The Chromatic Filtration and the Steenrod Algebra

Transcription

1 Contemporary Mathematics Volume 00, 0000 The Chromatic Filtration and the Steenrod Algebra JOHN H PALMIERI Abstract Using Steenrod algebra tools (Pt s -homology, in particular), we develop machinery analogous to the chromatic spectral sequence (and associated paraphernalia) of Miller, Ravenel, and Wilson in [MRW77]; and for any p-local spectrum X we use the Steenrod algebra to construct a variant of the chromatic tower over X 1 Introduction Fifteen years ago in [MRW77], Miller, Ravenel, and Wilson introduced the word chromatic to describe structure present in the Adams-Noviov spectral sequence E 2 -term; their ideas have been carried over to the geometric setting, as by Ravenel in [Rav] and[rav87] (see also his seminal paper [Rav84]) and by Mahowald and Sadofsy in [MS] If one reads too much of this sort of thing, one gets the impression that the Steenrod algebra is not much help in trying to understand these structures; the purpose of this paper is to try to correct this misconception In particular, we describe how the chromatic structure manifests itself for modules over the Steenrod algebra, and we use Steenrod algebra tools to construct a tower of spectra which is related to the chromatic tower On the algebraic side, we show that Margolis Postniov tower for Pt s- homology in [Mar83, Section 222] is actually closely related to the chromatic resolution in [MRW77]; along these lines, we develop other algebraic tools similar to those used in [MRW77] to calculate the E 2 -term We have tried to mae all of the constructions loo just lie the BP versions; this has been a success for all of the algebraic structures except the cohomology of the Morava stabilizer algebras We give a partial replacement for this last piece On the geometric side, Margolis construction for spectra gives a (convergent) filtration of π (X) for any finite spectrum X; this is related to the various 1991 Mathematics Subject Classification Primary 55S10, 55P42; Secondary 55N22, 55T99 This paper is in final form and no version of it will be submitted for publication elsewhere c 0000 American Mathematical Society /00 $100 + $25 per page 1

2 2 JOHN H PALMIERI versions of the chromatic filtration, but it is not clear how closely Indeed, our method of construction will loo similar to the geometric or telescope chromatic filtration using the functors L n of [Rav] and[ms], and the map X L n X factors through our replacement for L nx; hence the same is true of the map X L n X,whereL n is the localization functor used in the algebraic chromatic filtration of [Rav87] The structure of the paper is as follows: in Section 2 we recall some facts about the Steenrod algebra, and we construct some modules and spectra for later use In Section 3 we present the algebraic constructions analogous to the chromatic spectral sequence and associated machinery of [MRW77], including the Margolis chromatic spectral sequence and Bocstein spectral sequences for computing each column, as well as a candidate for a replacement for the cohomology of the Morava stabilizer algebras In Section 4 we do geometric constructions, related to the chromatic filtrations in [Rav87], [Rav] and[ms] Many of the results in this paper have appeared elsewhere in some form or other; to preserve continuity, we have relegated the few proofs that we give to Section 5 The author would lie to than Haynes Miller for providing the original inspiration for this wor; as a result, some of these ideas appeared in the author s thesis We also than Hal Sadofsy for numerous helpful conversations about chromatic things and the Steenrod algebra 2 Recollections We recall some facts about the Steenrod algebra Let p be a prime number, and let A be the mod p Steenrod algebra As an algebra, the dual of A is isomorphic to F p [ξ 1,ξ 2,] E(τ 0,τ 1,)ifp is odd, and to F 2 [ξ 1,ξ 2,]if p = 2 see [Mil58] Let the Milnor basis of A be the dual to the monomial basis, and let Pt s be the Milnor basis element dual to ξps t, and, when p is odd, let Q n be dual to τ n (when p =2,letQ n = Pn+1 0 ) Onecanchecthat(P t s ) p =0 when s<tand (Q n ) 2 =0foralln, and so we can define homology with respect to these elements; eg, given an A-module M, let H(M,Pt s )= er(p t s : M M) im((pt s)p 1 : M M) Hence, we will refer to these elements as differentials (see [Mar83], [MW81], or [Pal92], for instance) Given a differential x, define its slope, s(x), by s(pt s )= p P t s and s(q n )= Q n 2 (This definition is motivated by the vanishing line theorems in [AD73] and [MW81]) The differentials are linearly ordered by slope; let x 0,x 1,x 2, be the differentials in this ordering (at the prime 2 these are P1 0,P2 0,P2 1,; at odd primes

