Nilpotence and Stable Homotopy Theory II Michael J. Hopkins * Jeffrey H. Smith y Massachusetts Institute of Technology, Cambridge Massachusetts Purdue University, West Lafayette Indiana Contents Introduction 2 1 Morava K-theories 8 1.1 Construction : : : : : : : : : : : : : : : : : : : : : : 9 1.2 Spectra related to BP : : : : : : : : : : : : : : : : : 10 1.3 Fields in the category of spectra : : : : : : : : : : : 11 1.4 Morava K-theories and duality : : : : : : : : : : : : 15 2 Proofs of Theorems 3 and 7 15 3 vn self-maps 21 4 Construction of vn self-maps 25 4.1 Preliminaries : : : : : : : : : : : : : : : : : : : : : : 25 4.2 Vanishing lines : : : : : : : : : : : : : : : : : : : : : 28 4.3 Morava K-theories and the Adams spectral sequence 31 4.4 Examples of self maps : : : : : : : : : : : : : : : : : 32 4.5 Proof that Theorem 7 implies the Nilpotence Theorem 36 _________________________* supported by the National Science Foundation and the Sloan Foundation ysupported by the National Science Foundation 1 Nilpotence II 2 5 Endomorphisms, up to nilpotents 37 5.1 N-endomorphisms : : : : : : : : : : : : : : : : : : : 37 5.2 Classification of N-endomorphisms : : : : : : : : : : 39 5.3 Some technical tools : : : : : : : : : : : : : : : : : : 41 5.4 N-endomorphisms and thick subcategories : : : : : : 43 5.5 A spectrum with few nonnilpotent self maps : : : : : 46 5.6 Proof of Theorem 5.4 : : : : : : : : : : : : : : : : : 48 A Proof of Theorem 4.12 49 Introduction This paper is a continuation of [7]. Since so much time has lapsed since its publication a recasting of the context is probably in order. In [15] Ravenel described a series of conjectures getting at the structure of stable homotopy theory in the large. The theory was organized around a family of "higher periodicities" generalizing Bott periodicity, and depended on being able to determine the nilpotent and non-nilpotent maps in the category of spectra. There are three senses in which a map of spectra can be nilpotent: Definition 1. i) A map of spectra f : F ! X is smash nilpotent if for n 0 the map f(n) : F (n)! X(n) is null. ii) A self map f : kF ! F is nilpotent if for n 0 the map fn : knF ! F is null. Nilpotence II 3 iii)A map f : Sm ! R; from the sphere spectrum to a ring spectrum, is nilpotent if it is nilpotent when regarded as an element of the ring ss*R. The main result of [7] is Theorem 2. In each of the above situations, the map f is nilpo- tent if the spectrum F is finite, and if MU*f = 0. In case the range of f is p-local, the condition MU*f = 0 can be replaced with the condition BP*f = 0. The purpose of this paper is to refine this criterion and to pro- duce some interesting non-nilpotent maps. Many of the results of this paper were conjectured by Ravenel in [15]. Let K (n) be the nth Morava K-theory at the prime p (see x1). Theorem 3. i) Let R be a (p-local) ring spectrum. An element ff 2 ss*R is nilpotent if and only if for all 0 n 1, K(n)*(ff) is nilpotent. ii)A self map f : kF ! F , of the p-localization of a finite spectrum, is nilpotent if and only if K (n)*f is nilpotent for all 0 n < 1. iii)A map f : F ! X from a finite spectrum to a p-local spectrum is smash nilpotent if and only if K (n)*f = 0 for all 0 n 1. Of course, the hypothesis "p-local" can be dropped if the condition on the Morava K theory is checked at all primes. At first, the criterion of this theorem seems less useful than the one provided by [7]. Using Theorem 3 to decide whether a map is nilpotent or not requires infinitely many computations. On the other hand, Morava K-theories are often easier to use than complex cobordism. Theorem 3 also determines which cohomology theories detect the nonnilpotent maps in the category of spectra. Definition 4. A ring spectrum E is said to detect nilpotence if, equivalently, i) for any ring spectrum R, the kernel of the Hurewicz homomor- phism E* : ss*R ! E*R consists of nilpotent elements; Nilpotence II 4 ii) a map f : F ! X from a finite spectrum F to any spectrum X is smash nilpotent if 1E ^ f : E ^ F ! E ^ X is null homotopic. Corollary 5. A ring spectrum E detects nilpotence if and only if K (n)*E 6= 0 for all 0 n 1 and for all primes p. Now let C0 be the homotopy category of p-local finite spectra and let Cn C0 be the full subcategory of K (n - 1)-acyclics. The Cn fit into a sequence . . .Cn+1 Cn . . .C0: This is a nontrivial fact. That there are inclusions Cn+1 Cn is essentially the Invariant Prime Ideal Theorem. See [15]. That the inclusions are proper is a result of Steve Mitchell [12]. Definition 6. A full subcategory C of the category of spectra is said to be thick if it is closed under weak equivalences, cofibrations and retracts, ie_, i) An object weakly equivalent to an object of C is in C. ii) If X ! Y ! Z is a cofibration, and two of {X; Y; Z} are in C then so is the third. iii)A retract of an object of C is in C. Theorem 7. If C C0 is a thick subcategory, then C = Cn for some n. Theorem 7 is in fact equivalent to the Nilpotence Theorem (the proof is sketched at the end of Section 4). It is often used in the following manner. Call a property of p-local finite spectra generic if the full sub- category of C0 consisting of the objects with P is closed under cofi- brations and retracts. To show that X 2 Cn has a generic property Nilpotence II 5 P it suffices (by Theorem 7) to show that any object of Cn \ Cn+1 has P. The proofs of the next few results use this technique. Theorem 3 limits the nonnilpotent maps in C0-they must be detected by some Morava K-theory. The simplest type is a vn self-map. Definition 8. Let X be a p-local finite spectrum, and n > 0. A self map v : kX ! X is said to be a vn-self map if 8 >> an isomorphism ifm = n 6= 0 >: nilpotent ifm 6= n: It turns out that the property of admitting a vn self-map is generic. Theorem 9. A p local finite spectrum X admits a vn self-map if and only if X 2 Cn. If X admits a vn self-map, then for N 0, X admits a vn self-map N 2(pn-1) v : p X ! X satisfying ( pN (*) K(m)*v = vn if m = n. 0 otherwise: The vn self-maps turn out to be distinguished by another prop- erty. Definition 10. A ring homomorphism f : A ! B is an F-isomorphism if i) the kernel of f consists of nilpotent elements, and Nilpotence II 6 ii) given b 2 B, bpn is in the image of f for some n. Two rings A and B are F -isomorphic (A F B) if there is an F -isomorphism between them. Theorem 11. Let X 2 Cn \ Cn+1 . The K (n)-Hurewicz homomor- phism gives rise to an F-isomorphism ( (11.1) center [X; X]* F Z(p) (n = 0) Fp[vn] (n 6= 0): The description of spectra as cell complexes encourages the in- tuition that the endomorphism rings of finite spectra approximate matrix algebras over the ring ss*S0. This would suggest that the centers of these rings are generated by the maps obtained by smash- ing the identity map with a map between spheres - an impossibil- ity by Theorem 11. A more accurate description might be that the `Morita' equivalence classes of these rings are determined by the integer n of Theorem 11. This integer invariant can also be thought of as determining the `birational' equivalence classes of finite spectra. For more on this analogy see [9]. There is a less metaphorical interpretation of the integer which occurs in Theorems 9 and 11. Definition 12. Let X be a spectrum. The Bousfield class of X (denoted ) is the collection of spectra Z for which X ^ Z is not contractible. The Bousfield classes of spectra are naturally ordered by inclu- sion (though the relation is indicated with , rather