ISSN 364-0380 (on line) 465-3060 (pinted) 55 Geomety & Topology Volume 3 (999) 55 65 Published: 2 July 999 G G G G T T T G T T T G T G T GG TT G G G G GG T T T TT Vanishing lines in genealized Adams spectal sequences ae geneic MJ Hopkins J H Palmiei J H Smith Depatment of Mathematics, Massachusetts Institute of Technology Cambidge, MA 0239, USA Depatment of Mathematics, Univesity of Note Dame Note Dame, IN 46556, USA Depatment of Mathematics, Pudue Univesity West Lafayette, IN 47907, USA Email: mjh@math.mit.edu, palmiei@membe.ams.og, jhs@math.pudue.edu Abstact We show that in a genealized Adams spectal sequence, the pesence of a vanishing line of fixed slope (at some tem of the spectal sequence, with some intecept) is a geneic popety. AMS Classification numbes Pimay: 55T5 Seconday: 55P42 Keywods: Adams spectal sequence, vanishing line, geneic Poposed: Haynes Mille Received: 7 Mach 998 Seconded: Ralph Cohen, Gunna Calsson Revised: 27 May 999 c Geomety & Topology Publications
56 M J Hopkins and J H Palmiei and J H Smith Intoduction Ou main esult, Theoem.3, was motivated by seveal questions. The thick subcategoy theoem of Hopkins and Smith [4, Theoem 7] descibes some impotant stuctue in stable homotopy theoy: one can get a good deal of infomation about a finite p local spectum X by knowing its type that is, knowing the unique numbe n so that its nth Moava K theoy K(n) X is nonzeo while K(n ) X = 0. While one cannot in geneal detemine the type of a finite p local spectum by knowing just its mod p cohomology, one might ask whethe the type is eflected somehow in the behavio of the classical Adams spectal sequence. Ou main theoem shows that the answe is yes, and we explain this behavio (the existence of a vanishing line with a paticula slope) as ou fist application afte the statement of the theoem. Spectal sequences ae often sequences of bigaded abelian goups {E }, indexed stating at = o = 2. The E + tem is a subquotient of the E tem, so if one has some sot of vanishing esult at E, such as E s,t = 0 when s 3t + 4, then the same esult holds at E +k fo all k 0. As one might imagine, such esults can be quite useful. In a typical spectal sequence, one does not have an explicit desciption of E when 3, so most vanishing theoems have been poven at the E o E 2 tem; the main esult of [5] is a good example of this. One might ask how to poduce vanishing lines at late tems of a spectal sequence. Ou main theoem povides a way of doing this; see the second and thid applications afte the statement of the theoem fo examples. The following definition is at the heat of the pape. Definition. A popety P of specta is geneic if wheneve a spectum X satisfies P, then so does any etact of X; and if X Y Z is a cofibation and two of X, Y and Z satisfy P, then so does the thid. In othe wods, a popety is geneic if the full subcategoy of all specta satisfying it is thick. Fo example, fo any spectum E, the popety that E (X) = 0 is a geneic popety of X. We ecall thee othe concepts: fist, given a connective spectum W, we wite W fo its connectivity; that is, W is the lagest numbe w so that π n W = 0
Vanishing lines in Adams spectal sequences 57 when n w. Second, we assume that ou ing spectum E satisfies the standad assumptions fo the constuction and convegence of the E based Adams spectal sequence in othe wods, the assumptions necessay fo Theoem 5.(iii) in [, Pat III]; see also Assumptions 2.2.5(a) (c) and (e) in [8]. Moe pecisely, we assume that E satisfies the following. Condition.2 E is a ing spectum, associative and commutative up to homotopy. π E = 0 fo < 0. The map π 0 E π 0 E π 0 E, induced by multiplication on E, is an isomophism. Let R be the lagest subing of the ationals to which the ing map Z π 0 E extends; then H n (E;R) is finitely geneated ove R fo all n. Thid, in ode to state the main theoem pecisely, we need to wok with E complete specta: E complete is a technical condition on a spectum X which ensues that the E based Adams spectal sequence conveges to π X. We give a caeful definition below (Definition 3.); fo now, we point out that if the spectum X is connective, then E completeness is a mild estiction. Fo example, if π 0 E = Z, then evey connective X is E complete; if π 0 E = Z (p), then evey connective p local X is E complete. See [], [8], and [2] fo moe infomation. This is ou main esult. Theoem.3 Let E be a ing spectum satisfying Condition.2, and conside the E based Adams spectal sequence E (X) π (X). Fix a numbe m. The following popeties of an E complete spectum X ae geneic: (i) Thee exist numbes and b so that fo all s and t with s m(t s)+b, then E s,t (X) = 0. (ii) Thee exist numbes and b so that fo all finite specta W and fo all s and t with s m(t s W ) + b, then E s,t (X W) = 0. Remak.4 (a) One usually daws Adams spectal sequences E s,t with s on the vetical axis and t s on the hoizontal; in tems of these coodinates, the popeties say that E s,t is zeo above a line of slope m, with s intecept b in (i), and s intecept b m W in (ii).