3 THE CHROMATIC FILTRATION AND THE STEENROD ALGEBRA 3 these are Q 0,Q 1,P1 0,Q 2,) Given integers m and n with 0 m n, saythat a module M is of type x m,x n (also written M = M x m,x n )ifh(m,x )=0 whenever <mor >n We have similar notation for spectra, of course Let C(x ) be the subalgebra of A generated by x,andletu be the polynomial generator in Ext C(x ) (F p, F p )(soifp =2orx = Q n,thenu is in bidegree (1,s(x )); otherwise, u is in bidegree (2, 2s(x ))) Remar 21 For convenience of notation, throughout this paper we wor at the prime 2; the main consequence of this is that u then always has homological degree 1 The only change necessary at odd primes is that some numbers will need to be adjusted Given an A-module V,wesaythatf Ext A (V,V )isaui -map of V if f restricts to u i 1 V Ext C(x ) (V,V ) Similarly, a ui -map of a spectrum X is an element g [X, X] which is represented at the E 2 -term of the Adams spectral sequence by a u i -map of H (X) In both the algebraic and geometric settings, we will denote such maps by u i Recall the following theorem: Theorem 22 ([AM71], [MP72]) Let M be a bounded below module over A ThenM is free if and only if H(M,x )=0for all This provides some motivation for our next definition: since x -homology ignoresfree A-modules (and hence mod p Eilenberg-MacLane spectra), so should we We wor in the stable categories of Margolis see [Mar83], Chapters 14 and 17 for more details We give a brief overview of the relevant facts The stable category of A-modules has bounded below modules for objects, and the maps are the following bigraded group: given A-modules L and M, set{l, M} 0, = Hom A (L, M)/, where is stable equivalence: f 0iff factors through a projective module To define {L, M}, in general, we need a bit more notation: for any A-module M, define ΩM by the short exact sequence 0 ΩM PM M 0 where PM is a projective module; then one can chec that ΩM is well-defined up to stable equivalence Set { {Ω {L, M} s, s L, M} 0, if s 0, = {L, Ω s M} 0, if s 0 One has Ext s, A (L, M) = {L, M} s, if s>0(see[mar83, Proposition 148]), so an advantage of this viewpoint is that elements of Ext are equivalence classes of maps between actual modules, using Ω (We will usually apply this to a u i - map; note, for example, that a u i -map of a module V induces an isomorphism H(Σ is(x) Ω i V,x ) = H(V,x )) Another advantage of woring in this category is that one can turn any map into a surjection: given an A-module map f : L M, one can replace f by

4 4 JOHN H PALMIERI f : L PM M This is analogous to turning any map of spaces into a fibration Let K be the ernel of f; wesaythat 0 K L M 0 is a stable short exact sequence The module K is the stable ernel of f; itis well-defined up to stable equivalence There are analogous constructions for spectra; we won t give the details Briefly, we wor in the stable category of p-local spectra, in which a map is stably trivial if it factors through a locally finite mod p generalized Eilenberg-Mac Lane spectrum; there is an analogous construction to Ω which we will denote l (so that for any spectrum X there is a cofibration X f KX lx, where KX is a mod p generalized Eilenberg-Mac Lane spectrum and H (f) is surjective; ie, up to suspension, l X is the th stage in an Adams tower for X) We use {, } for maps in this category, as well We shall also use the notion of stable cofibration sequence frequently Lastly, we define the subcategory of stably finite A-modules to be the smallest category containing all finite A-modules, which is closed under stable equivalences and taing summands and stable ernels Similarly, a spectrum is stably finite if it is in the smallest subcategory of stable p-local spectra containing finite spectra, closed under stable equivalences, retracts, and stable cofibers We have the following results Theorem 23 (a) ([Pal92, Theorem 22]) If M is a stably finite A- module of type x n, with H(M,x n ) 0,thenM has a non-nilpotent u i n-map for some i>0 (b) ([PS, Proposition 21]) IfX is a stably finite spectrum of type x n, with H(H (X),x n ) 0,thenX has a u i n-map for some i>0 We refer to the following as the illing construction for x -homology groups Theorem 24 ([Mar83], [Pal92], [PS]) (a) Given a bounded below A- module M and an integer n, then there is a stable short exact sequence of bounded below modules 0 M x 0,x n 1 f M g M x n, 0, where H(M x 0,x n 1,x )=0if n and H(f,x ) is an isomorphism if <n(and similarly for M x n, and g) (b) We can realize this geometrically: given a connective spectrum X and an integer n, there is stable cofibration sequence of connective spectra X x n, γ X φ X x 0,x n 1,

