THE SITE R+G FOR A PROFINITE GROUP G DANIEL G. DAVIS Abstract. Let G be a non-finite profinite group and let G - Setsdfbe th* *e canonical site of finite discrete G-sets. Then the category R+G, defined by Devina* *tz and Hopkins, is the category obtained by considering G - Setsdftogether with the prof* *inite G-space G itself, with morphisms being continuous G-equivariant maps. We show th* *at R+Gis a site when equipped with the pretopology of epimorphic covers. Also, we e* *xplain why the associated topology on R+Gis not subcanonical, and hence, not canonical.* * We note that, since R+Gis a site, there is automatically a model category structure on* * the category of presheaves of spectra on the site. Finally, we point out that such presh* *eaves of spectra are a nice way of organizing the data that is obtained by taking the hom* *otopy fixed points of a continuous G-spectrum with respect to the open subgroups of * *G. 1. Introduction Let G be a profinite group that is not a finite group. Let R+Gbe the category w* *ith objects all finite discrete left G-sets together with the left G-space G. The morphisms* * of R+Gare the continuous G-equivariant maps. Since G is not finite, the object G in R+Gi* *s very different in character from all the other objects of R+G. In this paper, we sho* *w that R+Gis a site when equipped with the pretopology of epimorphic covers. As far as the author knows, the category R+Gis first defined and used in the* * paper [Devinatz and Hopkins, 2004], by Ethan Devinatz and Mike Hopkins. Let Gn be the* * profi- nite group Sn o Gal(Fpn=Fp), where Sn is the nth Morava stabilizer group. In [D* *evinatz and Hopkins, 2004, Theorem 1], Devinatz and Hopkins construct a contravariant f* *unctor - that is, a presheaf - F: (R+Gn)op ! (E1 )K(n), to the category (E1 )K(n) of K(n)-local commutative S-algebras (see [Elmendorf * *et. al., 1997]), where K(n) is the nth Morava K-theory (see [Rudyak, 1998, Chapter 9] fo* *r an exposition of K(n)). The functor F has the properties that, if U is an open sub* *group of Gn, then F(Gn=U) = EdhUn, and F(Gn) = En, where En is the nth Lubin-Tate spectrum (* *for salient facts about En and its importance in homotopy theory, see [Devinatz and* * Hopkins, 1995, Introduction]), and EdhUnis a spectrum that behaves like the U-homotopy f* *ixed point spectrum of En with respect to the continuous U-action. Since Hom R+G (Gn, Gn) * *~=Gn, n functoriality implies that Gn acts on En by maps of commutative S-algebras. In * *Section _____________ 2000 Mathematics Subject Classification: 55P42, 55U35, 18B25. Key words and phrases: site, profinite group, finite discrete G-sets, preshe* *aves of spectra, Lubin-Tate spectrum, continuous G-spectrum. Oc Daniel G. Davis, 2006. Permission to copy for private use granted. 1 2 5, we will give several related examples of presheaves of spectra that illustra* *te the utility of the category R+G. The pretopology of epimorphic covers on a small category`C is the pretopolog* *y K given by all covering families {fi:Ci ! C| i 2 I} such that OE: i2ICi ! C is onto, * *where Ci, C 2 C, fi2 Mor C(Ci, C), and I is some indexing set. (Of course, one must p* *rove that these covering families actually give a pretopology on C.) We note that we do n* *ot require that OE be a morphism in C; for our purposes, C = R+Gand we only require that O* *E be an epimorphism in the category of all G-sets (so that OE does not have`to be conti* *nuous). This assumption is important for our work, since, for example, G G is not in * *R+G. The pretopology K is a familiar one. For example, for a profinite group G, * *K is the standard basis used for the site G - Setsdfof finite discrete G-sets ([Jard* *ine, 1997, pg. 206]). However, there is an important difference between R+Gand G - Setsd* *f: the latter category is closed under pullbacks, but it is easy to see that R+Gdoes n* *ot have all pullbacks (this point will be discussed later). But in a category with pul* *lbacks, the canonical topology, the finest topology in which every representable presheaf i* *s a sheaf, is given by all covering families of universal effective