58 M J Hopkins and J H Palmiei and J H Smith (b) Assuming that X is E complete ensues that the spectal sequence conveges, which we need to pove the theoem. We do not need to identify the E 2 tem of the spectal sequence, so we do not need to know that E is a flat ing spectum one of the othe standad assumptions on ing specta in discussions of the Adams spectal sequence. We mention thee applications of the theoem. Since fo each pime p thee is a classification of the thick subcategoies of the categoy of finite p local specta (see [4, Theoem 7]), then in this setting one may be able to identify all specta with vanishing line of a given slope. Conside, fo example, the classical mod p Adams spectal sequence. This is based on the ing spectum E = HF p, and evey finite p local tosion spectum is HF p complete. When p = 2, fo instance, since the mod 2 Mooe spectum has a vanishing line of slope 2 at the E 2 tem, then the mod 2 n Mooe spectum, and indeed any type spectum, has a vanishing line of slope 2 at some E tem. Similaly, pat of the poof of the thick subcategoy theoem is the constuction of a type n spectum with a vanishing line of slope 2 n+ 2 at the E 2 tem, and hence any type n spectum has a vanishing line of slope 2 n+ 2 at some E tem. At odd pimes, any type n spectum has a vanishing line of slope 2p n 2 at some E tem of the classical mod p Adams spectal sequence. Theoem.3 gives no contol ove the tem o the intecept b of the vanishing line. Since the poof of Theoem.3 is fomal, it also applies in any categoy which satisfies the axioms of a stable homotopy categoy, as given in [3]. The second autho has used this esult in an appopiate categoy of modules ove the Steenod algeba to pove a vesion of Quillen statification fo the cohomology Ext A (F 2,F 2 ) of the mod 2 Steenod algeba A (see [6]). A key pat of the agument is to show that fo a paticula ing R, thee is a map φ: Ext A (F 2,F 2 ) R satisfying these two popeties: Evey element in ke φ is nilpotent. Fo evey element y in R, thee is an n so that y 2n im φ. To pove this, one uses a cetain genealized Adams spectal sequence in the setting of A modules. This spectal sequence conveges to Ext A (F 2,F 2 ), and the ing R is the zeo line of the E 2 tem. The map φ is then the edge homomophism. To pove that φ has the desied popeties, one uses Theoem.3 togethe with some specific computations to show that fo any m > 0, the spectal sequence eventually has a vanishing line of slope m. Evey element in ke φ is epesented by an element above the zeo line, and hence must be nilpotent its powes lie along a line of some slope, and some E tem has a
Vanishing lines in Adams spectal sequences 59 vanishing line of smalle slope. Fo evey y R, one can show that fo all n, the tagets of the possible diffeentials on y 2n all lie above a cetain line of positive slope, and since some E tem has a vanishing line of smalle slope, then y 2n is a pemanent cycle fo all sufficiently lage n. See [6, 7] fo details. A thid application is the computation of localized homotopy goups. Suppose that E is a ing spectum satisfying Condition.2, and fix v π (E). Say that an E complete spectum X has a nice v action if X has a self-map which induces multiplication by some powe of v on E X, and if thee is an n so that the E based Adams spectal sequence conveging to π X is a spectal sequence of π 0 (E)[v n ] modules, compatibly with the v self map on X. Note that multiplication by v n does not incease Adams filtation: it acts hoizontally in this spectal sequence. Given a spectum X with a nice v (X), one gets a localized spectal sequence v E (X). A natual question is, can one compute v π X fom the localized spectal sequence? It is not had to show that if the unlocalized spectal sequence E (X) has a hoizontal vanishing line at some E tem, then the localized spectal sequence will convege to v π X: the vanishing line foces both the diffeentials and the extensions to behave action, if one invets v n in the E based Adams spectal sequence E well. Hence if one can find a spectum X fo which E (X) has a hoizontal vanishing line, one can conclude fom Theoem.3 that fo all specta with nice v actions in the thick subcategoy geneated by X, the localized spectal sequence conveges and computes the localized homotopy goups. 