than ). For a finite spectrum X, let Cl(X) N x P denote the set of pairs (n; p) for which K(n)*X 6= 0 at p. Here N is the set of nonnegative integers and P is the set of primes. Theorem 13. Let X and Y be finite spectra. Then if and only if Cl(X) Cl(Y ). Theorem 13 affirms Ravenel's Class Invariance conjecture ([15]). Nilpotence II 7 Proof of Theorem 13: Since if and only if for all primes p, we may localize everything at a prime p. For a fixed Y , the property (of X) is a generic property. It follows that the class { X | } is equal to Cm for some m. Suppose that Y 2 Cn \ Cn+1 . We need to show that m = n. Since = , m n. But if X =2Cn then 6 (since K (n + 1) =2), so m 6 n. __|_ | Acknowledgements Most of the results of this paper date from 1985, and there have been many people who helped shaped the course of the results. Special thanks are due to Emmanuel Dror-Farjoun whose prodding eventually led to the formulation of Theorem 7, and to Clarence Wilkerson for helpful conversations concerning the proof of Theo- rem 4.12. Even deeper debts are owed to Doug Ravenel for formu- lating such a beautiful body of conjectures, and to Mark Mahowald for placing in the hands of the authors the tools for proving results like these. Finally, the first author would like to dedicate his con- tributions to this paper to Ruth, Randi and Rose. Notation and conventions For the most part, we will work in the homotopy category of spec- tra. Of course, to form things like "the induced map of cofibers" requires choosing a diagram in a model for the category of spectra, introducing a certain ambiguity into the resulting map. This am- biguity plays only a very small role in this paper, and is dealt with each time it comes up. The cofiber of a map X ! Y will be written with the "cone coordinate" on the right a Y [f CX = Y X x I= ~ : Nilpotence II 8 With this convention the cofiber of Z ^ X ! Z ^ Y is Z ^ (Y [f CX) (modulo associativity of the smash product) not just isomorphic to it. With this convention, the cofiber of Y ! Y [f CX is X ^ S1, which is isomorphic, but not equal to X. This avoids encountering the troublesome sign that can crop up when trying to relate the connecting homomorphism in a cofi- bration, with the connecting homomorphism in some suspension of the cofibration. The assumption that a spectrum is finite is made several times. In contexts when the the category in mind is the category of p-local spectra, this term is used to refer to a spectrum which is weakly equivalent to the p-localization of a finite spectrum. The only prop- erty of finite spectra that is used is that the set of homotopy classes of maps from a finite spectrum to a directed colimit is the colimit of the maps. [X; lim-!Yff] = lim-![X; Yff]: In general, and object of a category with this property is said to be small. It can be shown that the small objects of the category of p- local spectra are precisely the objects which are weakly equivalent to the p-localizations of a finite spectrum. A spectrum X is connective if sskX = 0 for k 0. It is connected if sskX = 0 for k < 0. Thus "connected" and "(-1)- connected" are synonymous. Similarly, a graded abelian is con- nective if the homogeneous part of degree k is zero for k 0. A graded abelian group is connected if the homogeneous component of degree k is zero for k < 0. The Eilenberg-MacLane spectrum with coefficients in an abe- lian group A will be denoted HA. To be consistent with this, the homology of a spectrum X with coefficients in A will be denoted HA*X. Finally, the suspension of a map will always be labeled with the same symbol as the map. 1. Morava K-theories Nilpotence II 9 1.1. Construction The study of a ring is often simplified by passage to its quotients and localizations. The same is true of ring spectra, though con- structing quotients and localizations can be difficult. In good cases the following constructions can be made: Quotients Suppose that E is a ring spectrum and that ss*E = R is commu- tative. Given x 2 R, define the spectrum E=(x) by the cofibration |x|E x.!E ! E=(x): If x is a non-zero divisor then ss*E=(x) is isomorphic to the ring R=(x). In good cases E=(x) will still be a ring spectrum, and the map E ! E=(x) will be a map of ring spectra. Given a regular sequence {x1; : :;:xn; : :}: R one can hope to iterate the above construction and form a ring spectrum E=(x1; : :;:xn; : :):with ss*E=(x1; : :;:xn; : :): R=(x1; : :;:xn; : :):; and such that the natural map E ! E=(x1; : :;:xn; : :): is a map of ring spectra. Localizations Let E and R be as above, and suppose that S R is a multiplica- tively closed subset. Since S-1 R is a flat R-module, the functor S-1 R E*( __) R is a homology theory, S-1 E. In good cases it is represented by a ring spectrum, and the localization map by a map of ring spectra E ! S-1 E: Nilpotence II 10 1.2. Spectra related to BP When the ring spectrum in question is BP , the above constructions can always be made, using the Baas-Sullivan theory of bordism with singularities. See [4, 13, 17] for the details. Recall that BP* Z(p)[v1; : :v:n: :]:with |vn| = 2pn - 2. To fix notation, take the set {vn} to be the Hazewinkle generators [8]. For 0 < n < 1 the ring spectra K(n) and P (n) are defined by the isomorphisms K(n)* Fp[vn; v-1n] P (n)* Z(p)[vn; vn+1 : :]:; with the understanding that they are constructed from BP using a combination of the above methods. It is also useful to set K(0) = HQ K(1) = HFp : There are maps P (n) ! P (n + 1), and the limit lim-!nP (n) is the Eilenberg-MacLane spectrum HFp . Proposition 1.1. The Bousfield classes of K(n) and P (n) are re- lated by

= _

: Consequently, = _ . ._. _

: Proof: The proposition follows from the next two results of Ravenel [15]. __|_ | Proposition 1.2. Let v : kX ! X be a self map of a spectrum X. Let X=vX and v-1 X denote the cofiber of v and the colimit of the sequence : :-:k|v|X -v!-(k+1)|v|X ! : : : Nilpotence II 11 respectively. Then there is an equality of Bousfield classes = _ : __|_ | Proposition 1.3. There is an equality of Bousfield classes = : __|_ | 1.3. Fields in the category of spectra The coefficient ring K (n)* is a graded field in the sense that all of its graded modules are free. This begets a host of special properties of the Morava K-theories. Proposition 1.4. For any spectrum X, K (n) ^ X has the homo- topy type of a wedge of suspensions of K (n). Proof: Choose a basis {ei}i2I of the free K (n)*-module K (n)*X, and represent it as a map _ S|ei|! K (n) ^ X: i2I The composition _ K (n) ^ S|ei|! K (n) ^ K (n) ^ X: ! K (n) ^ X i2I is then a weak equivalence. __|_ | Proposition 1.5. For any two spectra X and Y , the natural map (1.5.1) K (n)*X K (n)*K (n)*Y ! K (n)*X ^ Y is an equivalence. Nilpotence II 12 Proof: Consider the map (1.5.1) as a transformation of func- tors of Y . The left side satisfies the Eilenberg-Steenrod axioms since K (n)*Y is a flat (in fact free) K (n)*-module. The right side satisfies the Eilenberg-Steenrod axioms by definition. The trans- formation is an isomorphism when Y is the sphere, hence for all Y . __|_ | Proof of Corollary 5: If for some n, K(n)*E = 0, then E does not detect the nonnilpotent map : S0 ! K(n): If K(n)*E 6= 0, then by Proposition 1.4 E*ff = 0 ) K(n)*ff = 0; so the result reduces to Theorem 3. __|_ | Propositions 1.4 and 1.5 portray the Morava K-theories as being a lot like fields. One formulation of Theorem 3 is that they are the prime fields of the category of spectra. A (skew) field is a ring, all of whose modules are