EXPLICIT FIBRANT REPLACEMENT FOR DISCRETE G-SPECTRA DANIEL G. DAVIS Abstract.If C is the model category of simplicial presheaves on a site w* *ith enough points, with fibrations equal to the global fibrations, then it i* *s well- known that the fibrant objects are, in general, mysterious. Thus, it is* * not surprising that, when G is a profinite group, the fibrant objects in the* * model category of discrete G-spectra are also difficult to get a handle on. Ho* *wever, with simplicial presheaves, it is possible to construct an explicit fibr* *ant model for an object in C, under certain finiteness conditions. Similarly, in t* *his paper, we show that if G has finite virtual cohomological dimension and X is a * *discrete G-spectrum, then there is an explicit fibrant model for X. Also, we give* * several applications of this concrete model related to closed subgroups of G. 1.Introduction In this paper, G always denotes a profinite group and, by "spectrum," we mean a Bousfield-Friedlander spectrum of simplicial sets. In particular, a discrete* * G- spectrum is a G-spectrum such that each simplicial set Xk is a simplicial objec* *t in the category of discrete G-sets (thus, the action map on the l-simplices, G x (Xk)l! (Xk)l, is continuous when (Xk)lis regarded as a discrete space, for all l 0). The ca* *tegory of discrete G-spectra, with morphisms being the G-equivariant maps of spectra, * *is denoted by SptG. As shown in [3, Section 3], SptGis a simplicial model category, where a morph* *ism f in SptG is a weak equivalence (cofibration) if and only if f is a weak equiva* *lence (cofibration) in Spt, the simplicial model category of spectra. Given X 2 SptG,* * the homotopy fixed point spectrum XhG is the total right derived functor of fixed p* *oints: XhG = (Xf,G)G , where X ! Xf,G is a trivial cofibration and Xf,G is fibrant, all in SptG. This definition generalizes the classical definition of homotopy fixed* * point spectrum, in the case when G is a finite group (see [3, pg. 337]). Notice that we can loosen up the requirements on Xf,G. If X ! Xf is a weak equivalence, with Xf fibrant, all in SptG, then, by the right lifting property * *of a fibrant object, there is a weak equivalence Xf,G! Xf in SptG, so that XhG = (Xf,G)G ! (Xf)G is a weak equivalence. Thus, we can identify XhG and (Xf)G , and, hence, we only need a fibrant replacement Xf to form XhG . Henceforth, we relabel any such Xf as Xf,G and refer to it as a globally fibrant model for X. (Thus, from now on, X ! Xf,Gdoes not have to be a cofibration.) The preceding discussion shows that a globally fibrant model Xf,Gis an impor- tant object. Of course, the model category axioms guarantee that Xf,G always 1 2 DANIEL G. DAVIS exists. But it is reasonable to ask for more. For example, in Spt, there is a f* *unctor Q: Spt! Spt, Z 7! Q(Z) = Zf, where Zf is a fibrant spectrum, (Zf)k = colimn n(Ex 1(Zk+n)), and there is a natural weak equivalence Z ! Zf (see, for example, [13, pg. 524]* *). Hence, for the model category Spt, there is always an explicit model for fibrant replacement. Similarly, it is natural to wonder if an explicit model for Xf,G * *is available. But there is a difficulty with this. Let G-Setsdfbe the Grothendieck site of * *finite discrete G-sets (e.g., see [9, Section 6.2]). There is an equivalence between S* *ptG and the category of sheaves of spectra on G-Setsdf (the discrete G-spectrum X corresponds to the sheaf of spectra Hom G(-, X); see [3, Section 3] for details* *), and it is well-known that, in general, for categories of simplicial presheaves, pre* *sheaves of spectra, and sheaves of spectra, there is no known explicit model