epimorphisms (see Expo* *se IV, 4.3 of [Demazure, 1970]). This implies that G - Setsdfis a site with the canonical * *topology when equipped with pretopology K. However, due to the lack of sufficient pullba* *cks, we cannot conclude that K gives R+Gthe canonical topology. In fact, we will show * *that K does not generate the canonical topology, since K does not yield a subcanonical* * topology. Note that R+Gis built out of the two subcategories G-Sets dfand the groupoid* * G. Since each of these categories is a site via K (for G, this is verified in Lemm* *a 2 below), it is natural to think that R+Gis also a site via K. Our main result (Theorem 3.1)* *, verifies that this is indeed the case. As discussed earlier, F is a presheaf of spectra on the site R+G. More gener* *ally, there is the category PreSpt(R+G) of presheaves of spectra on R+G. Furthermore, since* * R+Gis a site, the work of Jardine (e.g., [Jardine, 1987], [Jardine, 1997]) implies that* * PreSpt(R+G) is a model category. We recall the definition of this model category in Section* * 5. In [Davis, 2006], the author showed that, given a continuous G-spectrum Z, t* *hen, for any open subgroup U of G, there is a homotopy fixed point spectrum ZhU , define* *d with respect to the continuous action of U on Z. In Examples 5.7 and 5.8, we see tha* *t there is a presheaf that organizes in a functorial way the following data: Z, ZhU for all * *U open in G, and the maps between these spectra that are induced by continuous G-equivariant* * maps between the G-spaces G and G=U. Thus, PreSpt(R+G) is a natural category within * *which to work with continuous G-spectra. It is our hope that the model category struc* *ture on PreSpt(R+G) can be useful for the theory of homotopy fixed points for profinite* * groups, though we have not yet found any such applications. Acknowledgements. When I first tried to make R+Ga site, and was focusing on an abstract way of doing this, Todd Trimble helped me get started by suggesting th* *at I extend K to all of R+G. Also, I thank him for pointing out Lemma 2.1. I thank* * Paul Goerss for discussions about this material. Also, I appreciate various conversa* *tions with Christian Haesemeyer about this work. 3 2. Preliminaries Before we prove our main results, we first collect some easy facts which will b* *e helpful later. As stated in the Introduction, G always refers to an infinite profinite * *group. (If the profinite group G is finite, then R+G= G-Sets dfand there is nothing to prove.) 2.1. Lemma. Let f :C ! G be any morphism in R+Gwith C 6= ?. Then C = G. Proof. Choose any c 2 C and let f(c) = fl. Choose any ffi 2 G. Then ffi = (ffifl-1)fl = (ffifl-1) . f(c) = f((ffifl-1) . c), by the G-equivariance of f. Thus, f is onto and |im (f)| = 1, so that C cannot* * be a finite set. * * | 2.2. Lemma. For a topological group G, let G be the groupoid with the single ob* *ject G and morphisms the G-equivariant maps G ! G given by right multiplication by some el* *ement of G. Then G is a site with the pretopology K of epimorphic covers. f g Proof. Any diagram G ! G G, where f and g are given by multiplication by fl* * and ffi, respectively, can be completed to a commutative square f0 G ____//_G |g0| g|| fflffl|ffflffl| G ____//_G, where f0 and g0 are given by multiplication by ffi-1 and fl-1, respectively. Th* *is property suffices to show that G is a site with the atomic topology, in which every siev* *e is a covering sieve if and only if it is nonempty. It is easy to see that the only nonempty s* *ieve of G is Mor G(G, G) itself. Thus, the only covering sieve of G is the maximal sieve. Si* *nce every morphism of G is a homeomorphism, in the pretopology K, the collection of cover* *s is exactly the collection of all nonempty subsets of Mor G(G, G). Then it is easy* * to check that K is the maximal basis that generates the atomic topology. * * | Observe that if f :G ! G is a morphism in R+G, then by G-equivariance, f is * *the map given by multiplication by f(1) on the right. As mentioned earlier, we have 2.3. Lemma. The category G-Setsdf, a full subcategory of R+G, is closed under p* *ullbacks. Proof. The