THE E2-TERM OF THE DESCENT SPECTRAL SEQUENCE FOR CONTINUOUS G-SPECTRA DANIEL G. DAVIS1 Abstract.Let {Xi} be a tower of discrete G-spectra, each of which is fib* *rant as a spectrum, so that X = holimiXiis a continuous G-spectrum, with homo- topy fixed point spectrum XhG. The E2-term of the descent spectral seque* *nce for ss*(XhG) cannot always be expressed as continuous cohomology. Howeve* *r, we show that the E2-term is always built out of a certain complex of spe* *c- tra, that, in the context of abelian groups, is used to compute the cont* *inuous cochain cohomology of G with coefficients in limiMi, where {Mi} is a tow* *er of discrete G-modules. 1.Introduction In this note, G always denotes a profinite group. Let H*c(G; M) denote the continuous cohomology of G with coefficients in the discrete G-module M. This cohomology is defined as the right derived functors of G-fixed points. Then we always assume that G has finite virtual cohomological dimension; that is, there exists an open subgroup H and a non-negative integer m, such that Hsc(H; M) = 0, for all discrete H-modules M and all s m. All of our spectra are Bousfield-Friedlander spectra of simplicial sets. In p* *artic- ular, a discrete G-spectrum is a G-spectrum such that each simplicial set Xk is* * a simplicial object in the category of discrete G-sets (thus, for any l 0, the * *action map on the l-simplices, G x (Xk)l! (Xk)l, is continuous when (Xk)l is regarded as a discrete space). The category of discrete G-spectra, with morphisms being G-equivariant maps of spectra, is denoted by SptG. Discrete G-spectra are considered in more detail in [3], which shows (see [3, Theorem 3.6]) that SptG is a model category, where a morphism f in SptG is a weak equivalence (cofibration) if and only if f is a weak equivalence (cofibrat* *ion) in Spt, the category of spectra. Given a discrete G-spectrum X, the homotopy fixed point spectrum XhG is obtained as the total right derived functor of fixed poin* *ts: XhG = (Xf,G)G , where X ! Xf,G is a trivial cofibration and Xf,G is fibrant, all in SptG. Let X0 X1 X2 . .b.e a tower of discrete G-spectra, such that each Xi is a fibrant spectrum. As explained in [3, Lemma 4.4], there exists a tower {X0* *i} of discrete G-spectra, such that there are weak equivalences holimiXi-'! holimiX0i'- limiX0i. ____________ Date: February 10, 2006. 1The author was supported by an NSF grant. Most of this paper was written du* *ring a visit to the Institut Mittag-Leffler (Djursholm, Sweden). 1 2 DANIEL G. DAVIS (In this paper, holimalways denotes the version of the homotopy limit of spectra that is constructed levelwise in the category of simplicial sets, as defined in* * [1] and [7, 5.6].) Since the inverse limit of a tower of discrete G-sets is a topol* *ogical G-space, and because holimiXican be identified with limiX0i, in the eyes of hom* *o- topy, holimiXi is a continuous G-spectrum. Notice that, under this identificati* *on, the continuous G-action respects the topology of both G and all the Xi together. Continuous G-spectra and examples of such in chromatic stable homotopy theory are considered in [3, 2]. Given the continuous G-spectrum holimiXi, there is the homotopy fixed point spectrum (holimiXi)hG = holimi(Xi)hG. This construction is called homotopy fixed points because it is equivalent to t* *he usual definition when G is a finite group and it is the total right derived fun* *ctor of fixed points in the appropriate sense (see [3, Remark 8.4]). By [3, Theorem 8.8], thanks to the assumption of finite virtual cohomological dimension, there is a descent spectral sequence (1.1) Es,t2) sst-s((holimiXi)hG), where (1.2) Es,t2= ssssst(holimi( oG(Xi)f,G)G ), and, if the tower of abelian groups {sst(Xi)} satisfies the Mittag-Leffler cond* *ition for every integer t, then Es,t2~=Hscont(G; {sst(Xi)}), which is continuous coho* *mology in the sense of Jannsen. (This cohomology is obtained by taking the right deriv* *ed functors of limi(-)G , a functor from towers of discrete G-modules to abelian g* *roups; see [5].) In expression (1.2), since