5 THE CHROMATIC FILTRATION AND THE STEENROD ALGEBRA 5 so that if M = H (X), then applying cohomology gives the short exact sequence in (a) Remar 25 (a) The modules (and spectra) with x -homology groups illed are unique up to stable equivalence; for instance, for any A-module M, one can get the stable short exact sequence for M (up to stable equivalence) by tensoring M with the stable short exact sequence for F p The same goes for the spectrum result; see [Mar83] (b) If M is of finite type (resp, X is locally finite), then so are M x 0,x n 1 and M x n, (resp, X x 0,x n 1 and X x n, ) Margolis [Mar83] proves this at the prime 2; we prove it in general in Section 5 Definition 26 Given integers n and an A-module M, define M x,x n to be any A-module stably equivalent to (M x, ) x 0,x n ; note that this is stably equivalent to (M x 0,x n ) x, WewillalsowriteM x n for M x n,x n We will use the construction of these modules and spectra below; see Proposition 28 for our ey result To describe this construction, and for other reasons, we define certain stably finite A-modules and spectra; first we need some notation Notation 27 We use capital letters to represent multi-indices; for example, I denotes (i 0,,i 1 ) (the subscript on I indicates the length of the multiindex) Given such a multi-index I of length and an integer r, we let I r =(i 0,,i r 1 ) Let I 0 = φ Also, with U =(u 0,,u 1 ), we let U I =(ui0 0,,ui 1 1 ) For each integer, we partially order the multi-indices I of length as follows: set I J if i r j r for all r, 0 r 1 For each 0 and suitable I, we inductively define stably finite A-modules V (U I ), so that V (U I ) is of type x, with non-zero x l -homology for l see [Pal92, Lemma 32] Start by setting V (U I0 0 )=F p Now, assume that we have constructed V (U I ), stably finite, of type x,, with H(V (U I ),x l) 0 for all l By Theorem 23, V (U I )hasaui -map for some i, so define V (U I ) by the stable short exact sequence 0 V (U I ) Σ i s(x ) Ω i V (U I ) u i V (U I ) 0 Then V (U I ) is stably finite, of type x +1,, and with non-zero x l -homology for all l +1 Note that for each, there are infinitely many multi-indices I for which we can define such a module V (U I ), since one can choose different powers of the maps u ir r at each stage One has a similar construction for spectra (for which the exponents i may need to be chosen larger); see [PS, Theorem 33] The end result is a collection of stably finite spectra W (U I ), each indexed by an integer and a -tuple of