free. Definition 1.6. A ring spectrum E is a field if E*X is a free E*- module for all spectra X. This property also admits a geometric expression. Lemma 1.7. If E is a field, then E ^ X has the homotopy type of a wedge of suspensions of E Proof: This is very similar to the proof of 1.7. __|_ | Proposition 1.8. Let E be a field. Then E has the homotopy type of a wedge of suspensions of K(n) for some n. Nilpotence II 13 Proof: Since 1 2 ss*E is non-nilpotent, for some prime p and for some n 1, K(n)*E 6= 0: Since K(n) and E are both fields, it follows from Lemma 1.7 that K(n) ^ E is both a wedge of suspensions of K(n) and a wedge of suspensions of E. In particular, E is a retract of a wedge of suspensions of K(n). The result therefore follows from the next proposition. __|_ | Proposition 1.9. Let M have the homotopy type of a wedge of suspensions of K (n) (fixed n). If E is a retract of M, then E itself has the homotopy type of a wedge of suspensions of K (n). Lemma 1.10. The homotopy homomorphism induced by the Hur- ewicz map ^ 1M : M S0 ^ M ! K (n) ^ M is a homomorphism of K(n)*-modules. Proof: The map in question is a wedge of suspensions of the map jR : K (n) S0 ^ K (n) ! K (n) ^ K (n); so it suffices to prove the claim when M is K (n). In this case the result is a consequence of the formula [16], jR (vn) = vn: __|_ | Lemma 1.11. Let f : M ! N be a map of wedges of suspensions of K (n). The homotopy homomorphism ss*f : ss*M ! ss*N is a map of K (n)*-modules. Nilpotence II 14 Proof: Consider the following commutative diagram: M --f--! N ?? ? y ?y 1^f K (n) ^ M ----! K (n) ^ N: The right vertical arrow is the inclusion of a wedge summand since N admits the structure of a K (n)-module spectrum. It therefore suffices to prove that the composition induces a map of K (n)*- modules. The left vertical arrow does by Lemma 1.10 and the bottom horizontal arrow is a map of K (n)-module spectra. __|_ | Proof of Proposition 1.9: Since M has the homotopy type of a wedge of suspensions of K(n)'s, it can be given the structure of a K(n)-module spectrum. Let i : E ! M, and p : M ! E be the inclusion and retraction mappings respectively. By Lemma 1.11, the composite i O p induces a homomorphism of K(n) *-modules ss*M ! ss*M: Choose a basis {ei} of the image of this map, and represent it by _ S|ei|! M: The map i_ j N = K(n) ^ S|ei| ! K(n) ^ M ! M then gives rise to an isomorphism ss*N image ofss*(i O p); since it sends the obvious basis of ss*N to the basis {ei}. The composite N ! M -p!E is the desired homotopy equivalence. __|_ | Nilpotence II 15 1.4. Morava K-theories and duality We will often use the device of replacing a self map of a finite spectrum f : nX ! X with its Spanier-Whitehead dual Df : Sn ! X ^ DX; a map from the n-sphere to the ring spectrum X ^ DX. If V is the finite dimensional K(n)* vector space K(n)*X, then the ring K(n)*(X ^ DX) is naturally isomorphic to the ring V V * End (V ): The effect in Morava K-theory of the duality map X ^ DX flip!DX ^ X duality!S0 is to send an endomorphism to its trace (in the graded sense). Let {ei} V be a basis of V , and {e*i} V *the corresponding dual basis. The effect of the other duality map S0 ! X ^ DX P is to send 1 2 K(n)* to ei e*i2 V V *. In particular, Lemma 1.12. The duality map S0 ! X ^DX induces a non-zero homomorphism in K(n)-homology if and only if K(n)*X 6= 0. 2. Proofs of Theorems 3 and 7 Some of the conditions in Theorem 3 require the case n = 1, and some of them don't. When the target spectrum is finite, the case n = 1 is superfluous. Lemma 2.1. Let X and Y be finite spectra. For m 0 Nilpotence II 16 i) K(m)*X HFp *X K(m)* ii)K(m)*Y HFp *Y K(m)* iii)K(m)*f = HFp *f 1K(m)* for every f : X ! Y: Proof: This follows from the Atiyah-Hirzebruch spectral se- quence, using the fact that |vm | ! 1 as m ! 1. __|_ | Corollary 2.2. If f is either a self-map of a finite spectrum or an element in the homotopy of a finite ring spectrum, the following are equivalent: i) K(m)*f is nilpotent for m 0 ii)HFp *f is nilpotent: If |f| 6= 0 then both of these conditions hold. Proof: If |f| 6= 0 then, from dimensional considerations, HFp *fi = 0 for i 0. It then follows from 2.1 that K(m)*fi = 0 for i; m 0. When |f| = 0, part (3) of 2.1 applies to every power of f. The result follows easily from this. __|_ | Let f : S0 ! X be a map of spectra. Consider the homotopy direct limit T of the sequence (2.2.1) S0 ! X ! X ^ X ! X ^ X ^ X ! : :;: in which the map X(n)! X(n+1) is given by f ^ 1X(n) : X(n) S0 ^ X(n)! X(n+1): The n-fold composition S0 ! . .!.X(n) is the iterated smash product f(n) = f ^ . .^.f: Nilpotence II 17 The map f(1) : S0 ! T can be thought of as the infinite smash product of f. Lemma 2.3. Let E be a ring spectrum with unit : S0 ! E. The following are equivalent: i) E ^ T is contractible; ii) ^ f(1) : S0 ! E ^ T is null; iii) ^ f(n) : S0 ! E ^ X(n) is null for n 0; iv) 1E ^ f(n) : E E ^ S0 ! E ^ X(n) is null for n 0. Proof: i))ii) and iv))i) are immediate. Since lim-!E ^ X(n) E ^ T; and since homotopy groups commute with direct limits, a null ho- motopy of S0 ! E ^ T must occur at some S0 ! E ^ X(n) for n 0. This gives ii)) iii). The implication iii))iv) is the only one requiring E to be a ring spectrum. If S0 ! E ^ X(n) is null then so is the first map in the following factorization of 1E ^ f(n): E ^ S0 ! E ^ E ^ X(n)! E ^ X(n) This completes the proof. __|_ | Proof of Theorem 3: Part i) follows from part iii), since the iterated multiplication factors through iterated smashing. Part ii) follows from part i) since multiplication in the rings ss*X ^ DX and K(n)*X ^ DX corresponds, under Spanier-Whitehead duality, to composition in [X; X]* and End K(n)*(K(n)*X): Replacing f : F ! X Nilpotence II 18 with Df : S0 ! DF ^ X in part iii) if necessary, we may assume that F = S0. The result reduces to Theorem 2 once it is shown that 1BP ^ f(m) is null for m 0. From Lemma 2.3 (with the obvious notation) this is equivalent to showing that BP ^ T is contractible. In view of the Bousfield equivalence = _ . ._. _

: it is enough to show that P (n) ^ T is contractible for n 0. Again from 2.3 this equivalent to showing that S0 ! P (n) ^ T is null for n 0. Now let n grow to infinity. Since lim-!P (n) HFp ; the map S0 ! lim-!P (n) ^ T is null by assumption. Since homotopy commutes with direct lim- its, the nullhomotopy arises at some S0 ! P (n) ^ T: This completes the proof of Theorem 3. __|_ | The proof of Theorem 7 requires a slight modification of the third assertion of Theorem 3, and a useful cofibration (2.6). Corollary 2.4. Let F and Z be finite spectra, E a ring spectrum, and X an arbitrary spectrum. i) If a map f : F ! X ^ E satisfies K(n)*(f) = 0 for all0 n 1; then for m 0, the composite (m) 1^ F (m)-f--! (X ^ E)(m) X(m) ^ E(m) --! X(m) ^ E is null. Nilpotence II 19 ii)A map f : F ! X has the property that f(m) ^ 1Z : F (m)^ Z ! X(m) ^ Z is null for m 0 if and only if K(n)*(f ^ 1Z ) = 0 for all 0 n 1. Proof: In part i), the map f(m) is already null for m 0 by part iii) of Theorem 3. The only if part of ii) is clear. Letting E be the ring spectrum Z ^ DZ and replacing f ^ 1Z : F ^ Z ! X ^ Z with its Spanier-Whitehead dual F ! X ^ Z ^ DZ reduces the if part to i). __|_ | Lemma 2.5. Let X -f!Y -g!Z be a sequence of maps. The map Cf ! CgOf induced by g gives rise to a cofibration Cf ! CgOf ! Cg: Proof: Consider the following diagram in which the rows and columns are cofibrations: X ---f-! Y ----! Cf flfl ? ? fl g?y ?y X --gOf--!Z ----! CgOf ?? ? y ?y Cg ----! ? Nilpotence