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 " W and H0 ⊇ H is a subgroup of G, then H0 " 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 " Γ(U; P ) and P " 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 = fG=H × i j i a cofibration of PSh(Sm=k0); H " Wg, (ii) the generating trivial cofibrations are of the form Equivariant motivic homotopy theory 3 JG = fG=H × j j j a trivial cofibration of PSh(Sm=k0); H ⊆ G; H " Wg 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 " 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 " 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 = fG=H j H " Wg. A morphism G=H ! G=K corresponds to γ " 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 ) = fΦ(F )(G=H) = F g and Oo Θ : PSh G ! PShc(Sm=k0; G);M 7! Θ(M) = lim M(G=H). ! fHjH " Wg 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 " 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].