pullback of a diagram in G-Setsdfis formed simply by regarding the d* *iagram as being in the category TG of discrete G-sets. The category TG is closed under* * pullbacks, as explained in [Mac Lane and Moerdijk, 1994, pg. 31]. * * | 4 We recall the following useful result and its proof. ` n __ 2.4. Lemma. Let X be any finite set in R+G. We write X = i=1xi, the disjoint * *union of`all the distinct orbits __xi, with each xi a representative. Then X is home* *omorphic to n i=1G=Ui, where Ui= Gxiis the stabilizer in G of xi. Proof. Let f :G=Ui ! __xibe given by f(flUi) = fl . xi. Since X is a discrete G* *-set, the stabilizer Uiis an open subgroup of G with finite index, so that G=Uiis a finit* *e set. Then f is open and continuous since it is a map between discrete spaces. Also, it is* * clear that f is onto. Now suppose flUi= ffiUi. Then fl-1ffi 2 Ui, so that (fl-1ffi) . xi= (f* *l-1) . (ffi . xi) = xi. Thus, fl . xi = ffi . xi and f is well-defined. Assume that fl . xi = ffi . xi.* * Then fl-1ffi 2 Gxi so that f is a monomorphism. * * | 2.5. Lemma. Let X be a finite discrete G-set in R+G and let _ :G ! X be any G- equivariant function. Then _ is a morphism in R+G. ` n Proof. As in Lemma 2.4, we identify X with i=1G=Ui. Since _ is G-equivariant * *and _(fl) = fl . _(1), _ is determined by _(1). Let _(1) = ffiUj for some ffi 2 G a* *nd some j. Then for any fl in G, flUj = (flffi-1ffi)Uj = (flffi-1) . _(1) = _(flffi-1), so that im _ = G=Uj. Since X is discrete, _ is continuous, if, for any x 2 X, _* *-1(x) is open in G. It suffices, by the identification, to let x = flUj, for any fl 2 G.* * Then _-1(flUj)= {i 2 G| _(i) = flUj} = {i 2 G| i . (ffiUj) = flUj} = {i 2 G| ffi-1i-1fl 2 Uj} = flUjffi-1. Since Uj is open and multiplication on the left or the right is always a homeom* *orphism in a topological group, we see that _-1(x) is an open set in G. * * | 3. The proof of the main theorem With these lemmas in hand, we are ready for 3.1. Theorem. For any profinite group G, the category R+Gequipped with the pret* *opology K of epimorphic covers is a small site. Before proving the theorem, we first make some remarks about pullbacks in R+* *Gand how this affects our proof. In a category C with sufficient pullbacks, to prov* *e that a pretopology is given by a function K, which assigns to each object C a collecti* *on K(C) of families of morphisms with codomain C, one must prove the stability axiom, whic* *h says the following: if {fi:Ci ! C| i 2 I} 2 K(C), then for any morphism g :D ! C, the family of pullbacks {ssL :D xC Ci! D| i 2 I} 2 K(D). Let us examine what this axiom would require of R+G. 5 3.2. Example. The map G ! * forms a covering family and so the stability axiom requires that Gx{*}G = G x G be in R+G. 3.3. Example. Let C be any finite discrete G-set with more than one element and* * with trivial G-action, g :G ! C any morphism, and consider the cover {fi:Ci! C| i 2 I} 2 K(C), where Cj = C and fj: C ! C is the morphism mapping C to g(1), for some j 2 I. Because the action is trivial, fj is G-equivariant. There certainly exist cover* *s of C of this form, since one could let fk = idC, for some k 6= j in I, and then let the othe* *r fi be any morphisms with codomain C. Then the stability axiom requires that G xC C exist* *s in R+G, but this is impossible, since G xC C = {(fl, c)| g(fl) = fj(c)} = {(fl, c)| fl . g(1) = g(1)} = Gg(1)x C * *= G x C. Thus, the stability axiom for a pretopology must be altered so that one stil* *l obtains a topology. We list the correct axioms for our situation below. They are take* *n from [Mac Lane and Moerdijk, 1994, Exercise 3, pg. 156]. 1.If f :C0! C is an isomorphism, then {f :C0! C} 2 K(C). 2.(stability axiom) If {fi:Ci! C| i 2 I} 2 K(C), then for any morphism g :D * *! C, there exists a cover {hj: Dj ! D| j 2 J} 2 K(D) such that for each j, g O * *hj factors through some fi. 3.