6 6 JOHN H PALMIERI integers I =(i 0,,i 1 ), so that W (U I )hasaui -map for some i > 0and there are stable cofibration sequences W (U I ) u i Σ i s(x ) l i W (U I ) W (U I ) Note that for both the module and the spectrum cases, the maps u ir r can be chosen compatibly for different multi-indices I, so that, for example, if I is smaller than J, then there is a map V (U J ) V (U I ); for the spectrum case one needs to use the nilpotence theorem to see this see [PS] Also, by construction each V (U I ) is deloopable times, and F p maps to Ω V (U I ); let V (U I ) be the stable ernel Similarly, let W (U I )bethestable cofiber of the map l W (U I ) S0 We have the following proposition; parts (a) and (b) are the ey results in the proofoftheorem24 Proposition 28 ([Pal92], [PS]) (a) Taing the inverse limit over the partial ordering of multi-indices I n of length n of the stable short exact sequence 0 V (U In n ) F p Ω n V (U In n ) 0 gives the stable short exact sequence in Theorem 24(a) (b) Similarly, taing the direct limit over the I n s of l n W (U In n ) S 0 W(U In n ) gives the stable cofibration sequence of Theorem 24(b) (c) If M is a stably finite module of type x n,, then M x n lim ( u in n Σ ins(xn) Ω in M uin n M) (d) If X is a stably finite spectrum of type x n,, then X x n lim(x u in n Σ ins(xn) l in X uin n ) 3 The algebraic constructions In this section we describe the analogs in the Steenrod algebra setting to the results in [MRW77] In particular, we have analogs for the chromatic resolution, for the chromatic spectral sequence, and for the Bocstein spectral sequences used to compute each column of the chromatic spectral sequence; we also have a partial replacement for the role played by the cohomology of the Morava stabilizer algebras

7 THE CHROMATIC FILTRATION AND THE STEENROD ALGEBRA 7 31 The Margolis chromatic resolution In this subsection we construct a resolution of F p by A-modules analogous to the chromatic resolution of BP by BP BP-comodules The end result will be the tower constructed in Section 222 of [Mar83] (see Theorem 31) Our construction and our point of view are the new results here We define modules N m and M m for m 0, which play roles analogous to those of the BP BP-comodules of the same name in [MRW77] We give two equivalent descriptions The quic way to define them is to let N m =Ω m F p x m, and M m = Ω m F p x m ;thenforeachm we have a stable short exact sequence 0 N m+1 M m N m 0 Splicing these short exact sequences together yields M 2 M 1 M 0 F p 0 Ω 2 F p x 2 Ω 1 F p x 1 F p x 0 F p 0 As in the BP BP picture, we have resolved the sphere by monochromatic objects Call this the Margolis chromatic resolution of F p ; the terms and maps are well-defined up to stable equivalence Note that this is the same as the tower constructed in [Mar83, Section 222] Also, given any bounded below A-module K, we can tensor the tower with K to get a Margolis chromatic resolution for K Another way to describe this tower bears stronger resemblance to the construction in [MRW77] We define N m and M m inductively, starting with N 0 = F p We have, for any i 0 > 0, 0 V (u i0 0 ) u i0 Σ i0 Ω i0 F 0 p Fp 0 u i V (u 2i0 0 ) u 2i0 Σ 2i0 Ω 2i0 0 F p F p 0 Since all the modules involved are of finite type, taing inverse limits over i 0 gives a short exact sequence, which we use to define M 0 and N 1 : 0 N 1 M 0 N 0 0 By Proposition 28, we have N 1 ΩF p x 1, and M 0 F p x 0

8 8 JOHN H PALMIERI Now we iterate this, as in [MRW77]: for each m 0 and suitable I m,we have 0 V (U Im+1 m+1 ) Σ ims(xm) Ω im V (U Im u im m m ) 0 V (U I m+1 m+1 ) Σ 2ims(xm) Ω 2im V (U Im m ) u im m V (Um Im ) 0 u 2im m V (U I m m ) 0 (where i = i for <m,andi m =2i m ) Taing inverse limits over the vertical maps yields the same short exact sequence as above, 0 Ω m+1 F p x m+1, Ω m F p x m Ω m F p x m, 0 32 The Margolis chromatic spectral sequence Let K and L be A- modules Applying {,L} to the Margolis chromatic resolution for K gives a trigraded spectral sequence which we call the Margolis chromatic spectral sequence: Theorem 31 ([Mar83], Theorem 224) Given bounded below A-modules K and L, there is a spectral sequence with E s,t,u 1 = {K M s,l} t,u = {K xs,l} s+t,u and d r : Er s,t,u Er s+r,t+1 r,u, abutting to {K, L} s+t,u If K is of finite type, then this spectral sequence converges Margolis proves this at the prime 2; his proof carries over easily to the odd prime case This is our analog of the chromatic spectral sequence of [MRW77]; the sth column {K x s,l} contains information which should be, in some sense, periodic with respect to u s and torsion with respect to all of the periodicity operators u when <s; the spectral sequence tells us how to assemble these various inds of periodic information to get, essentially, the Adams E 2 -term Remar 32 If K is finite, then {K, L} s, = Ext s, A (K, L) for all s 0, and {K, L} s, =0ifs<0 see [Pal92, Proposition 19] So in this case we get precisely the Adams E 2 -term 33 Bocstein spectral sequences Another ingredient of the BP chromatic picture is the following: for each n, there is a collection of Bocstein spectral sequences, in which the E 1 -term of the first is purely v n -periodic and computable (as the cohomology of the nth Morava stabilizer algebra), each converges to the E 1 -term of the next, and the last converges to the nth column of the chromatic spectral sequence We have the same machinery here, constructed in the same fashion as the BP version Fix n and a suitable I n =(i 0,,i n 1 ) Recall that for n we have I =(i 0,,i 1 ) The V (U I ) s with n fit together in n Bocstein short