2 Composites of maps in towes In this section, we pove a lemma about geneicity and composites of maps in towes. In the next section, we will apply this lemma to the E based Adams towe to pove Theoem.3. Let F be an exact functo fom specta to towes of specta; in othe wods, given a spectum X, we have a diagam, functoial in X: F n+ X F n X F n X. By exact, we mean that if X Y Z is a cofibation, then so is F n X F n Y F n Z fo each n. We want to show that the following popety is geneic in X: all fold composites of maps in the towe F X ae zeo on homotopy, ie, ae null afte composition with any map fom a sphee. Actually, we efine this condition in
60 M J Hopkins and J H Palmiei and J H Smith two ways: fist, athe than using sphees as test specta, we use an abitay set of specta; and second, we only look at fold composites F s+ X F s X when s satisfies an inequality which may depend on the paticula test specta used. Lemma 2. Fix a numbe m and an exact functo F fom specta to towes of specta. Let T = {(W α,n α )} be a set of pais of the fom (spectum, numbe). The following condition is geneic in X: ( ) Thee exist numbes and b so that fo all (W α,n α ) T, all s with s mn α + b, and all degee zeo maps W α F s+ X, the composite W α F s+ X F s X is null. In the next section, we will apply this in the two cases T = {(S n,n) : n Z}, T = {(W, DW ) : W finite}, whee DW is the Spanie Whitehead dual of W. Poof If Y is a etact of X, then the towe F Y is a etact of F X, so if F s+ X F s X is null afte mapping in each W α, then so is F s+ Y F s Y. (Given W α F s+ Y, then conside W α F s+ Y F s Y F s+ X F s X F s+ Y F s Y Then the composite W α F s Y factos though W α F s+ X F s X.) The condition ( ) involves numbes and b, and we wite ( ),b if we want to specify the numbes. Given a cofibation sequence X f Y g Z in which X and Z satisfy conditions ( ),b and ( ),b, espectively, we show that Y satisfies ( ) +,max(b,b ). Con-
Vanishing lines in Adams spectal sequences 6 side the following commutative diagam, in which the ows ae cofibations: F s++ X F s++ Y β F s++ (g) F s++ Z γ F s+ X F s+ Y F s+ Z δ ε F s X F s Y F s Z We fix (W α,n α ) T and assume that s mn α + max(b,b ), so that we have s mn α + b, s + mn α + b. Given any map ζ : W α F s++ Y, then since γ F s++ (g) ζ is null, the composite β ζ factos though F s+ X, giving ζ : W α F s+ X. Since δ ζ is null, though, then the composite is null. W α ζ Fs++ Y β F s+ Y ε F s Y 3 Adams towes and the poof of Theoem.3 The difficulty in poving a esult like Theoem.3 is that the E tem of an Adams spectal sequence does not have nice exactness popeties if 3 a cofibation of specta does not lead to a long exact sequence of E tems, fo instance. So we pove the theoem by showing that the pupoted geneic conditions ae equivalent to othe conditions on composites of maps in the Adams towe, and we apply Lemma 2. to conclude that those othe conditions ae geneic. We stat by descibing the standad constuction of the Adams spectal sequence, as found in [, III.5], [8, 2.2], and any numbe of othe places. Given a ing spectum E, we let E denote the fibe of the unit map S 0 E. Fo any intege s 0, we let F s X = E s X, K s X = E E s X.