(transitivity axiom) If {fi:Ci! C| i 2 I} 2 K(C), and if for each i 2 I th* *ere is a family {gij:Dij! Ci| j 2 Ii} 2 K(Ci), then the family of composites {fiO gij:Dij! C| i 2 I, j 2 Ii} is in K(C). Proof of Theorem 3.1. It is clear that the pretopology of epimorphic covers sat* *isfies axiom (1) above. Also, it is easy to see that axiom (3) holds. Indeed, using * *the above notation, choose any c 2 C. Then there is some ci2 Ci for some i, such that fi(* *ci) = c. Similarly, there must be some dij 2 Dij`for some j, such that gij(dij) = ci. H* *ence, (fiO gij)(dij) = fi(ci) = c, so that i,jDij! C is onto. This verifies (3). We* * verify (2) by considering five cases. Case (1 ): Suppose that D and each of the Ci are finite sets in R+G. By Lemm* *a 2.1, C must be a finite set. Consider the cover {ssL(i): D xC Ci! D| i 2 I}, where ssL(i) is the obvious map and g O ssL(i) factors through fi via the canon* *ical map ssR(i). Now choose any d 2 D and let g(d) = c 2 C. Then there exists some i suc* *h that 6 ` fi(ci) = c for ci2 Ci. Thus, (d, ci) 2 D xC Ci, so that ID xC Ci! D maps (d, * *ci) to d and is therefore an epimorphism. This shows that {ssL(i)} is in K(D). Case (2 ): Suppose that D = G and that each`Ciis a finite set in R+G. By Lem* *ma 2.1, C is a finite set and we identify it with ni=1G=Ui, where Ui= Gxi, the stabil* *izer of xiin G.`The map g is determined by g(1) = ffiUk for some ffi 2 G and some stabilizer* * Uk. Since ICi ! C is onto and im(g) = G=Uk , there exists some cl 2 Cl such that fl(cl)* * = Uk. Since Cl is a finite set, we can identify cl with some ~Gz, where ~ 2 G and Gz * *is the stabilizer of some element z 2 Cl. Then define the cover to be {~: G ! G}, where ~(fl) = flffi-1. Define ffl:G * *! Cl to be the G-equivariant map given by 1 7! ~Gz. By Lemma 2.5, ffl is continuous and* * is a morphism in R+G. Since ~ is a homeomorphism, the cover {~} is in K(D). Now, (g O ~)(1) = g(ffi-1) = ffi-1 . g(1) = Uk = ~ . fl(Gz) = ~ . fl(~-1 . ffl(1)* *) = (flO ffl)(1). This shows that g O ~ factors through flvia ffl. Case (3 ): Suppose not all the Ci are finite sets and that D = G. Also, assu* *me that C = G. This implies that Ci = G for all i 2 I. Choose any k 2 I, let ffk = id* *G, and define ~: G ! G to be multiplication on the right by fk(1)g(1)-1. Then the diag* *ram idG G ____//_G ~|| |fk| fflffl|fflffl|g G ____//_G is commutative, since (g O ~)(1) = g(fk(1)g(1)-1) = fk(1)g(1)-1 . g(1) = fk(1) = (fk O ffk)(1* *). Thus, g O ~ factors through fk via ffk, so that the stability axiom is verified* * by letting the covering family be {~}. Case (4 ): Suppose that not all the Ci are finite sets, D = G, and C is a fi* *nite set. With C as in Lemma 2.4, let g(1) = ffiUk 2 C, as in Case (2). Then there exists* * some l such that fl(cl) = Uk, for some cl2 Cl. Now we consider two subcases. Case (4a ): Suppose that Cl is a finite set in R+G. Just as in Case (2), we* * construct maps ~ and ffl, so that g O ~ factors through flvia ffland {~} 2 K(D). Case (4b ): Suppose that Cl = G. By G-equivariance, fl(1) = c-1lUk. Then * *define ~: G ! G by 1 7! ffi-1 and ffl:G ! G by 1 7! cl. Then g O ~ factors through fl * *via ffl, since (g O ~)(1) = g(ffi-1) = ffi-1 . g(1) = Uk = fl(cl) = (flO ffl)(1). Thus, the cover {~}, as a homeomorphism, is in K(D). This completes Case (4). Now we consider the final possibility, Case (5 ): suppose that not all of t* *he Ci are finite sets and suppose that D is a finite set. This implies that C is a finite* * set. This case 7 is more difficult than the others because the cover consists of more than one m* *orphism and it combines the previous constructions. For each d 2 D, we make`a choice of* * some cl2 Cl for`some l, such that clis in the preimage of g(d) under Ci ! C. Then * *write D = Ddf DG , where Ddfis the set of all d such that the corresponding Clis in * *a finite set, and DG is the set of all d`such that the corresponding Cl = G. Now consid* *er the cover {hd: Dd ! D| d 2 D = Ddf DG }, where ( D xC Cd if d 2 Ddf, Dd = G if d 2 DG. If d 2 Ddf, then hd = ssL and ffd: D xC Cd ! Cd is the canonical map ssR; it is* * clear that the required square commutes. Now suppose d 2 DG . Then there exists cl2 Cl= G * *for some l, such that g(d) = fl(cl). We write fl(1) = `Uk 2 C for some ` 2 G and fo* *r some stabilizer Uk. Then we define ffd: G ! Cl= G by 1 7! `-1. Also, we define hd: G* * ! D by 1 