sst(holimi(-)) is not necessarily limisst(-), the * *E2-term of descent spectral sequence (1.1), in general, can not be expressed as continu* *ous cohomology, and, in general, it has no compact algebraic description. However, * *in this note, we show that the E2-term (1.2) can always be described in an interes* *ting way. In more detail, Theorem 2.5 gives a particular cochain complex C* for computi* *ng the continuous cochain cohomology of G for a topological G-module limiMi, where {Mi} is a tower of discrete G-modules. In Corollary 3.4, we show that the E2-te* *rm of (1.2) can always be given by taking the cohomology of the homotopy groups of the complex C*, where limiMiis replaced by the continuous G-spectrum holimiXi, in an appropriate sense. This presentation of the E2-term shows that E*,*2always takes into account the topology of the continuous G-spectrum, even when it cann* *ot be expressed as continuous cohomology. Acknowledgements. I thank Paul Goerss and Halvard Fausk for helpful com- ments. 2.The pro-discrete cochain complex and continuous cohomology We begin this section with some terminology. If C is a category, then tow(C) * *is the category of towers C0 C1 C2 . . . THE E2-TERM OF THE DESCENT SPECTRAL SEQUENCE 3 in C. The morphisms {fi} are natural transformations such that each fi is a mor- phism in C. In this note, we will be working with tow(DMod (G)), where DMod (G) is the category of discrete G-modules, and tow(SptG). If A is an abelian group with the discrete topology, let Map c(G, A) be the a* *belian group of continuous maps from G to A. If X is a spectrum, one can also define Map c(G, X), where the l-simplices of the kth simplicial set (Map c(G, X)k)l are given by Map c(G, (Xk)l), where (Xk)lis given the discrete topology. Consider the functor G :SptG ! SptG, X 7! G (X) = Map c(G, X), where the action of G on Map c(G, X) is induced on the level of sets by (g . f)* *(g0) = f(g0g), for g, g02 G and f 2 Map c(G, (Xk)l), for each k, l 0. As explained i* *n [3, Definition 7.1], the functor G forms a triple and there is a cosimplicial disc* *rete G-spectrum oGX. Also, it is clear that G :DMod (G) ! DMod (G) can be defined as above, so that, given a discrete G-module M, oGM is a cosimplicial discrete G-module. We do not claim any originality in the following definition, where here and e* *lse- where colimNis a colimit indexed over all open normal subgroups N of G. Definition 2.1. Let {Xi} be an object in tow(DMod (G)) or in tow(SptG). Then the pro-discrete cochain complex is defined to be the complex C*(G; {Xi}) = limicolimN(( *G=N((Xi)N ))G=N ), where *G=N((Xi)N ) is the canonical complex associated to oG=N((Xi)N ). The p* *ro- discrete cochain complex is a complex of abelian groups or spectra, respectivel* *y, and the limit and colimit are both formed in abelian groups or spectra, respect* *ively. Lemma 2.2. Let {Xi} be in tow (DMod (G)) or in tow (SptG). Then there is a natural isomorphism C*(G; {Xi}) ! limi( *GXi)G . Remark 2.3. In the isomorphism above, if the limiis removed from both the source and target, then Lemma 2.2 is basically a version of the isomorphism in * *the first sentence of [6, proof of Proposition (1.2.6)], which is in the context of* * discrete G-modules only. We give a proof of Lemma 2.2 below because this is easier than translating from [op. cit.]. Proof of Lemma 2.2.Let X be a discrete G-module or a discrete G-spectrum. Note that Mapc(G, X)G ~=X ~=colimNXN ~=colimNMapc(G=N, XN )G=N . Similarly, notice that Map c(G, Mapc(G, X))G~=Map c(G, X) ~=colimNMapc(G=N, XN ) ~=colimMap (G=N, Map (G=N, XN ))G=N . N c c Also, we have Map c(G, Mapc(G, Mapc(G, X)))G ~=Map c(G2, X) ~=colimNMapc((G=N)2, XN ), 4 DANIEL G. DAVIS and the last expression is isomorphic to colimNMapc(G=N, Mapc(G=N, Mapc(G=N, XN )))G=N . This verifies the isomorphism for 0-, 1-, and 2-cochains. By tracing through and iterating the above ingredients, we see that there is a natural isomorphism colimN(( oG=N(XN ))G=N ) ! ( oGX)G of cosimplicial objects. Thus, there is an isomorphism of associated cochain co* *m- plexes colimN(( *G=N(XN ))G=N ) ! ( *GX)G . Now let {Xi} be as described in the statement of the theorem. Then there is an isomorphism limicolimN(( *G=N((Xi)N ))G=N ) ! limi( *GXi)G . Let M be any topological G-module. Then the continuous cochain cohomology of G with coefficients in M, H*cts(G; M), is the cohomology of a cochain complex that has the form (2.4) M ! Map c(G, M) ! Map c(G2, M) ! . . . (see [6, pg. 106] for details). We note that, by [5, Theorem (2.2)], if {Mi} is* * a tower of discrete G-modules that satisfies the Mittag-Leffler condition, then Hscts(G; limiMi) ~=Hscont(G; {Mi}), for all s 0, but, in general, these two versions of continuous cohomology nee* *d not be isomorphic. Also, if M is a discrete G-module, then Hscts(G; M) = Hsc(G; M). Now we show that the pro-discrete cochain complex can be used to compute continuous cochain cohomology. Theorem 2.5. If {Mi} is a tower of discrete G-modules, then Hscts(G; limiMi) ~=Hs[C*(G; {Mi})]. Proof.As explained in [6, pg. 106], for a topological G-module M, the chain com- plex in (2.4) is defined by taking the G-fixed points of the complex (2.6) X*(G; M) = [Map c(G, M) ! Map c(G2, M) ! . .]., where Xn(G; M) = Map c(Gn+1, M) has a G-action that is defined by (g . f)(g1, ..., gn+1) = g . f(g-1g1, ..., g-1gn+1). Now let M be a discrete G-module. Then it is a standard fact that the cochain complex (X*(G, M))G is naturally isomorphic as a complex to the cochain complex ( *GM)G . This isomorphism uses the fact that the abelian group of n-cochains of ( *GM)G is isomorphic to Map c(Gn+1, M)G , where Map c(Gn+1, M) has a G-action that is given by (g . f)(g1, g2, g3, ..., gn+1) = f(g1g, g2, g3, ..., gn+1). THE E2-TERM OF THE DESCENT SPECTRAL SEQUENCE 5 Since (Xn(G; limiMi))G ~=limi((Xn(G; Mi))G ) ~=limi( n+1GMi)G , we have: Hscts(G; limiMi) = Hs[(X*(G; limiMi))G ] = Hs[limi( *GMi)G ], where we used the aforementioned fact that (X*(G, Mi))G and ( *GMi)G are nat- urally isomorphic cochain complexes. The proof is completed by applying Lemma 2.2. 3.The E2-term and the pro-discrete cochain complex In this section, we show that the E2-term of (1.2) can be built out of the sa* *me complex that computes continuous cochain cohomology. More precisely, given a continuous G-spectrum holimiXi, there exists a tower {X0i} of discrete G-spectr* *a, such that (3.1) Es,t2~=Hs[sst(C*(G; {X0i}))]. We find the expression on the right-hand side in (3.1) interesting for the fo* *llowing reason. The homotopy fixed point spectrum is defined with respect to a continuo* *us action of G on the spectrum. Thus, homotopy fixed points take into account the topology of the spectrum. Similarly, since the E2-term is built out of the pro- discrete cochain complex of spectra, the E2-term is always taking into account * *the topology of the spectrum. By [4, VI, Proposition 1.3], tow(SptG) is a model category, where {fi} is a w* *eak equivalence (cofibration) if and only if each fi is a weak equivalence (cofibra* *tion) in SptG. Theorem 3.2. The E2-term (1.2) of descent spectral sequence (1.1) has the form (3.3) Es,t2~=ssssst(limi( oGX0i)G ), where {Xi} ! {X0i} is a trivial cofibration with {X0i} fibrant, all in tow(SptG* *). Proof.Let {X0i} be as stated in the theorem. By [4, VI, Remark 1.5], each X0iis fibrant and each map X0i! X0i-1is a fibration, all in SptG. For any k 0, we consider the expression holimi(( oGX0i)G )k = holimi(Map c(G, Mapc(G, . .,.Mapc(G, X0i) . .).))G , where Map c(G, -) appears k + 1 times. By [3, Section 3], the forgetful functor U :SptG ! Spt, Map c(G, -): Spt ! SptG, where Map c(G, X) = G (X), and the functor (-)G :SptG ! Spt all preserve fibrations. Thus, {X0i} is a tower of fibrations of fibrant spectra, all in Spt. This implies that {Map c(G, X0i)} is* * a tower of fibrations of fibrant spectra, in SptG, and hence, in Spt. By iteration, {Map c(G, Mapc(G, . .,.Mapc(G, X0i) . .).)