62 M J Hopkins and J H Palmiei and J H Smith We use these to constuct the following diagam of cofibations, which we call the Adams towe fo X: g g g X F 0 X F X F 2 X. K 0 X K X K 2 X This constuction satisfies the definition of an E Adams esolution fo X, as given in [8, 2.2.] see [8, 2.2.9]. Note also that F s X = X F s S 0, and the same holds fo K s X the Adams towe is functoial and exact. We pause to define E completeness. Definition 3. A spectum X is E complete if the invese limit of its Adams towe is contactible. Given the Adams towe fo X, if we apply π, we get an exact couple and hence a spectal sequence. This is called the E based Adams spectal sequence. Moe pecisely, we let D s,t = π t s F s X, E s,t = π t s K s X. If we let g: F s+ X F s X denote the natual map, then g = π t s (g) is the map D s+,t+ D s,t. Then we have the following exact couple (the pais of numbes indicate the bidegees of the maps): (, ) D, (0,0) (,0) D, E, This leads to the following th deived exact couple, whee D s,t of g, and the map D s+,t+ D, D s,t is the estiction of g : (, ) D, is the image (, ) (,0) E,
Vanishing lines in Adams spectal sequences 63 Unfolding the th deived exact couple leads to the following exact sequence: E s,t+ D s+,t+ D s,t E s+,t+. (3.2) Fix a numbe m. With espect to the E based Adams spectal sequence E ( ), we have the following conditions on a spectum X: () Thee exist numbes and b so that fo all s and t with s m(t s)+b, then D s,t (X) = 0. (In othe wods, the map g : π t s (F s+ X) π t s (F s X) is zeo. In othe wods, fo all maps f : S t s F s+ X, the composite g f is null.) (2) Thee exist numbes and b so that fo all s and t with s m(t s)+b, then E s,t (X) = 0. (3) Thee exist numbes and b so that fo all finite specta W, all s with s m DW +b, and all maps W F s+ X, then the composite W F s+ X F s X is null. (Hee, DW denotes the Spanie Whitehead dual of W.) (4) Thee exist numbes and b so that fo all finite specta W and fo all s and t with s m(t s W ) + b, then E s,t (X W) = 0. As with the condition in the poof of Lemma 2., each condition involves a pai of numbes and b, and we wite (),b to indicate condition () with the numbes specified, and so foth. If m = 0, fo instance, then condition (3) says that F s+ X F s X is a phantom map wheneve s b. If m = 0, then condition () says that F s+ X F s X is a ghost map (zeo on homotopy) wheneve s b. Lemma 3.3 Fix a spectum X, and fix numbes m,, and b. We have the following implications: (a) If m, then (),b (2),b+. (2),b+m. If < m, then (),b (b) Suppose that X is E complete. If m, then (2),b (),b+m. If < m, then (2),b (),b +. (c) If m, then (3),b (4),b+. (4),b+m. If < m, then (3),b (d) Suppose that X is E complete. If m, then (4),b (3),b+m. If < m, then (4),b (3),b +. (Obviously, (3),b (),b and (4),b (2),b, but we do not need these facts.)
64 M J Hopkins and J H Palmiei and J H Smith Poof As above, we wite g fo the map F s+ X F s X and g fo the map D s+,t+ D s,t, so that Ds,t is the image of g : π t s F s+ X π t s F s X. (a) Assume that if s m(t s) + b, then D s,t = 0. In the case m, if s m(t s) + b, then s + m((t + ) (s + )) + b; so we see that D s+,t+ = 0. By the long exact sequence (3.2), we conclude that E s+,t+ = 0 when s m(t s) + b. Reindexing, we find that E p,q = 0 when p m(q p)+b+ ; ie, condition (2),b+ holds. The case < m is simila; in this case, the long exact sequence implies that E s,t = 0. (b) Assume that m. If E s,t (X) = 0 wheneve s m(t s) + b, then E s+,t+ 2 (X) = 0 when s m(t s) + b. So by the exact sequence (3.2), we see that D s+,t D s,t is an isomophism unde the same condition. This map is induced by g : π t s F s+ X π t s F s X, so we conclude that when s m(t s) + b, we have lim q π t s F q X = D s,t lim qπ t s F q X = 0., But since X is E complete, then lim qπ t s F q X = 0, so D s,t = im g = 0. Reindexing gives D p,q = 0 when p m(q + p) + b; ie, (2),b implies (),b+m. If < m, then a simila agument shows that D s +,t + = 0. Pats (c) and (d) ae simila. Poof of Theoem.3 This follows immediately fom Lemmas 3.3 and 2.. Moe pecisely, to show that condition (i) is geneic, one applies Lemma 2. to the set T = {(S n,n) : n Z}. Fo condition (ii), one applies it to the set T = {(W, DW ) : W finite}.
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