7! (`-1c-1l) . d. Lemma 2.5 shows that hd is a morphism in R+G. Then we ha* *ve the required commutative diagram ffd G _____//G hd|| |fl| fflffl|fflffl|g D ____//_C, since (g O hd)(1)= g((`-1c-1l) . d) = (`-1c-1l) . g(d) = (`-1c-1l) . fl(cl) = fl(`-1) = (flO ffd)(1). The`only remaining detail is to show that {hd} 2 K(D); that is, we must show* * that OE: D Dd ! D is an epimorphism. Let d be any element in D. Suppose d 2 Ddf. Then, using our choice above, there exists some cl2 Cl, a finite set for some l* *, such that fl(cl) = g(d). Then (d, cl) 2 D xC Cl and OE(d, cl) = ssL(d, cl) = d. Now suppo* *se d 2 DG . With cland ` as above, cl` 2 Dd = G and OE(cl`) = hd(cl`) = (cl`) . hd(1) = d. * *Therefore, OE is an epimorphism. * * | 4. The site R+G does not have the canonical topology Now that we have established that R+Gis a site with pretopology K, we begin wor* *king to show that, contrary to what typically happens with this pretopology, it does no* *t give the canonical topology. We start with a definition. 4.1. Definition. If T is some collection of morphisms with codomain C, where C * *is an object in the category C, then (T ) denotes the sieve generated by T . Thus, (T ) = {f O g| f 2 T, dom (f) = cod(g)}. 8 4.2. Lemma. Let K be a pretopology on a category C. Let J be the Grothendieck t* *opology generated by K. Then for any C 2 C, J(C) consists exactly of all (R) [ (T ) su* *ch that R 2 K(C) and T is some collection of morphisms with codomain C. Proof. Let S be a covering sieve of C. Then there exists some R 2 K(C) such th* *at R S. We will prove that S = (R) [ (S), verifying the forward inclusion. To * *prove equality it suffices to show that (R) [ (S) S. If f 2 (R), f = g O h for some* * g 2 R and some h with dom (g) = cod(h). Since g 2 S, f 2 S. Similarly, if f 2 (S), then f* * 2 S. Now consider any family of morphisms (R) [ (T ) as described in the statement o* *f the lemma. Since R (R) [ (T ), (R) [ (T ) 2 J(C) if it is a sieve. Since (R) an* *d (T ) are sieves, it is clear that (R) [ (T ) is also a sieve. * * | This result is useful for understanding the topology of a site, when the sit* *e is defined in terms of a pretopology. For example, G-Sets dfis a site by the pretopology * *K and its category of sheaves of sets is equivalent to the category of sheaves on the* * site S(G) consisting of quotients of G by open subgroups (the morphisms are the G-equivar* *iant maps), where S(G) is given the atomic topology (see [Mac Lane and Moerdijk, 199* *4, Chapter 3, Section 9]). Thus, one might ask if G-Sets dfalso has the atomic top* *ology. However, the lemma allows us to see that K generates a topology`that is coarser* * than the atomic topology. To see this, let X = G=U and Y = G=U G=U, where U is a proper open subgroup of G. (Since G is an infinite profinite group, the cano* *nical way of writing G as an inverse limit guarantees the existence of such a U.) We* * define f :X ! Y by f(U) = U, where U lives in the factor on the left; f is the left in* *clusion. Now consider`the sieve S = ({f}). Clearly, S does not contain an epimorphic cov* *er, since im ( g2S(dom (g)) ! Y ) = G=U. The lemma indicates that every sieve of G-Setsd* *fmust contain an epimorphic cover, so that S is not a sieve for Y in the topology gen* *erated by K. Now we consider the site R+Gwith the pretopology K of epimorphic covers. We* * use Hom G(X, Y ) to denote continuous G-equivariant maps between continuous G-sets* * X and Y . Recall that a presheaf of sets P on a site (C, J) is a sheaf, if for each o* *bject C 2 C and each covering sieve S 2 J(C), the diagram Q _p__//_Q P (C)__e_//_f2SP (dom (f))____//_P (dom (g)) a is an equalizer of sets, where the second product is over all f, g, with f 2 S,* * dom (f) = cod(g). Here, e is the map e(x) = {P (f)(x)}f, p is given by {xf}f 7! {xfg}f,g, and a is given by {xf}f 7! {P (g)(xf)}f,g= {xf O g}f,g. Recall that a representable presheaf of R+Gis any presheaf which, up to isom* *orphism, has the form of Hom G(-, C) for some C 2 R+G. Also, the Yoneda embedding +)op R+G! Sets(RG , C 7! Hom G(-, C) 9 * * +)op is a full and faithful functor, so that one