} is a tower of fibrations of fibrant spectra, in SptG, so that {(Map c(G, Mapc(G, . .,.Mapc(G, X0i) . .).))G } is a tower of fibrations of fibrant spectra in Spt. Therefore, the canonical map limi(( oGX0i)G )k ! holimi(( oGX0i)G )k 6 DANIEL G. DAVIS is a weak equivalence. Since {(( oGXi)G )k} and {(( oG(Xi)f,G)G )k} are towers of fibrant spectra, t* *here is a zigzag of weak equivalences limi(( oX0i)G )k ! holimi(( oX0i)G )k holimi(( oXi)G )k ! holimi(( o(Xi)f,G)G* * )k, where = G . This zigzag of weak equivalences implies that ssssst(limi(( oGX0i)G )) ~=ssssst(holimi(( oG(Xi)f,G)G )). Corollary 3.4. Let {X0i} be as in Theorem 3.2. Then there is an isomorphism Es,t2~=Hs[sst(C*(G; {X0i}))], where Es,t2is the E2-term of (1.2). Proof.This follows immediately from applying Lemma 2.2 to Theorem 3.2. Remark 3.5. By Theorem 2.5, Hs[C*(G; {sst(X0i)})] ~=Hscts(G; limisst(Xi)). 4. The failure of other possible descriptions of the E2-term After studying the expression in (3.3) further, one recalls that limi(-)G is * *the functor used to define Hscont(G; -), and, if M is any discrete G-module, then 0 ! M ! *GM is a (-)G -acyclic resolution of M, so that Hs[( *GM)G ] = Hsc(G; M). Let {Mi} be a tower of discrete G-modules. If {0} ! {Mi} ! { *GMi} is a limi(-)G -acyclic resolution of {Mi} in tow(DMod (G)), then Hs[(limi(-)G )({ *GMi})] = Hscont(G; {Mi}). This would imply that Es,t2~=Hs[sst(limi( *GX0i)G )] is computed by taking the cohomology of the homotopy groups of a complex of spectra that, in the context of abelian groups, computes continuous cohomology. This would be an interesting presentation of the E2-term. However, it is not hard to show that {0} ! {Mi} ! { *GMi} need not be a limi(-)G -acyclic resolution of {Mi} in tow(DMod (G)), so that t* *he above interpretation of the E2-term does not work out. For example, by [5, (2.1* *)], there is a short exact sequence 0 ! lim1iHs-1c(G; G Mi) ! Hscont(G; { G Mi}) ! limiHsc(G; G Mi) ! 0, for each s 0, where H-1c(G; -) = 0. Therefore, when s 1, Hsc(G; G Mi) = 0, so that, for all s 2, Hscont(G; { G Mi}) = 0. But, the short exact sequence a* *lso implies that H1cont(G; { G Mi}) ~=lim1iMi, which need not vanish. Thus, { G Mi}, the first object in the complex { *GMi}, need not be limi(-)G -acyclic in tow(DMod (G)). THE E2-TERM OF THE DESCENT SPECTRAL SEQUENCE 7 Upon further consideration of the expression in (3.3), one notices that, for * *any k, l, m 0, ((limi( m+1GX0i)G )k)l= limi( m+1G((X0i)k)l)G ~=Map c(Gm , limi((X0i)k)l) is an isomorphism of sets. If one could promote this isomorphism to (4.1) limi( m+1GX0i)G ~=Map c(Gm , limiX0i), then one could use this to interpret the expression in (3.3) as being the cohom* *ology of homotopy groups applied to the complex of continuous cochains with target ("coefficients") the continuous G-spectrum limiX0i. But notice that, in this interpretation, the expression Map c(Gm , limiX0i) d* *oes not have the desired meaning. For isomorphism (4.1) to hold, limiX0imust be a spectrum whose simplicial sets have simplices with the pro-discrete topology. But, as a Bousfield-Friedlander spectrum, in the construction Map c(Gm , limiX0* *i), limiX0iconsists of simplicial sets whose simplices all have the discrete topolo* *gy, by default. This conflict means that this interpretation also fails to work. References [1]A. K. Bousfield and D. M. Kan. Homotopy limits, completions and localization* *s. Springer- Verlag, Berlin, 1972. Lecture Notes in Mathematics, Vol. 304. [2]Daniel G. Davis. Iterated homotopy fixed points for the Lubin-Tate spectrum.* * In preparation, Purdue University, 2006. [3]Daniel G. Davis. Homotopy fixed points for LK(n)(En ^ X) using the continuou* *s action. To appear in the Journal of Pure and Applied Algebra, accepted July 13, 2005. [4]Paul G. Goerss and John F. Jardine. Simplicial homotopy theory. Birkh"auser * *Verlag, Basel, 1999. [5]Uwe Jannsen. Continuous 'etale cohomology. Math. Ann., 280(2):207-245, 1988. [6]J"urgen Neukirch, Alexander Schmidt, and Kay Wingberg. Cohomology of number * *fields, vol- ume 323 of Grundlehren der Mathematischen Wissenschaften [Fundamental Princip* *les of Mathematical Sciences]. Springer-Verlag, Berlin, 2000. [7]R. W. Thomason. Algebraic K-theory and 'etale cohomology. Ann. Sci. 'Ecole N* *orm. Sup. (4), 18(3):437-552, 1985.