II 20 The upper right square is a pushout. It follows that the bottom arrow is a homotopy equivalence. This completes the proof. __|_ | Corollary 2.6. Let f : X ! Y and g : Z ! W be two maps. There is a cofibration X ^ Cg ! Cf^g ! Cf ^ G: Proof: Apply the lemma to the factorization f ^ g = f ^ 1Y O 1X ^ g: __|_ | Proof of Theorem 7: It suffices to establish (2.6.1) if X 2 C and X 2 Cn then Cn C, for it then follows that C = Cm , where m = min {n | Cn C }: Since everything has been localized at p, set Cl(X) = {n 2 N | K(n)*(X) 6= 0}: With this notation, (2.6.1) becomes: (2.6.2) if X 2 C and Cl(Y ) Cl(X), then Y 2 C. Suppose, then, that X 2 C. Then so is Z ^ X for any Z 2 C0. Let f : F ! S0 be the fiber of the duality map S0 ! X ^ DX. Then Y ^Cf 2 C. Setting g = f(m-1) in Corollary 2.6 and smashing with the identity map of Y gives a cofibration Y ^ F ^ Cf(m-1)! Y ^ Cf(m)! Y ^ Cf ^ F (m-1): It follows that Y ^ Cf(m)2 C for all m. By 1.12, K(n)*f 6= 0 if and only if n =2Cl(X), so that K(n)*(1Y ^ f) = 0 for all n, Nilpotence II 21 since Cl(Y ) Cl(X). Part ii) of Corollary 2.4 then gives that 1Y ^ f(m) is null for m 0. This means that Y ^ Cf(m) Y _ (Y ^ F (m)) for m 0 ; so Y 2 C. This completes the proof of Theorem 7. __|_ | 3. vn self-maps The purpose of the next two sections is to establish Theorem 9. The "only if" part, that X 62 Cn implies that X does not admit a vn self-map is easy: if for some j < n K(j)*X 6= 0, and if v is a vn self-map, then the cofiber Y of v is a finite spectrum satisfying K(n)*Y = 0 K(j)*Y 6= 0; contradicting the fact that Cn Cj. The proof of the "if" part falls into two steps. In this section it is shown that the property of admitting a vn self-map is generic. It then remains to construct for each n, a spectrum Xn with a vn self-map. This is done in section 4. For any spectrum X, the element p 2 [X; X]* is a v0 self-map satisfying condition (*) of Theorem 9. We therefore need only consider vn self-maps when n 1. Because of this, unless otherwise mentioned, in this and the next section, we will work entirely in the category C1. As mentioned in section 1.4 a self-map kF ! F of a finite spectrum corresponds, under Spanier-Whitehead duality, to a map from the k-sphere to the ring spectrum R = F ^ DF: Nilpotence II 22 Definition 3.1. Let R be a finite ring spectrum, n > 0: An ele- ment ff 2 ss*R is a vn-element if ( K(m)*ff is a unit if m = n nilpotent otherwise: Lemma 3.2. Let R be a finite ring spectrum, and ff 2 ss*R a vn- element. There exist integers i and j such that ( K(m)*ffi = 0 if m 6= n vjn if m = n Proof: It follows from lemma 2.2 that HFp *ff is nilpotent. Rais- ing ff to a power, if necessary, we may suppose that HFp *ff = 0. It then follows from lemma 2.2 that K(m)*ff = 0 for all but finitely many m. Raising ff to a further power, if necessary, it can be arranged that K(m)*ff = 0 for m 6= n. The assertion K(n)*ffi = vjnis equivalent to the assertion that ffi = 1 2 K(n)*R=(vn - 1). The ring K(n)*R=(vn - 1) has a finite group of units, and so i can be taken to be the order of this group. __|_ | Corollary 3.3. If f : kF ! F is a vn self-map, then there exist integers i; j with the property that ( K(m)*fi = 0 if m 6= n __|_ | multiplication by vjnif m = n: Lemma 3.4. Suppose that x and y are commuting elements of a Z(p)-algebra. If x-y is both torsion and nilpotent, then for N 0, N pN xp = y : Nilpotence II 23 Proof: Since we are working over a Z(p)-algebra it follows that pk(x - y) = 0 for some k. The result now follows by expanding N pN xp = (y + (x - y)) using the binomial theorem. __|_ | Lemma 3.5. Let R be a finite ring spectrum, and ff 2 sskR a vn- element. For some i > 0, ffi is in the center of ss*R. Proof: Raising ff to a power, if necessary, we may assume that K(m)*ff is in the center of K(m)*R for all m. Let l(ff) and r(ff) be the elements of End (ss*R) given by left and right multiplication by ff. Since R 2 C1 the difference l(ff)-r(ff) has finite order. Since K(m)* (l(ff) - r(ff))= 0 for all m; l(ff) - r(ff) is nilpotent by Theorem 3. The result now follows from 3.4. __|_ | Lemma 3.6. Let ff; fi 2 ss*R be vn-elements. There exist integers i and j with ffi = fij. Proof: Raising ff and fi to powers if necessary, we may assume that K(m)* (ff - fi)= 0 for all m. The result follows, as above, from 3.4. __|_ | Corollary 3.7. If f and g are two vn self-maps of F , then fi is homotopic to gj for some i and j. __|_ | Nilpotence II 24 Corollary 3.8. Suppose X and Y have vn self-maps vX and vY . There are integers i and j so that for every Z and every f : Z ^ X ! Y the following diagram commutes: Z ^ X --f--! Y ? ? 1^vXi?y ?yvYj Z ^ X --f--! Y: Proof: The spectrum DX ^ Y has two vn self-maps: DvX ^ 1Y and 1DX ^vY . By Corollary 3.7 there are integers i and j for which DvX i^1Y is homotopic to 1DX ^vY j. The result now follows from Spanier-Whitehead duality. __|_ | Corollary 3.9. The full subcategory of C1 consisting of spectra ad- mitting a vn self-map is thick. Proof: Call the subcategory in question C. Note that X 2 C if and only if X 2 C. To check that C is closed under cofibrations it therefore suffices to show that if (3.9.1) X ! Y ! Z is a cofibration with X and Y in C, then Z is in C. Using Corol- lary 3.8 choose the vn self-maps vX and vY of X and Y so that kX - ---! kY ----! kZ ? ? vX?y vY?y X - ---! Y ----! Z: commutes. The induced map vZ : kZ ! Z is easily seen to be a vn self-map. Nilpotence II 25 Now suppose that Y is a retract of X, and let i : Y ! X and p : X ! Y be the inclusion and retraction mappings respectively. Choose a vn self-map v of X which commutes with i O p. The map p O v O i is easily checked to be a vn self-map of Y . __|_ | Corollary 3.10. The full subcategory of C1 consisting of spectra admitting a vn self-map satisfying condition (*) of Theorem 9 is thick. Proof: This is similar to 3.9, and involves checking that the integers which arise in 3.6-3.8 are powers of p. In fact, the only place where an integer which is not a power of p comes up is in using 3.7 to arrange that K(m)*v is in the center of End K(m)*(K(m)*X). But this is guaranteed at the outset by condition (*). __|_ | 4. Construction of vn self-maps 4.1. Preliminaries The examples of self-maps needed for the proof of Theorem 9 are constructed using the Adams spectral sequence Exts;tA[H*Y; H*X] ) [X; Y ]t-s which relates the mod p cohomology of X and Y as modules over the Steenrod algebra to [X; Y ]*. The spectral sequence is usually displayed in the (t - s; s)-plane, so that the groups lying in a given vertical line assemble to a single homotopy group. With this con- vention the "filtration jumps" are vertical in the sense that the difference between two maps representing the same class in Ext s;tA[H*Y; H*X] represents a class in 0;t0* * ExtsA [H Y; H X]; Nilpotence II 26 with s0> s, and t - s = t0- s0. There are many criteria for convergence of the Adams spectral sequence. A simple one, which is enough for the present purpose is [1] Lemma 4.1. If X a finite spectrum and Y is a connective spec- trum with the property that each sskY is a finite abelian p-group, then the Adams spectral sequence converges strongly to [X; Y ]*: If B C are Hopf-algebras over a field k, the forgetful functor C-modules ! B-modules has both a left and a right adjoint. The left adjoint M 7! C M B carries projectives to projectives, and so prolongs to a change of rings isomorphism (4.1.1) Ext*C[C M; N] Ext*B[M; N]: B When M is a C-module this can be combined with the "shearing isomorphism" C M ! C==B