can identify C with an object of Se* *ts(RG . We now consider which objects of R+Gyield sheaves of sets on R+G. Noting that the empty set is a discrete G-set, we have 4.3. Lemma. The presheaf Hom G (-, ?) is a sheaf of sets on the site R+G. Proof. Let o : ? ! X denote the vacuous map, for any X 2 R+G. Since o : ? ! ? is vacuously an epimorphism, {o} is the unique covering sieve for ?. Let C = ?. Th* *en the desired equalizer diagram has the form __p_//_ Hom G(?, ?) = {o} _e__//_{o}__//_{o}. a It is clear that this is an equalizer diagram. Now let C be a nonempty finite set in G - Setsdf. Let S be any covering sie* *ve of C. There must exist a morphism in S with domain equal to a nonempty object in R* *+G. Therefore, since ? x Z = ? for any space Z, we have __p__// Hom G(C, ?) = ?__e__//?____//?. a Since the equalizer must exist and the vacuous map o : ? ! ? is the unique map * *with codomain ?, this must be an equalizer diagram. Finally, letting C = G, we get Q __p_//_ Hom G(G, ?) = ? __e_//_f2HomG(G,G)? = ? ____//_?. a Again, this is an equalizer diagram. * * | To prove the next theorem, we need the following lemma. 4.4. Lemma. If G is a compact topological group, U an open subgroup of G, and X* * 6= ? a finite discrete G-set, then Hom G(G=U, X) ~={x 2 X| U < Gx}, where Gx is the stabilizer of x in G. Proof. Let f : G=U ! X. It is clear that f is G-equivariant if and only if it i* *s completely determined by f(U) in the obvious way. Since U is an open subgroup, it has fini* *te index in G, so that G=U is a discrete space. Thus, any G-equivariant map G=U ! X is continuous. The key is that f is well-defined if and only if U < Gf(U). To see * *this, first assume that f is well-defined; let fl 2 U. Then flU = U, so that fl .f(U) = f(f* *lU) = f(U). Hence, fl 2 Gf(U)and U < Gf(U). Now suppose that U < Gf(U)and take any flU = ff* *iU. This implies that fl-1ffi 2 U and hence, in Gf(U). Thus, (fl-1ffi) . f(U) = f(* *U), so that fl . f(U) = ffi . f(U). Equivariance gives f(flU) = f(ffiU) and f is well-defin* *ed. Thus, Hom G(G=U, X) ~={f(U) 2 X| U < Gf(U)}. | 10 Henceforth, let J denote the topology of R+Ggenerated by K. 4.5. Theorem. Let X be any object in R+Gthat is not a finite discrete trivial G* *-set, where G is an infinite profinite group. Then the presheaf Hom G (-, X) is not a sheaf* * of sets on the site R+G. Proof. Suppose Hom G (-, X) is a sheaf of sets on the site R+G. The equalizer c* *ondition says that for every object C 2 R+Gand for every covering sieve S 2 J (C), Hom G (C, X) ~={{hf}f| hfg = hf O g, f, g, f 2 S, dom (f) = cod(g)}, where for f 2 S, hf 2 Hom G(dom (f), X). We will construct an example of some C* * and S such that this sheaf condition fails to`be true with X as above. Let C 2 G-Setsdf; we identify C with ni=1G=Ui, where each Uiis an open sub* *group of G. For each i, define fi:G ! C by 1 7! Ui. Thus, im(fi) = G=Ui and {fi} is* * an epimorphic cover of C. The preceding lemma tells us that S = ({fi}) is a coveri* *ng sieve of C. For this S, we will examine the sheaf condition. Let S = S0[ S00, where S* *0= {fi} and S00is the complement of S0in S. Thus, every k 2 S00has the form k = fiO g f* *or some g with dom (fi) = cod(g). Then {{hf}f| hfg= hf O g, f, g, f 2 S, dom (f) = cod(g)} = {{hfi}fix {hk}k2S00| hfg = hf O g, f, g, f 2 S, dom (f) = cod(g)} = {{hfi}fix {hfiO g}fiOg2S00| hfg = hf O g, f, g, f 2 S, dom (f) = c* *od(g)} = {{hfi}fix {hfiO g}fiOg2S00| hfi2Hom G(G, X), fi2S0, g, dom (fi) =* * cod(g)}. We verify the last equality. Suppose hfiis any morphism in Hom G(G, X). Now tak* *e any f and g with f 2 S and dom (f) = cod(g). If f = fi2 S0, then hfOg = hfiOg = hfiOg* *= hfOg, by construction. Now suppose f 2 S00. Then f = fiO k for some k :G ! G. Thus, hfg = hfiO(kOg)= (hfiO k) O g = hfiOkO g = hf O g. Since hfiO g is determined by hfiand fiO g, we see that the set {{hfi}fix {hfiO g}fiOg2S00| hfi2 Hom G(G, X), fi2 S0, g, dom (fi) = cod(* *g)} is isomorphic to the set {{hfi}fi| hfi2 Hom G(G, X), fi2 S0} = Hom G(G, X)n, where Hom G (G, X)n is the n-fold Cartesian product of Hom G (G, X). Now, ther* *e is an isomorphism Hom G (G, X)n ~=Xn. Therefore, for Hom G (-, X) to be a sheaf, it m* *ust be that Hom G(C, X) ~=Xn for every C 2 G - Setsdf. If X = G and C 6= ? is in G - S* *etsdf, then Hom G (C, G) = ?, whereas, since |C| 1, n 1 and Xn = Gn. Thus, Hom G (* *-, G) is not a sheaf. 