi, (Z/l^)n(E))

Here Hom denotes the internal hom in the category Spectra(C).

4.0.7. Properties of the completion functor. Here we list a sequence of key properties of the completion functor so as to serve as a reference. 30 Gunnar Carlsson and Roy Joshua

(i) The Z/l-completion, Z/l^∞(E) is Z/l-complete. i.e. (i) it is fibrant in Sptmot(k0, G) and for every map φ : ∼= A → B in Sptmot(k0, G) between cofibrant objects which induces an isomorphism H∗(A, Z/l)→H∗(B, Z/l) of ∗ the homology presheaves with Z/l-coefficients, the induced map φ : Map(B, Z/l^∞(E)) → Map(A, Z/l^∞(E)) is a weak-equivalence of simplicial sets.

(ii) If R ε Sptmot(k0, G) is a ring spectrum so are each Z/l^n(R) and Z/l∞(R). If R ε Sptmot(k0, G) is a ring

spectrum and M ε Sptmot(k0, G) is a module spectrum over R, then each Z/l^n(M) is a Z/l^n(R)-module

spectrum and Z/lg∞(M) is a Z/lg∞(R)-module spectrum.

(iii) If R ε Sptmot(k0, G) is a ring spectrum, each Z/l^n(R) is a module spectrum over Z/l∞(R). (This property may be deduced from the discussion in in the second paragraph of 4.0.4.) 1 (iv) For each E ε Sptmot(k0, G), each Z/l^n(E) is A -local and belongs to SptZ/l,mot(k0, G, ?). (v) For E ε Sptet(k0, G) and let Z/l(E) denote the free Z/l-vector space functor applied to E. Then Z/ln(E) = Z/l(U(Z/ln−1(E))) defines the n-th term of the tower defining the usual Bousfield-Kan Z/l-completion. One

obtains a natural map {Z/ln(E) → Z/l^n(E)|n} of towers compatible with the induced multiplication if E is a ring spectrum.

In this paper completions play a key role. This is to be expected since even the ´etalehomotopy type of schemes has good properties only after completion away from the residue characteristics. We will adapt the Bousfield-Kan completion (see [B-K]) to our framework as follows.

First recall that the Bousfield-Kan completion is defined with respect to a commutative ring with 1: in our framework, this ring will be always Z/l for a fixed prime l different from char(k0). Therefore, we will work with category SptZ/l,mot(k0, G). Al our arguments and constructions work also on SptZ/l,et(k0, G), but we do not discuss this explicitly.

1 The last propositions show that each Z/l(E) is already A -local in the ´etalesetting, i.e. with k0 separably closed and where the ´etaletopology is used and E is a constant sheaf. In the motivic framework, i.e. when the topology used is the Nisnevich topology, and/or when E is not constant, one needs to modify the above definitions so that each Z/l(E) is also A1-local. We proceed to do this presently, but we will digress to develop a bit of the technicalities.

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Department of Mathematics, Stanford University, Building 380, Stanford, California 94305 E-mail address: [email protected]

Department of Mathematics, Ohio State University, Columbus, Ohio, 43210, USA. E-mail address: [email protected]