M C==B = C k B B X c m 7! c0i c00im X (c)= c0i c00i; to give another "change of rings" isomorphism Ext *C[C==B M; N] Ext*B[M; N]: The difference between ExtC and ExtB can therefore be measured by the augmentation ideal ______ C==B = ker{ffl : C==B ! k}; Nilpotence II 27 using the long exact sequence coming from ______ C==B M ae C==B M i M: Recall that for p = 2, the dual Steenrod algebra is A* = F2[1; 2; : :]: |i| = 2i- 1 and for p odd A* = [o0; o1; : :]: Fp[1; 2; : :]: |oi| = 2pi- 1 |i| = 2(pi- 1): The subalgebra of the Steenrod algebra generated by Sq1; : :;:Sq2n when p = 2 fi; P1; : :;:Pn-1when p is odd, and n 6= 1 fi when p is odd and n = 0 is denoted An. It is the finite sub Hopf-algebra which is annihilated by the ideal n+1 2n (21 ; 2 ; : :;:n+1 ; n+2 ; : :):p = 2 n (p1; : :;:n; n+1 ; on+1 ; : :):p 6= 2: The augmentation ideal of A==An is 2pn(p-1)-connected. The fact that the connectivity goes to infinity with n plays an important role in the Approximation Lemma 4.5. It is customary to give the dual Steenrod algebra the basis of monomials in the 's and o's. With this convention, the Adams- Margolis elements are s Ptsdual to pt (s < t) ae Qn dual to on p odd n+1p = 2: Each Qn is primitive, and together they generate an exterior sub Hopf-algebra of the Steenrod algebra. The Ptsall satisfy (Pts)p = 0; Nilpotence II 28 but are primitive only when s = 0. The Adams-Margolis elements are naturally ordered by degree |Pts|= 2ps(pt- 1) |Qn| = 2pn - 1: 4.2. Vanishing lines Given an A-module M, and an Adams-Margolis element d, the Margolis homology of M, H(M; d), is the homology of the complex (M*; d*), with Mn = M n 2 Z d2n = d ( p-1 s d2n+1 = d ifd = Pt d ifd = Qn: When X is a spectrum the symbol H(X; d) will be used to denote H(H*X; d). The Margolis homology groups are periodic of period 1 if p even, or if d = Qn, and are periodic of period 2 otherwise. Definition 4.2. Let M be an A-module. A line y = mx + b is a vanishing line of Ext*;*A[M; Fp] if Exts;tA[M; Fp] = 0 for s > m(t - s) + b: The following result, due to Anderson-Davis [2] and to Miller- Wilkerson [10] relates vanishing lines to Margolis homology groups. It has not been stated in its strongest form. Nilpotence II 29 Theorem 4.3. If M is a connective A-module with H(M; d) = 0 for |d| n; then Ext*;*A[M; Fp] has a vanishing line of slope 1=n. __|_ | In general, there is no way to predict the intercept of the van- ishing line, but there is the following: Proposition 4.4. Suppose that M is a connective A-module, and that y = mx + b is a vanishing line for Ext*;*A[M; Fp]. If N is a (c - 1)-connected A-module, then y = m(x - c) + b is a vanishing line for Ext*;*A[M N; Fp]: Proof: Let Nk be the quotient of N by the elements of degree greater than k, and Nkj Nk the submodule consisting of elements of degree > j. There is an exact sequence Nkj! Nk ! Nj: Since M is connective, M N = lim-!M Nk k and Exts;tA[M N; Fp] = lim-!Exts;tA[M Nk ; Fp]; k so it suffices to prove the result for each Nk . This is trivial for k < c, so suppose k c, and by induction, that the result is true for k0< k. Suppose that (s; t) satisfies s > m(t - s - c) + b Nilpotence II 30 and consider the exact sequence M Nkk-1! M Nk ! M Nk-1 : By induction, Exts;tA[M Nk-1 ; Fp] = 0: The module Nkk-1is just a sum of copies of kF p_the A-module which consists of F p in degree k, and zero elsewhere. It follows that Exts;tA[M Nkk-1; Fp] is a product of copies of Exts;tA[M kF p; Fp] Exts;t-kA[M; Fp]; which is zero since s > m(t - s - c) + b > m((t - k) - s) + b: __|_ | Lemma 4.5 (Approximation lemma). Let M be a connective A-module, and suppose that Ext*;*A[M ; Fp] has a vanishing line of slope m. Given b, for n 0 the restriction map Exts;tA[M; Fp] ! Exts;tAn[M; Fp] is an isomorphism when s m(t - s) + b: Proof: The result follows from the exact sequence ______ A==An M ae A==An M i M; ______ Proposition 4.4, and the fact that the connectivity of A==An can be made arbitrarily large by taking n to be large. __|_ | Nilpotence II 31 4.3. Morava K-theories and the Adams spectral sequence We need to be able to examine the K(n)-Hurewicz homomorphism from the point of view of the Adams spectral sequence. This can be done, but it is a little easier to work with the connected cover k(n) of K(n). The spectrum k(n) is a ring spectrum, with k(n)* = Fp[vn] K(n)* = Fp[vn; v-1n]: Lemma 4.6. The transformation k(n)*X ! K(n)*X extends to a natural isomorphism v-1nk(n)*X K(n)*X: Proof: Since localization is exact, both sides satisfy the exact- ness properties of a homology theory. They agree when X is the sphere, hence for all X. __|_ | Corollary 4.7. If k(n)*X is finite then K(n)*X = 0. Proof: If k(n)*X is finite, then for j 0, k(n)jX = 0. This means that for each x 2 k(n)*X, vmnx = 0 for m 0. The result then follows from lemma 4.6. __|_ | Since k(n) is a ring spectrum, the mod p cohomology H*k(n) is a coalgebra over the Steenrod algebra. It has been calculated by Baas and Madsen [5] Proposition 4.8. As a coalgebra over the Steenrod algebra, H*k(n) A==E[Qn]: It follows that the E2-term of the Adams spectral sequence for ss*k(n) ^ X is isomorphic to Exts;tE[Qn][H*X; Fp]; and that the map of E2-terms induced by the Hurewicz homomor- phism is the natural restriction. __|_ | Nilpotence II 32 Corollary 4.9. If X is a finite spectrum and H(X; Qn) = 0, then K(n)*X = 0. Proof: The group Ext *;*E[Qn][H*(X); Fp] is the cohomology of the complex H*X Qn!H*X Qn!H*X Qn!: : :: This means that for s > 0, the graded abelian group Exts;*E[Qn][H*X; Fp] is isomorphic to the Margolis homology group H(X; Qn). The van- ishing of these groups implies that Ext *;*E[Qn][H*X; Fp] Ext*;0E[Qn][H*X; Fp] H*X is finite, and hence that k(n)*X is finite. The result then follows from Corollary 4.7. __|_ | 4.4. Examples of self maps The key to constructing self-maps is the following result of the second author [19]. An account appears in [14]. Theorem 4.10 (Smith). For each n = 1; 2; : :t:here is a finite spectrum Xn satisfying i) The Adams spectral sequence Exts;tE[Qn][H*Xn ^ DXn; Fp] ) k(n)*Xn ^ DXn collapses; Nilpotence II 33 ii)The Margolis homology groups H(Xn ^ DXn; d) are zero if |d| < |Qn|: Theorem 4.11. The spectrum Xn is in Cn \ Cn-1 and has a vn self-map satisfying (*) of Theorem 9. The proof of Theorem 4.11 uses the Adams spectral sequence and the following consequence of the results of Wilkerson [20]. The proof is in the appendix to this section. Theorem 4.12. Suppose that B C are finite, connected, graded, cocommutative Hopf-algebras over a field k of characteristic p > 0. If b 2 Ext*;*B[k; k]; then for N 0, bpN is in the image of the restriction map Ext*;*C[k; k] ! Ext*;*B[k; k]: __|_ | Proof of Theorem 4.11: That Xn is in Cn \ Cn-1 follows from Corollary 4.9. For the construction of the self-map it is slightly cleaner to work from the point of view of finite ring spectra. Thus let R be the finite ring spectrum Xn ^ DXn. The ring ss*R is an algebra over ss*S0, and the image of ss*S0 in ss*R is in the center (in the graded sense). Similarly, if B A is a sub Hopf-algebra, the ring Ext *;*B[H*R; Fp] is a central algebra over Ext*;*B[F p; Fp]. To show that Xn admits a vn self-map satisfying condition (*) of Theorem 9 it suffices to exhibit an element v 2 ss*R satisfying M pN (4.12.1) k(n)*vp = vn . 