11 Now we consider X 6= G and assume that Hom G(C, X) ~=Xn for every C 2 G-Sets* *df. This implies that ` n Xn ~=Hom G(C, X) ~=Hom G ( i=1G=Ui, X) ~=Q ni=1HomG(G=Ui, X) ~=Q ni=1{x 2 X| Ui< Gx} Xn. Therefore, it must be that {x 2 X| Ui< Gx} =`X, for all i = 1, ..., n. Thus, Ui* *< Gx for all x 2 X and each i. Now let us write X ~= mj=1G=Gxj, where each xj is a repr* *esentative from a distinct orbit of X. Let C be a trivial G-set so that every stabilizer o* *f c 2 C in G is equal to G. This implies that G < Gxj for all j. Thus, each Gxj = G. This in* *dicates that X must be a trivial G-set. This contradiction shows that every X violates * *the sheaf condition for some C and S. * * | This result immediately yields 4.6. Corollary. For an infinite profinite group G, the site R+Gwith the pretopo* *logy K of epimorphic covers is not subcanonical. Proof. There exists a proper open subgroup U of G satisfying`[G : U] > 1. Thus,* * the representable presheaves Hom G (-, G) and Hom G (-, ni=1G=U), for any n 1, * *are not sheaves. * * | Since a canonical topology is, by definition, subcanonical, we obtain 4.7. Corollary. For an infinite profinite group G, the site R+G, with the preto* *pology K, is not canonical. The next result is an elementary fact about profinite groups that helps us u* *nderstand "how often" representable presheaves fail to be sheaves in R+Gand what such "fa* *iling" presheaves can look like, based on what we know from Theorem 4.5. 4.8. Lemma. If G is an infinite profinite group, then G contains an infinite nu* *mber of distinct proper open subgroups. Proof. We have already seen that G has at least one proper open subgroup. Suppo* *se that G has only a finite number of distinct proper open subgroups. Then G has a* * finite numberTof distinct proper open normal subgroups N1, ..., Nk. Since G is profini* *te, N = k i=1Ni = {1}. Because N is an open subgroup with finite index, it has uncount* *able order. This contradiction gives the conclusion. * * | 4.9. Remark. Since any topology finer than J would contain the covering sieve (* *{fi}) that was the key to Theorem 4.5, no topology finer than J can be subcanonical. 5. Presheaves of spectra on the site R+G Let Ab be the category of abelian groups, and let Spt denote the model categor* *y of Bousfield-Friedlander spectra of pointed simplicial sets. We refer to the obje* *cts of Spt 12 as simply "spectra." Now that R+Gis a site, we can consider the category PreSpt* *(R+G) of presheaves of spectra on the site R+G. By applying the work of Jardine ([Jardin* *e, 1987], [Jardine, 1997, Section 2.3]), PreSpt (R+G) is a model category. We recall the* * critical definitions that give the model category structure and then we state Jardine's * *result, when it is applied to R+G. 5.1. Definition. Let P :(R+G)op ! Spt be a presheaf of spectra. Then, for each * *n 2 Z, ssn(P ): (R+G)op ! Ab , C 7! ssn(P (C)), is a presheaf of abelian groups. Then the associated sheaf "ssn(P ) of abelian * *groups is the sheafification of ssn(P ). Let f :P ! Q be a morphism of presheaves of spectra on R+G. Then f is a weak equivalence if the induced map "ssn(P ) ! "ssn(Q) of sheaves is an isomorphism,* * for all n 2 Z. The map f is a cofibration if f(C) is a cofibration of spectra, for all * *C 2 R+G. Also, f is a global fibration if f has the right lifting property with respect to all* * morphisms which are weak equivalences and cofibrations. 