9 THE CHROMATIC FILTRATION AND THE STEENROD ALGEBRA 9 exact sequences, starting with V (φ) =F p Applying x n is exact (eg, tensor with F p x n ), so we get new Bocstein short exact sequences: 0 V (U I ) x n Σ i s(x ) Ω i V (U I ) x n u i V (U I ) x n 0 Applying {,L} to these gives a collection of Bocstein spectral sequences, in which the th has E 1 -term {V (U I n +1 n +1 ) x n,l} and abuts to E 1 of the next, {V (U I n n ) x n,l} So we have a cascade of BSS s, ending with {F p x n,l}, as desired (And, of course, we can tensor the short exact sequences with K to compute {K x n,l},instead) Proposition 33 If L is stably finite, then these spectral sequences all converge Proof This follows from [MRW77, Remar 311] and a vanishing line argument; namely, if we have a spectral sequence with E 1 I = {V (U ) x n,l}, abutting to V (U I ) x n,l}, then to show that the spectral sequence converges, it suffices to show that {V (U I ) x n,l} is torsion under multiplication by u i This follows from the vanishing line theorem of [MW81]: {V (U I ) x n,l} has a vanishing line of slope s(x n )(see[pal92] for the vanishing line theorem in this generality), and multiplication by u i then s(x ) <s(x n ) acts at a slope of s(x ) Since <n, 34 The start of the Bocstein cascade The last piece of the chromatic machine for BP is the computation of the E 1 -term of the first BSS for each n, as the cohomology of the nth Morava stabilizer algebra (see [MRW77], [MR77]) We don t have a precisely analogous result, but we have a (partial) replacement The salient feature of the E 1 -term of the first BSS in our cascade is that V (Un In ) is stably finite of type x n,, so we have the following corollary of Theorem 24 and Proposition 28 Lemma 34 ([Pal92], Theorem 22) If L is stably finite, then {V (Un In ) x n,l} = u 1 n Ext In A (V (Un ),L) So regardless of whether we can compute this, it is purely u n -periodic, as in the BP case There are some tools available for calculating localized Ext groups; we will discuss one here, the Margolis Adams spectral sequence Theorem 35 Fix a differential x n A; let A(x n ) be the algebra of operations for x n -homology (defined below) LetK be stably finite of type x n,, L stably finite; then there is a spectral sequence with E 2 = Ext (H(K, x A(xn) n),h(l, x n )) F p [u ±1 n ], abutting to u 1 n Ext A (K, L) If p =2and x n = Q j, then the spectral sequence converges