1; for some M; N > 0; (4.12.2) the map k(m)*v is nilpotent when m 6= n. Nilpotence II 34 Step 1: First to find an approximation to a vn self-map in the E2-term of the Adams spectral sequence. Let n-1 vn 2 Ext1;2pE[Qn][F p; Fp] be the element represented by vn 2 k(n)*. We need to find a N ;pN (2pn-1)* w 2 ExtpA [H R; Fp] restricting to vpNn. 1, for N 0. By 4.3, the bigraded group Ext *;*A[H*R; Fp] has a vanishing line of slope 1=2(pn - 1). Using the approximation lemma, an integer n can be chosen for which the restriction map (4.12.3) Ext s;tA[H*R; Fp] ! Exts;tAn[H*R; Fp] is an isomorphism if s > ____1____2(pn(-t1)- s): By Theorem 4.12 there is an element w"2 Ext*;*An[F p; Fp] restricting to vpNn2 Ext*;*E[Qn][F p; Fp]. The class w can be taken to be the image of "w. 1 under the isomorphism (4.12.3). Step 2: This construction of the class w actually gives some- thing more. Since Ext *;*An[F p; Fp] is in the center (in the graded sense) of Ext*;*An[H*R; Fp]; the class w commutes with every ff 2 Exts;tA[H*R; Fp] with (4.12.4) s ____1____2(pn(-t1)- s): Nilpotence II 35 Step 3: Now to choose a power of w which survives the Adams spectral sequence. The differentials in the Adams spectral sequence are derivations, and the values of drw lie in the region (4.12.4). This means that dr-1w = 0 ) drwp = 0: Sincebd1w = 0 it follows that dbwpb = 0. The possible values of drwp for r > b lie in the region s > ____1____2(pn(-t1)- s); which is above the vanishing line. This means that the class wpb is a permanent cycle. Step 4: For simplicity, replaceNw with wpb, and adjust the in- teger N so that w restricts to vpn . 1. Let v 2 ss*R be a representative of w. We willNsee that this is the desired class. The difference k(n)*(v - vpn ) is represented by a class in Ext s;tE[Qn][H*R; Fp] with s > 1=2(pn - 1) (t - s). Some power of k(n)*(v - vpNnb) is therefore represented by a class above the vanishing line of Ext *;*E[Qn][H*R; Fp] (which has slope 1=2(pn - 1)), and hence is zero. Lemma 3.4 then gives that that M pMN k(n)*vp = vn M 0: This proves property (4.12.1) Property (4.12.2) is trivial when m < n, since R 2 Cn. When m > n, it is a consequence of the fact that the Adams spectral sequence k(m)*R has a vanishing line of slope 1=2(pm - 1), and that the powers of v are represented by classes lying on the line s = ____1____2(pn(-t1)- s) which has a larger slope. This completes the proof. __|_ | Nilpotence II 36 4.5. Proof that Theorem 7 implies the Nilpotence Theo- rem This subsection is included to satiate any curiosity aroused by the claim made after the statement of Theorem 7. Since the argument is not necessary for establishing any of the results of this paper, it is included only as a sketch. In [7, Section 1] the Nilpotence Theorem is reduced to showing that if R is a connective, associative ring spectrum, and ff 2 ss*R is in the kernel of the MU-Hurewicz homomorphism then ff is nilpotent. This in turn is easily reduced to the case when R is localized at p and MU is replaced with BP . The case |ff| 0 is easy, so it may be assumed that |ff| > 0. Let __ffbe the map (4.12.5) |ff|R ff^1!R ^ R ! R: The map (4.12.5) induces multiplication by BP*ff = 0 in BP ho- mology. This means that from the point of view of the Adams- Novikov spectral sequence, composition withe __ffmoves the homo- topy to the right along a line of positive slope. Step 1: The construction used to produce the spectra Xn of this section can be used to construct finite torsion free spectra Yn with the property that H*Yn, as a module over An is free over An==E, where E is the sub-Hopf-algebra [Q0; : :;:Qn]: See [19]. Step 2: Use the spectral sequence of [16, Theorem 4.4.3] to show that Exts;tBP*BP[BP*; BP*R ^ Yn] has a vanishing line with slope tending to zero as n ! 1. Nilpotence II 37 Step 3: It follows from the vanishing line that for n 0 the spectrum Yn ^ ff-1 R is contractible. Step 4: Now use Theorem 13 to conclude that Yn is Bousfield equivalent to the sphere, hence that ff-1 R is contractible, hence that ff is nilpotent. 5. Endomorphisms, up to nilpotents 5.1. N-endomorphisms The vn self maps form an endomorphism (up to nilpotent elements) of the category Cn. It turns out that these are the only endomor- phisms of this kind that can occur in the category of finite spectra. Definition 5.1. Let C be a full subcategory of C0 which is closed under suspensions. A collection v, of self-maps vX : kX X ! X X 2 C satisfying is an N-endomorphism of C if i) The map vX is the composite kX --flip--!kX ^ S1 ?? y vX ^1S1 X ^ S1 --flip--!X ii) for each f : X ! Y in C there are integers i and j with ikX = jkY , such that the following diagram commutes: N X --f--! N Y ? ? vXi?y ?yvYj X ----! Y: f Nilpotence II 38 An N-endomorphism is an F -endomorphism if the integers i and j can be taken to be powers of p. Two N-endomorphisms v and v0 will be identified if for each X 2 C there are integers i and j with vX i = v0Xj. Two F - endomorphisms v and v0 will be identifiedjif for each X 2 C there are integers i and j with vX pi= v0Xp. Remark 5.2. (1) If v is an N-endomorphism of a category C, and f : X ! Y an isomorphism with X 2 C, then defining vY to be kf kY - ---! kX ?? y vX X - ---! Y f-1 extends v to the full subcategory obtained from C by adjoining the suspensions of Y . Because of property ii), the resulting N- endomorphism is independent of the choice of isomorphism f. In this way an N-endomorphism can always be extended uniquely to a full subcategory which is closed under suspensions and isomor- phisms. This procedure will be used without comment, so once an N-endomorphism has been defined on a subcategory C of fi- nite spectra, it will be taken to be extended to the smallest full subcategory containing C, which is closed under suspensions and isomorphisms. Among other things, this means that if vX is de- fined, so is vX^S1 and vX^S1 = vX ^ 1S1: (2) An N-endomorphism is of degree zero if all of the integers kX are zero. If an N-endomorphism is not of degree zero, then none of the integers kX is zero, and the maps vX can all be chosen to have finite order. Given two spectra X; Y 2 C, the maps vX and vY can be chosen in such a way that the integers kX and kY coincide. With this arrangement, given a map f 2 [X; Y ]*; Nilpotence II 39 if there are integers i and j for which viYO f = f O vjX; then it must be the case that i = j. This same discussion applies to any finite collection of elements of C. Example 5.3. (1) Taking vX to be nilpotent defines an F -endomorphism. (2) Taking each vX to be a multiple of the identity defines an actual endomorphism . (3) Suppose C Cn. Taking vX to be a vn self-map defines an N-endomorphism . Taking vX to be a vn self-map satisfying con- dition (*) of Theorem 9 defines an F -endomorphism . 