5.2. Theorem. [Jardine, 1997, Theorem 2.34] The category PreSpt(R+G), together * *with the classes of weak equivalences, cofibrations, and global fibrations, is a mod* *el category. Now we give some interesting examples of presheaves of spectra on the site R* *+G. 5.3. Example. In the Introduction, we saw that the Devinatz-Hopkins functor F i* *s an example of an object in PreSpt(R+Gn). For the next example, if X is a spectrum, then, for each k 0, we let Xk be* * the kth pointed simplicial set constituting X, and, for each l 0, Xk,lis the poin* *ted set of l-simplices of Xk. 5.4. Example. Let X be a discrete G-spectrum (see [Davis, 2006] for a definitio* *n of this term), so that each Xk,lis a pointed discrete G-set. If C 2 R+G, then let Hom G* * (C, X) be the spectrum, such that Hom G(C, X)k = Hom G(C, Xk), where Hom G(C, X)k,l= Hom G(C, Xk)l= Hom G(C, Xk,l). Above, the set Xk,lis given the discrete topology, since it is naturally a disc* *rete G-set. Then Hom G (-, X) is an object in PreSpt(R+G). It is easy to see that if U is * *an open subgroup of G, then Hom G (G=U, X) ~= XU , the U-fixed point spectrum of X. Al* *so, Hom G(G, X) ~=X. Now we recall part of [Behrens and Davis, 2005, Proposition 3.3.1], since th* *is result (and its corollary) will be helpful in our next example. We note that this resu* *lt is only a slight extension of [Jardine, 1997, Remark 6.26]: if U is normal in G, then t* *he lemma below is an immediate consequence of Jardine's remark. 13 5.5. Lemma. Let X be a discrete G-spectrum. Also, let f :X ! Xf,Gbe a trivial c* *ofibra- tion, such that Xf,Gis fibrant, where all this takes place in the model categor* *y of discrete G-spectra (see [Davis, 2006]). If U is an open subgroup of G, then Xf,G is fibr* *ant in the model category of discrete U-spectra. 5.6. Corollary. Let X and U be as in the preceding lemma. Then XhU = (Xf,G)U. Proof. Let f be as in the above lemma. Since f is G-equivariant, it is U-equiva* *riant. Also, since f is a trivial cofibration in the model category of discrete G-spec* *tra, it is a trivial cofibration in the model category of spectra. The preceding two facts i* *mply that f is a trivial cofibration in the model category of discrete U-spectra. By the * *lemma, Xf,G is fibrant in this model category. Thus, XhU = (Xf,G)U. * * | 5.7. Example. Let X be a discrete G-spectrum. Then Hom G (-, Xf,G) is a preshea* *f in PreSpt(R+G). In particular, notice that Hom G(G=U, Xf,G) ~=(Xf,G)U = XhU and Hom G(G, Xf,G) ~=Xf,G' X. 5.8. Example. For any unfamiliar concepts in this example, we refer the reader * *to [Davis, 2006]. Let Z = holimiZibe a continuous G-spectrum, so that {Zi}i 0is a tower of* * discrete G-spectra, such that each Zi is a fibrant spectrum. Then P (-) = holimHom G(-, (Zi)f,G) 2 PreSpt(R+G), i where P (G=U) ~=holim((Zi)f,G)U = holim(Zi)hU = ZhU i i and P (G) ~=holim(Zi)f,G' Z. i References Mark J. Behrens and Daniel G. Davis, The homotopy fixed point spectra of * *profi- nite Galois extensions, 33 pp., current version of manuscript is * *available at http://dgdavis.web.wesleyan.edu, 2005. Daniel G. Davis, Homotopy fixed points for LK(n)(En ^ X) using the continuous a* *ction, J. Pure Appl. Algebra 206 (2006), no. 3, 322-354. Michel Demazure, Topologies et faisceaux, Sch'emas en groupes. SGA 3, I: Propri* *'et'es g'en'erales des sch'emas en groupes, Springer-Verlag, Berlin, 1970, pp. xv+* *564. 14 Ethan S. Devinatz and Michael J. Hopkins, The action of the Morava stabilizer g* *roup on the Lubin-Tate moduli space of lifts, Amer. J. Math. 117 (1995), no. 3, 669* *-710. Ethan S. Devinatz and Michael J. Hopkins, Homotopy fixed point spectra for clos* *ed sub- groups of the Morava stabilizer groups, Topology 43 (2004), no. 1, 1-47. A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May, Rings, modules, and alg* *ebras in stable homotopy theory, American Mathematical Society, Providence, RI, 1997* *, With an appendix by M. Cole. J. F. Jardine, Stable homotopy theory of simplicial presheaves, Canad. J. Math.* * 39 (1987), no. 3, 733-747. J. F. Jardine, Generalized 'etale cohomology theories, Birkh"auser Verlag, Base* *l, 1997. Saunders Mac Lane and Ieke Moerdijk, Sheaves in geometry and logic, Springer-Ve* *rlag, New York, 1994, A first introduction to topos theory, Corrected reprint of * *the 1992 edition. Yuli B. Rudyak, On Thom spectra, orientability, and cobordism, Springer Monogra* *phs in Mathematics, Springer-Verlag, Berlin, 1998, With a foreword by Haynes Mille* *r. Department of Mathematics Wesleyan University 265 Church St. Middletown, CT 06459-0128 Email: dgdavis@wesleyan.edu