10 10 JOHN H PALMIERI (Note that if x n = Q j,thenu n = v j ) Margolis constructed this spectral sequence in [Mar]; independently, Eisen used it to compute certain v 1 j Ext groups in his thesis [Eis87] We prove Theorem 35 in [Pal]; we give a brief description here, for completeness First of all, we need to define the algebra of operations for H(,x n ); to do this we use the following lemma Lemma 36 ([Mar83], 193) For any bounded below A-module K, we have H i (K, x n ) = {A/Ax n,k} 0, i Define the algebra of operations for x n -homology, A(x n ), to be the opposite algebra to H (A/Ax n,x n )={A/Ax n,a/ax n } 0, Then for any A-module K, H(K, x n )isalefta(x n )-module The A(x n ) s can be computed when x n = Q j (eg, for p =2see[Mar83, 1925]) The spectral sequence is constructed in a standard fashion: one taes an A/Ax n -resolution of K so that applying H(,x n )givesana(x n )-free resolution of H(K, x n ); then one applies u 1 n Ext A (,L) One uses Lemma 36 to identify the E 2 -term One can show that convergence follows from a certain generic condition; convergence then follows by exhibiting a nice enough module which satisfies this condition (for p =2andx n = Q j, this module is 1 2 A j = A j /A j Q j ; here A j is the sub-hopf algebra of A generated as an algebra by the set {Pt s : s + t j +1}, with Mitchell s A-module structure see [Mit85]) 4 The geometric construction In this section we give the Steenrod algebra analog of the chromatic filtration for spectra; as for the Margolis chromatic resolution of F p, this first appeared for p =2in[Mar83, Section 222] As above, our construction and our point of view are the new results here This filtration bears some relation to the chromatic filtrations of [Rav84], [Rav] and[ms], but it is not clear exactly what this relation is Our main tool here is the self-map theorem for spectra, Theorem 23(b) As in the algebraic case, we give two descriptions: according to [Mar83, Section 222], we let N m = l m S 0 x m, and M m = l m S 0 x m ; then we have stable cofibration sequences (as in Theorem 24) N m M m N m+1 Splicing these together gives a resolution of S 0 ; smashing this with a connective spectrum Z gives a tower over Z: Z Z M 0 Z M 1 Z Z x 0 lz x 1

11 THE CHROMATIC FILTRATION AND THE STEENROD ALGEBRA 11 We can also rewrite this as Z Z x 0,x Z x 0,x 1 Z x 0, as in [Mar83, Section 222] (where the maps are the ones given in Theorem 24) Call this the Margolis chromatic tower for Z Instead we can construct the M m s and N m s using direct limits, as in the algebraic case As before, start with N 0 = S 0 For any i 0 > 0, we have this diagram of stable cofibration sequences: S 0 u i 0 0 Σ i 0s(x 0) l i0 S 0 W (u i0 u i 0 0 S 0 u 2i 0 0 Σ 2i0s(x0) l 2i0 S 0 Taing direct limits gives 0 ) W (u 2i0 0 ) N 0 M 0 N 1, with M 0 S 0 x 0 and N 1 ls 0 x 1, Now iterate this: for each m 0 and suitable I m,wehave W (U Im m ) W (U Im m ) u im m Σ ims(xm) l im W (U Im u im m m ) W (U Im+1 m+1 ) u 2im m Σ 2i ms(x m) l 2im W (U Im m ) W (U I m+1 m+1 ) (where i m =2i m ) As in the algebraic case, we tae limits (direct, this time), and end up with stable cofibration sequences l m S 0 x m, l m S 0 x m l m+1 S 0 x m+1,, as desired This construction clearly bears some resemblance to that of the geometric chromatic filtration of [MS] and[rav87]; in fact, Mahowald and Sadofsy use a modification of this construction to construct L nx and to study it via a localized Adams spectral sequence In any case, given a spectrum Z, we get a tower over Z; now we can apply {S 0, } Theorem 41 ([Mar83], Theorem 225, Proposition 172) Let Z be a connective p-local spectrum Z is the limit (in the stable category of spectra) of its Margolis chromatic tower; hence, applying {S 0, } to the Margolis chromatic tower for Z yields a Hausdorff, complete filtration of {S 0,Z}; furthermore, if Z is finite, then {S 0,Z} = π (Z) As in the module case, Margolis proves this at the prime 2; his proof wors at odd primes, too Note that this theorem says that Z is stably equivalent to the limit of its Margolis chromatic tower; in other words, the limit of the Margolis