for a glob* *ally fibrant object. In fact, the situation is such that [7, pg. 1049] says that "* *[t]he fibrant objects in all of these theories continue to be really quite mysterious* *" (a similar statement appears in [10, between Corollary 19 and Definition 20]). Nevertheless, under certain hypotheses, explicit models for globally fibrant * *ob- jects are available in the cases of simplicial presheaves and presheaves of spe* *ctra. Such results are based on Jardine's result in [11, Proposition 3.3], which cons* *tructs an explicit globally fibrant model for a simplicial presheaf P on the site 'et * *|S, where P and the scheme S must satisfy certain finiteness conditions (and other hypoth* *e- ses). For example, under similar finiteness conditions, [12, Proposition 3.20] * *follows the proof of Jardine's result to obtain a concrete globally fibrant model for a* * presheaf of spectra on a site with enough points. Now suppose that G has finite virtual cohomological dimension (see Definition 4.1) and that X is a discrete G-spectrum. In this paper, we show that there is * *an explicit model for Xf,G, by expressing the homotopy limit for diagrams in SptG in terms of the homotopy limit for diagrams in Spt (see Theorem 2.3) and by modifying the proof of [3, Theorem 7.4] (which applies the two results cited ab* *ove, [11, Proposition 3.3] and [12, Proposition 3.20]). We refer the reader to Theor* *em 4.2 for the precise statement of our main result; its formulation depends on defini* *tions that are given in Section 3. Let H be a closed subgroup of G. If Y ! Z is a weak equivalence in SptH, such that Y is a globally fibrant model for X in SptG and Z is a globally fibrant mo* *del for X in SptH, then we label the map Y ! Z as rGH. Note that the map Xf,G! (Xf,G)f,H, a weak equivalence in SptH, can be labelled as rGH, so that rGHalways exists. * *In Corollary 4.7, we show that the explicit globally fibrant model constructed in * *The- orem 4.2 yields an explicit model for rGH(where, as before, we assume that G has finite virtual cohomological dimension). Section 5 explains that, when H is a closed normal subgroup of G and X is a discrete G-spectrum, there are cases when XhH , unlike XH , is not known to be a G=H-spectrum. In Corollary 5.4, we point out that, if G has finite virtual cohomological dimension, then Theorem 4.2 implies that XhH can always be taken to be a G=H-spectrum. EXPLICIT FIBRANT REPLACEMENT FOR DISCRETE G-SPECTRA 3 Throughout this paper, U 0, by [3, Section 7]. Thus, the spec* *tral sequence collapses, so that the map b_is a weak equivalence. Now let H be a closed subgroup of G. Then X is a discrete H-spectrum, so that X ~=colimKCoH XK . Composing this isomorphism with the map colimKCoH(_b)K gives the H-equivariant map : X ! colimKC(holim oGbX)K . oH EXPLICIT FIBRANT REPLACEMENT FOR DISCRETE G-SPECTRA 9 Now we show that if G has finite vcd, then is a weak equivalence. As mentio* *ned in the Introduction, the proof (below) closely follows the proof of [3, Theorem* * 7.4], so that our proof will be somewhat abbreviated. Also, we should mention that the proof of [3, Theorem 7.4] follows the arguments given in [11, proof of Proposit* *ion 3.3] and [12, Proposition 3.20]. Theorem 4.2. Let G have finite vcd, let X be a discrete G-spectrum, and let H be a closed subgroup of G. Then the map : X ! colimKC(holim oGbX)K oH is a weak equivalence in the category of discrete H-spectra, such that the targ* *et is a fibrant discrete H-spectrum. Proof.Because of the earlier Theorem 3.5, we only have to prove that is a weak equivalence of spectra. Since H is closed in G, H also has finite vcd. Hence, H