5.2. Classification of N-endomorphisms The above list of examples turns out to be complete. Theorem 5.4. Suppose that v is an N-endomorphism of a full subcategory C C0 which is closed under suspensions. Then vX is nilpotent for every X, some power of vX is a multiple of the identity, or C Cn for some n and vX is a vn self-map. Of course, these possibilities aren't exclusive. If X 2 Cn+1 Cn any vn self-map of X is nilpotent. Corollary 5.5. Suppose the X 2 C0, and that v 2 [X; X]* is in the center. Then v is nilpotent, a power of v is a multiple of the identity, or v is a vn self-map. Proof: Let C be the full subcategory of C0 consisting of the suspensions of X. The map v determines an N-endomorphism of C, so the result follows from Theorem 5.4. __|_ | Nilpotence II 40 Theorem 11 is an immediate consequence of Corollary 5.5 and Theorem 9. The proof of Theorem 5.4 falls into two parts. First it is shown that an N-endomorphism extends uniquely to a thick subcategory. It then suffices to construct, for each n, a spectrum Xn 2 Cn \ Cn+1 whose only non-nilpotent self-map is a vn self-map. First to dispense with the N-endomorphisms of degree zero. Proposition 5.6. If X is in C0, and v : X ! X is in the center of [X; X]* = ss*X ^ DX then there are integers m and n for which vn = multiplication by m. Proof: Since ss*X ^ DX Q HQ*X ^ DX End HQ*X; The map HQ*v must be in the center of End HQ*X which consists of the endomorphism "multiplication by a constant". Since the Hurewicz map HQ* factors through HZ*, this constant must be an integer k. The map w = v - k then has finite order. Since all of the eigenvalues of HFp *wp-1 are equal to 0 or 1, the map HFp *w(p-1)pN is an idempotent for N 0. Replace w with w(p-1)pN. The map w still has finite order, and is in the center of [X; X]*. Define connective spectra A1 and A2 by A1 = w-1 X A2 = (1 - w)-1 X The map X ! A1 _ A2 induces an isomorphism on both mod p and rational homology, hence on homology with coefficients in Z(p). It is therefore a ho- motopy equivalence, and in particular A1 and A2 are finite. The ring of self-maps [X; X]* can be written as a ring of 2 x 2 matrices, in which the ij-entry is in [Aj; Ai]. The map w is represented by the matrix w|A1 0 0 0 ; Nilpotence II 41 whose (1; 1) entry is an equivalence. Given a map f : kA2 ! A1, let "fbe the map 0 f 0 0 : Then ad (w)f"= w|A1f0 00 : Since w is central, and w|A1 is an equivalence this means that f is null. By Lemma 5.7 below, it follows that one of A1 and A2 is contractible. If A1 is contractible, then w is nilpotent, and the result follows from Lemma 3.4. If A2 is contractible, then 1 - w is nilpotent, HQ*X = 0, andNwe may assume that the integer k is 0, soMthat w = v(p-1)p . It then follows from Lemma 3.4 that v(p-1)p = 1 for M 0. This completes the proof. __|_ | We have used Lemma 5.7. If A and B are non-contractible p-local finite spec- tra, then [A; B]* 6= 0. Proof: Since HFp *DA ^ B = hom [HFp *A; HFp *B] 6= 0; the spectrum DA ^ B is not-contractible. It therefore has a non- zero homotopy group. Now use the isomorphism ss*DA ^ B [A; B]*: __|_ | 5.3. Some technical tools The next few results are a bit technical, but they come up several times. Nilpotence II 42 Lemma 5.8. Suppose that M is a bimodule over the ring Z(p)[v], and for m 2 M let ad (v)m = vm - mv: If there are integers i, j and k, for which i) k ad(vi)m = 0, and ii)ad(vj) ad(vi)m = 0, then ad vijk m = 0: Clearly, k can be taken to be a power of p, so that if i and j are powers of p, then so is ijk. Lemma 5.9. Suppose M is a bimodule over the ring Z[v]. Let ad (v) : M ! M be the operator ad(v)m = vm - mv. Then there is a formula X n (5.9.1) ad (vn)m = adi(v)m . vn-i: i 0 there exists a sequence k_= (k0; : :;:kn-1 ); and a finite spectrum M(k_) 2 Cn \ Cn-1 , satisfying: k0 pkn-1 i) BP*M(k_) = BP*=(vp0 ; : :;:vn-1 ) (v0 = p); Nilpotence II 47 ii)If v : jM(k_) ! M(k_) is a non-nilpotent self-map, then some power of v is the identity map, or v is a vn self-map. Proof: Suppose by induction on n that a sequence k_= (k0; : :;:kn-1 ) and a spectrum M = M(k_) have been found, satisfying condition (1). When n = 1 the sequence can be taken to be (1), and the spectrum M, the mod p Moore spectrum S0 [p e1: Let I(k_) BP* be the ideal k0 pkn-1 (vp0 ; : :;:vn-1 ): If v is a non-nilpotent self-map of M(k_) then the BP -Hurewicz image BP*v : BP*=I(k_) ! BP*=I(k_) must be non-nilpotent. The map BP*v must also be a map of BP*-modules, and of BP*BP -comodules, and so is an element of Hom BP*BP [BP*=I(k_); BP*=I(k_)] BP*=I(k_): Modulo the ideal (5.12.1) (p; v1; : :;:vn-1 ) this group is just [16, Theorem 4.3.2] Fp[vn]: Since (5.12.1) is nilpotent modulo I(k_) BP*v vkn mod I(k_) k 2 Z; 2 Fp: Nilpotence II 48 It then follows from Lemma 3.4 that N k(p-1)pN BP*v(p-1)p = vn N 0: Replace v with v(p-1)pN. If k = 0, then BP*(v - 1) = 0, and so v - 1 is nilpotent. It then follows from Lemma 3.4 that N vp = 1M N 0: Suppose then that k 6= 0, and let w be a vn self-map of M. By the above discussion applied to w, there are integers i and j, for which BP*vi = BP*wj. But this means that vi - wj is nilpotent, so by Lemma 3.4 some power of v is homotopic to some power of w, and v is a vn self-map. This proves (2). For the rest of the induction step, let n N w : 2(p -1)P M ! M be a vn self-map satisfying condition (*) of Theorem 9. The integer kn can be taken to be N, and M(k0; : :k:n); the cofiber of the map w. __|_ | 5.6. Proof of Theorem 5.4 Let v be an N-endomorphism of C C0. Then v extends uniquely to the smallest thick subcategory Cn C0 containing C. Let k_= (k0; : :;:kn-1 ) and M = M(k_) be as in Lemma 5.12, and let D be the full sub- category of Cn consisting of the suspensions of M. Then v is also uniquely determined by its restriction to D, ie by the map vM . By Proposition 5.12, there are three possibilities for vM , and these are the restrictions of the nilpotent, identity, and vn self-map N- endomorphisms. This completes the proof of Theorem 5.4. __|_ | Nilpotence II 49 A. Proof of Theorem 4.12 The purpose of this appendix is to prove (rather, deduce from [20]) Theorem 4.12. All of the techniques used here can be found in [20]. Throughout this appendix, all Hopf-algebras will be over a field of characteristic p > 0. They will be connected, graded, cocommu- tative, and finite dimensional. The dual of a Hopf-algebra will be graded in such a way that the dual of the homogeneous component of degree k has degree -k. This convention enables the co-action map (A.0.3) to preserve degrees. The action of a Hopf-algebra B on a module M can be expressed as an "action" (A.0.2) B M ! M or as a "coaction" (A.0.3) M ! B* M: A module M which happens to be an algebra is an algebra over B if the multiplication map M M ! M is a map of B-modules. This is equivalent to the requirement that the coaction map (A.0.3) be multiplicative. All algebras over Hopf- algebras in this appendix will be graded and connected. If B C is normal, and M is a C-module, then the sub-module of elements invariant under B, MB = hom B[k; M]; inherits an action of the quotient Hopf-algebra C==B. In fact, so do all of the derived functors (A.0.4) Ext *B[k; M]: If M is an algebra over B, then (A.0.4) becomes an algebra over C==B [18]. In case B C is normal, the relationship between the co- homologies of B and C is given by the Lyndon-Hochschild-Serre spectral sequence Ext*C==B[k; Ext*B[k; M]]) Ext*C[k; M]: Nilpotence II 50 The main result of this appendix is Theorem A.1. Suppose that R is a Noetherian C-algebra, and that B C is normal. Then i) The algebra Ext*C[k; R] is Noetherian, hence finitely generated. ii)The Lyndon-Hochschild-Serre spectral sequence Ext*C==B[k; Ext*B[k; R]]) Ext*C[k; R]: terminates at a finite stage in the sense that there is an integer N with the property that all of the differentials dr are zero, if r > N. iii)There is an integer N with the property that drxpN is zero, for all x and all r. iv) The Lyndon-Hochschild-Serre spectral sequence is a spectral sequence of finitely generated modules over some connected, graded, Noetherian ring T . The parts of this theorem are closely related. Lemma A.2. In Theorem A.1, parts i), ii), and iii) follow from iv). Given i), parts ii), iii), and iv) are equivalent. Proof: Suppose first that iv) holds. Then part ii) follows from Lemma A.3 below. Part iii) follows from ii) since the differentials are derivations. That iv))i) follows from the fact that if a ring is complete with respect to an exhaustive filtration, and