12 12 JOHN H PALMIERI chromatic tower is Z K where K is some mod p generalized Eilenberg-Mac Lane spectrum Recall that L n is localization with respect to the spectrum E(n); equivalently, it is localization with respect to L n is localization with respect to K(0) K(n) T (0) T (n), where T () isthetelescopeofav -map on a finite spectrum of type ; L n is independent of the choice of these finite spectra and the choice of the self-maps (Note that L n is also nown as L f n see [Mil]) One has the factorization X L n X L nx for all X See [MS], [Mil], [Rav], [Rav84], and [Rav87] for more information on these functors We have the following observations; the second implies the first (recall that x 0 = Q 0 ): Proposition 42 ([Mar83], p434) Fix p = 2; then X L n X factors through X Q 0,Q n Proposition 43 For any p, X L nx factors through X Q 0,Q n (We prove this in Section 5) Indeed, we get a commutative diagram of towers X Q 0,Q n L nx L n X X Q 0,Q 1 L 1X L 1 X u uuu u uuu X X Q 0 L 0X L 0 X Note that the tower on the left has intermediate terms from the differentials between the Q n s This reflects the fact that X L nx factors through X X Q 0,x whenever s(x ) s(q n )

13 THE CHROMATIC FILTRATION AND THE STEENROD ALGEBRA 13 Assume X is finite The tower on the right converges, in the sense that X is the homotopy inverse limit of L n X; this is due to Hopins and Ravenel in [HR]; see also [Rav92] The tower on the left converges up to stable equivalence; this is the gist of Theorem 41 It is not nown whether the middle tower converges In any case, the Margolis chromatic filtration for spectra is clearly related to these chromatic filtrations; this connection should be of some use in studying chromatic phenomena, as evidenced by the wor in [MS] 5 A few proofs 51 If M is of finite type, so is M x m,x n In [Pal92] wegaveaproofof Margolis illing construction; here we prove that if M is an A-module of finite type (and bounded below, of course), then M x m,x n is of finite type, as well As in the proof of the illing construction, [Pal92, Theorem 33], it suffices to prove this for F p x, for each differential x A Proposition 51 Fix a differential x n in A The module F p x n, is stably equivalent to a module of finite type Corollary 52 Fix a differential x n in A ThespectrumS 0 x n, is stably equivalent to a locally finite spectrum Proof Recall (Proposition 28) that F p x n, is constructed as an inverse limit of modules Ω n V (Un In ) We get an inverse system of the Ω n V (Un In ) s as we increase I n ; we want to chec that we get an isomorphism through an initial range of dimensions, increasing with I n Since the Ω n V (Un In ) s are of finite type,thisisgoodenough Proposition 51 is a consequence of the following Lemma 53 Given d, for J n large enough (ie, each j 0) and I n >J n, Ω n V (Un In ) and Ω n V (Un Jn ) may be chosen so that the induced map Ω n V (Un In ) Ω n V (Un Jn ) is an isomorphism through degree d Proof The proof is by induction on Assume that we have stably finite modules V and W, both of type x, with non-zero x -homology Furthermore, assume that we have g : V W which is an isomorphism through degree d Then for suitable i and j with i>j, we have a commutative diagram 0 Σ is(x) Ω i u i V V V 0 Ω i g 0 Σ is(x) Ω i u i W W u i j 0 Σ js(x) Ω j u j W W W 0 g g