has a collection {U} * *of open normal subgroups such that (a) {U} is a cofinal subcollection of {K}KCoH (so, for example, H ~=limUH=U) and (b) for all U, Hsc(U; M) = 0, for all s m, where m is some natural number that is independent of U, and for all discrete U-modules M. Thus, (4.3) colimKC(holim oGbX)K ~=colim(holim oGbX)U , oH U so that, to show that is a weak equivalence, it suffices to show that the map b:X ! colim(holim o bX)U ~=colimholim( o bX)U , U G U G induced by and (4.3), is a weak equivalence. Notice that each U is a closed subgroup of G. Then, for each U, ( oGbX)U is a cosimplicial fibrant spectrum, so that there is a conditionally convergent homo* *topy spectral sequence (4.4) Es,t2(U) = ssssst(( oGbX)U ) ) sst-s(holim( oGbX)U ), with Es,t2(U) ~=Hsc(U; sst(X)) (these assertions are verified in the proof of [3, Lemma 7.12]). Since Es,*2(U) = 0 whenever s m, the E2-terms E*,*2(U) are uniformly bounded on the right. Therefore, by [12, Proposition 3.3], taking a colimit over {U} of* * the spectral sequences in (4.4) gives the spectral sequence (4.5) Es,t2= colimUHsc(U; sst(X)) ) sst-s(colimUholim( oGbX)U ). Notice that E*,t2~=H*c(limUU; sst(X))~=H*c({e}; sst(X)), which is isomorphic to sst(X), concentrated in degree zero. Thus, spectral sequ* *ence (4.5) collapses, so that, for all t, sst(colimUholim ( oGbX)U ) ~=sst(X), and, * *hence, b is a weak equivalence. Let X be a C-diagram of discrete G-spectra, where C is a small category. Then, by Theorem 2.3, there is a canonical map OE(X, G): holimGCX ~=colimNC(holimX)N ! holimX oG C C 10 DANIEL G. DAVIS that is G-equivariant. Corollary 4.6. Let G have finite vcd, let X be a discrete G-spectrum, and let H be a closed subgroup of G. Then the H-equivariant map OE( oGbX, H): holimH oGbX! holim oGbX is a weak equivalence in Spt. Proof.Notice that b_= OE( oGbX, H) O . Then the desired conclusion follows from the fact that b_and are weak equivalences, where the latter fact is from Theo* *rem 4.2. In the Introduction, we pointed out that a weak equivalence rGH:Xf,G! Xf,H in SptH always exists. The following result uses Theorem 4.2 to give a concrete model for rGH. Corollary 4.7. Let G have finite vcd, let X be a discrete G-spectrum, and let H be a closed subgroup of G. Then there is a weak equivalence rGH:colimNC(holim oGbX)N ! colim(holim oGbX)K oG KCoH in SptH , where the source of this map is a fibrant discrete G-spectrum and the target is a fibrant discrete H-spectrum. Proof.Let N be an open normal subgroup of G. Then N \ H is an open normal subgroup of H and, hence, there is a canonical map (holim oGbX)N ,! (holim oGbX)N\H ! colimKC(holim oGbX)K . oH These maps, as N varies, induce the desired map, which is easily seen to be H- equivariant. In SptH, the weak equivalence X ! colimKCoH(holim oGbX)K is the composition of the weak equivalence X ! colimNCoG(holim oGbX)N and rGH, so that rGHis a weak equivalence. The following result is a special case of the fact that, if H is open in G, t* *hen a fibrant discrete G-spectrum is also fibrant as a discrete H-spectrum (see [4, L* *emma 3.1] and [9, Remark 6.26]). Corollary 4.8. Let G have finite vcd and let X be a discrete G-spectrum. If H is an open subgroup of G, then colimNCoG(holim ( oGbX))N , a fibrant discrete G- spectrum, is also a fibrant discrete H-spectrum. Proof.By Theorem 4.2, the spectrum colimKCoH(holim ( oGbX))K is a fibrant dis- crete H-spectrum. Thus, to verify the corollary, it suffices to show that this * *fibrant discrete H-spectrum is isomorphic to colimNCoG(holim ( oGbX))N in SptH. Note that if U is an open subgroup of H, then U is also an open subgroup of G, so that {H \ V | V