if the associ- ated graded ring is Noetherian, then so is the original ring (see [6, 3.2.9 and Corollary 2 to Proposition 12] or [3, Corollary 10.25]). Now suppose that part i) holds. Then the the E2-term of the Lyndon-Hochschild-Serre spectral sequence is Noetherian, hence finitely generated over k. Given ii), part iii) follows as above. Given part iii), the algebra T in part iv) can be taken to be the algebra of pN th-powers in E2. The implication iv))ii) was established in the preceding paragraph. This completes the proof. __|_ | We have used: Nilpotence II 51 Lemma A.3. Let {Er; dr} be a spectral sequence of finitely gen- erated modules over a Noetherian ring T . There is an integer N with the property that all of the differentials dr are zero, if r > N. Proof: The modules Er are sub-quotients of E2. Define Br+1 Br . . .Zr Zr+1 . . .E2 with the property that Er+1 = Zr=Br: The graded T -modules Zr and Br can be thought of as the kernel and image of dr respectively. By the ascending chain condition, there is an integer N with the property that Br = BN if r N. But this implies, for r N + 1 that Er EN+1 , so image of dr is zero. __|_ | Wilkerson [20] has proved a special case of Theorem A.1. Theorem A.4 (Wilkerson). i) Suppose that B C is in the center, and that the action of C on R is trivial. Then i)-iv) of Theorem A.1 hold. ii)If B C is normal, the map Ext*C[k; k] ! Ext*B[k; k] is finite. The requirement that C act trivially on R turns out not to be much of a restriction. Lemma A.5. Suppose a Hopf-algebra A acts on a graded commu- tativeNring R. Given an element x 2 R, for N 0, the element xp is invariant under A. In particular, if R is Noetherian, then RA ,! R is finite. Nilpotence II 52 Proof: This is easiest to verify from the point of view of the coaction. By assumption, the coaction is given by X (x) = 1 x + ai xi; where |xi| 6= 0. Since A is finite dimensional, there is an N with the property that apN = 0 for all a 2 A* with |a| 6= 0. But then N pN X pN pN (xp )= 1 x + ai xi N = 1 xp : This completes the proof. __|_ | Lemma A.6. If A is a finite Hopf-algebra and R ! S is a finite map of Noetherian A-algebras, then Ext*A[k; R] ! Ext*A[k; S] is finite. Corollary A.7. If C is a Hopf-algebra, and R is a Noetherian C-module, then the cohomology algebra Ext*C[k; R] is Noetherian. Proof: By the above result, the map RC ! R is finite. Again by the above result, RC Ext*C[k; k] Ext*C[k; RC ] ! Ext*C[k; R] is finite. The result now follows from A.4. __|_ | Corollary A.8. It suffices to prove Theorem A.1 when R = k. Nilpotence II 53 Proof: It is enough to deduce part iv). Suppose that the Lyndon-Hochschild-Serre spectral sequence Ext *C==B[k; Ext*B[k; k]] ) Ext*C[k; k] consists of finitely generated modules over the Noetherian ring T . Then the spectral sequence Ext*C==B[k; Ext*B[k; RC ]] ) Ext*C[k; RC ] consists of finitely generated modules over the Noetherian ring RC T . By Lemma A.5, the map RC ! R is finite. It follows from Lemma A.6 that the map Ext*C==B[k; Ext*B[k; RC ]] ! Ext*C==B[k; Ext*B[k; R]] is finite, so the spectral sequence Ext*C==B[k; Ext*B[k; R]] ) Ext*C[k; R] is also a spectral sequence of finite modules over RC T . __|_ | The proof of A.8 is built out of a few special cases. Lemma A.9. Suppose that E is a Hopf-algebra of the form E[x], where ( 2 (A.9.1) E[x] = k[x]=x if |x| is odd k[x]=xp if |x| is even, and R ! S is a finite map of Noetherian E-algebras. If the action of E on R is trivial, then Ext*E[k; R] ! Ext*E[k; S] is finite. Proof: Let's take the case in which E = k[x]=xp with |x| even. The others are similar. If M is an E-module, the cohomology Ext*E[k; M] Nilpotence II 54 is the cohomology of the complex M [a] k[b] with differential d(m bk) = xp-1m a bk d(m a bk) = xm bk+1: When the action of E on M is trivial, the differential d is zero. The result now follows since the complex for calculating Ext*E[k; S] is already a finite module over the Ext*E[k; R]. __|_ | Lemma A.10. Suppose A is a Hopf-algebra and R ! S is a finite map of Noetherian A-algebras. If the action of A on R is trivial, then Ext*A[k; R] ! Ext*A[k; S] is finite. Proof: The proof is by induction on the dimension of A, the case in which the dimension of A is 1 being a tautology. Suppose that the dimension of A is greater than 1, and that the result is known to be true for Hopf-algebras of dimension less than that of A. Let E A be a central sub-Hopf-algebra of the form (A.9.1), and let {Er} ; and {E0r} be the associated Lyndon-Hochschild-Serre spectral sequences with coefficients in R and S, respectively. The spectral sequence {Er} is just the tensor product of R with the Lyndon-Hochschild-Serre spectral sequence with coefficients in k. By Theorem A.4 it is a spectral sequence of finite modules over a Noetherian ring of the form R T . It follows that the map E1 ! E01 is finite, and so the map Ext*A[k; R] ! Ext*A[k; S] Nilpotence II 55 is finite by [3, Proposition 10.24]. __|_ | Proof of Lemma A.6: Since S is finite over R and R is finite over RA by Lemma A.5, S is finite over RA . It follows from Lemma A.10 that Ext*A[k; RA ] ! Ext*A[k; S] is finite, so a fortiori Ext*A[k; R] ! Ext*A[k; S] is finite. This completes the proof. __|_ | Proof of Theorem A.1: By Corollary A.8 we may assumeNthat R = k. Choose an integer N with the property that xp is invariant under the action of C==B for every x 2 ExtB [k; k], and let S Ext*B[k; k] be the sub-algebra consisting of the (pN )th powers of the elements in the image of ExtC *[k; k]. Then the maps S ! Ext*B[k; k]; and Ext*C==B[k; S]! Ext*C==B[k; Ext*B[k; k]] are finite by A.5 and A.6. But Ext *C==B[k; S] = Ext*C==B[k; k] S is Noetherian, and consists of permanent cycles. Taking T to be this algebra establishes part iv), and completes the proof. __|_ | To deduce Theorem 4.12 requires Lemma A.11. Suppose that B C is an inclusion of finite Hopf- algebras. There is a sequence B = C0 C C1 . .C.Cn = C with each Ci C Ci+1 normal, and with the property that the Hopf- algebra Ci+1==Ci is of the form (A.9.1). Nilpotence II 56 Proof: It suffices, by induction on dim kC==B, to show that if B 6= C then there is a surjective map of Hopf-algebras C ! E[x] with the property that the composite B ! C ! E[x] is trivial (which means that it is the augmentation followed by the inclusion of the degree zero part). Since B 6= C the map of dual algebras C* ! B* is not a monomorphism. It follows from [11, Proposition 3.9] that the map of primitives is not injective. Let D be a primitive in the kernel. The element D can by thought of as a derivation from A to k with the property that D(b) = 0 when b 2 B. Give x the degree -|D|. The map to E[x] is then given by Taylor's formula: 8 P n < p-1n=0Dna x_n!|D| even a 7! : D0a + D(a) x |D| odd The powers of D are taken in the algebra C*. In particular, D0, being the unit of C*, is the augmentation. __|_ | Proof of Theorem 4.12: It suffices, by Lemma A.11 to deal with the case in which B C is normal. Let b 2 Ext*B[k; k] be a cohomology class.MBy Lemma A.5 there is an integer M with the property that bp is invariant under C==B. This gives a class in the E2-term of the Lyndon-Hochschild-SerreMspectral sequence. For convenience, replace bp with b.NBy Theorem A.1 there is an integer N with the property thatNdrbp = 0 for all r. The class in Ext *C[k; k] represented by bp is then the desired class. __|_ | Nilpotence II 57 References [1]J. F. Adams. On the structure and applications of the Steenrod algebra. 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