14 14 JOHN H PALMIERI The map Ω i g is an isomorphism through degree d; by a relative vanishing line argument (as in [Pal92, Theorem 22(d)]), if j 0thenu i j is an isomorphism through degree d; hence the induced map g : V W is, also Now apply the above argument to the case V =Ω ( 1) V (U I 1 1 )andw = Ω ( 1) V (U J 1 1 )(sov =Ω V (U I )andw =Ω V (U J )) This finishes the proof of Proposition X L nx factors through X X Q 0,Q n Proof Let x 1 = Q n ;letu =(u 0,u 1,,u 1,) We have a cofibration Z W (U I ) M(U I ), where W (U I ) is a stably finite spectrum, M(U I ) is finite, and Z has a finite Adams resolution In particular, M(U I ) is defined inductively by cofibrations M(U Ir r ) uir r Σ i r(s(x r) 1) M(U Ir Ir+1 ) M(U r r+1 ), for r =0, 1,, 1 (see[ps]) Note that M(U I ) is constructed so that its top cell is in dimension, and as with W (U I ), there is a map Σ M(U I ) S0 We define M(U I ) by the cofibration Σ M(U I ) S0 M(U I ), and we define W (U I ) by the stable cofibration l W (U I ) S0 W (U I ) In [MS], Mahowald and Sadofsy show that lim M(U I )=L n S0 ; in [PS], Sadofsy and the author show that lim W (U I )=S0 Q 0,Q n We want to construct a map S 0 Q 0,Q n L n S0 that factors the L n localization map Given a spectrum X, letf X denote the th term in an Adams tower for X Observe that for any, l X Σ F X We need to do two things: we need to show that the map W (U I ) M(U I ) factors through Σ l W (U I ), and we need to show that the resulting maps in the two direct systems are compatible We re given a map W (U I ) M(U I ) Now, for any finite spectrum Y and any connective spectrum X, anymaplx Y is zero on cohomology (cf [Pal92, Proposition 19]); hence it lifts to a map lx FY,andsotoamap ΣX Y Apply this observation times to the map W (U I ) M(U I ), using W (U I )=l (l W (U I )), to get the desired factorization

15 THE CHROMATIC FILTRATION AND THE STEENROD ALGEBRA 15 By applying [PS, Lemma 31], we see that (if I is sufficiently larger than J ) the map l W (U I ) l W (U J ) induces a map Σ M(U I ) Σ M(U J ), so we get a map of direct systems if we use these maps between the M s instead of the ones in [MS] But then the nilpotence theorem says that the two direct systems of M s have the same direct limit; this finishes the proof References [AD73] D W Anderson and D M Davis, A vanishing theorem in homological algebra, Comment Math Helv 48 (1973), [AM71] J F Adams and H R Margolis, Modules over the Steenrod algebra, Topology 10 (1971), [Eis87] D K Eisen, Localized Ext groups over the Steenrod algebra, PhD thesis, Princeton University, 1987 [HR] M J Hopins and D C Ravenel, to appear [Mar] H R Margolis, unpublished notes [Mar83] H R Margolis, Spectra and the Steenrod algebra, North-Holland, 1983 [Mil] H R Miller, Finite localizations, unpublished [Mil58] J W Milnor, The Steenrod algebra and its dual, Ann of Math (2) 67 (1958), [Mit85] S Mitchell, Finite complexes with A(n)-free cohomology, Topology 24 (1985), [MP72] J C Moore and F P Peterson, Modules over the Steenrod algebra, Topology 11 (1972), [MR77] H R Miller and D C Ravenel, Morava stabilizer algebras and the localization of Noviov s E 2 -term, Due Math J 44 (1977), [MRW77] H R Miller, D C Ravenel, and S Wilson, Periodic phenomena in the Adams- Noviov spectral sequence, Ann of Math (2) 106 (1977), [MS] M E Mahowald and H Sadofsy, v n-telescopes and the Adams spectral sequence, to appear [MW81] H R Miller and C Wilerson, Vanishing lines for modules over the Steenrod algebra, J Pure Appl Algebra 22 (1981), [Pal] J H Palmieri, TheMargolisAdamsspectralsequence, in preparation [Pal92] J H Palmieri, Self-maps of modules over the Steenrod algebra, J Pure Appl Algebra 79 (1992), [PS] J H Palmieri and H Sadofsy, Self maps of spectra, a theorem of J Smith, and Margolis illing construction, to appear in Math Zeit [Rav] D C Ravenel, Life after the telescope conjecture, toappear [Rav84] D C Ravenel, Localization with respect to certain periodic homology theories, Amer J of Math 106 (1) (1984), [Rav87] D C Ravenel, The geometric realization of the chromatic resolution, Algebraic Topology and Algebraic K-theory (W Browder, ed), 1987, Annals of Mathematics Studies, no 113, pp [Rav92] D C Ravenel, Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies, vol 128, Princeton University Press, Vincent Hall, School of Mathematics, University of Minnesota, Minneapolis, MN address: palmieri@mathumnedu