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2 Hopkins and Smith Nilpotence II 3,5 Endomorphisms up to nilpotents 37 iii A map. 5 1 N endomorphisms 37 f S m R, 5 2 Classification of N endomorphisms 39 from the sphere spectrum to a ring spectrum is nilpotent if it is. 5 3 Some technical tools 41 nilpotent when regarded as an element of the ring R. 5 4 N endomorphisms and thick subcategories 43, 5 5 A spectrum with few nonnilpotent self maps 46 The main result of 7 is. 5 6 Proof of Theorem 5 4 48, Theorem 2 In each of the above situations the map f is nilpo. tent if the spectrum F is finite and if M U f 0,A Proof of Theorem 4 12 49.

In case the range of f is p local the condition M U f 0 can be. replaced with the condition BP f 0, Introduction 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 is a continuation of 7 Since so much time has lapsed this paper were conjectured by Ravenel in 15. since its publication a recasting of the context is probably in order Let K n be the nth Morava K theory at the prime p see 1. In 15 Ravenel described a series of conjectures getting at the Theorem 3. structure of stable homotopy theory in the large The theory was. organized around a family of higher periodicities generalizing i Let R be a p local ring spectrum An element R is. Bott periodicity and depended on being able to determine the nilpotent if and only if for all 0 n K n is nilpotent. nilpotent and non nilpotent maps in the category of spectra There ii A self map f k F F of the p localization of a finite. are three senses in which a map of spectra can be nilpotent spectrum is nilpotent if and only if K n f is nilpotent for all. Definition 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. i A map of spectra, f F X Of course the hypothesis p local can be dropped if the condition. on the Morava K theory is checked at all primes,is smash nilpotent if for n 0 the map. At first the criterion of this theorem seems less useful than the. f n F n X n one provided by 7 Using Theorem 3 to decide whether a map. is nilpotent or not requires infinitely many computations On the. is null other hand Morava K theories are often easier to use than complex. cobordism Theorem 3 also determines which cohomology theories. ii A self map detect the nonnilpotent maps in the category of spectra. is nilpotent if for n 0 the map Definition 4 A ring spectrum E is said to detect nilpotence if. equivalently,f n kn F F, i for any ring spectrum R the kernel of the Hurewicz homomor. is null phism E R E R consists of nilpotent elements. 4 Hopkins and Smith Nilpotence II 5, ii a map f F X from a finite spectrum F to any spectrum X P it suffices by Theorem 7 to show that any object of Cn Cn 1.

is smash nilpotent if 1E f E F E X is null homotopic 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. Corollary 5 A ring spectrum E detects nilpotence if and only if self map. Definition 8 Let X be a p local finite spectrum and n 0 A. for all 0 n and for all primes p self map v k X X is said to be a vn self map if. Now let C0 be the homotopy category of p local finite spectra. multiplication by a if m n 0, and let Cn C0 be the full subcategory of K n 1 acyclics The rational number. Cn fit into a sequence K m v is if m n 6 0,an isomorphism. nilpotent 6 n,Cn 1 Cn C0, It turns out that the property of admitting a vn self map is. 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. Theorem 9 A p local finite spectrum X admits a vn self map if. and only if X Cn If X admits a vn self map then for N 0. Definition 6 A full subcategory C of the category of spectra is. X admits a vn self map, said to be thick if it is closed under weak equivalences cofibrations. and retracts ie N, i An object weakly equivalent to an object of C is in C.

satisfying, ii If X Y Z is a cofibration and two of X Y Z are in C. then so is the third vnp if m n, iii A retract of an object of C is in C 0 otherwise. The vn self maps turn out to be distinguished by another prop. Theorem 7 If C C0 is a thick subcategory then C Cn for erty. Theorem 7 is in fact equivalent to the Nilpotence Theorem the Definition 10 A ring homomorphism. proof is sketched at the end of Section 4 It is often used in the. following manner f A B, Call a property of p local finite spectra generic if the full sub. is an F isomorphism if, category of C0 consisting of the objects with P is closed under cofi. brations and retracts To show that X Cn has a generic property i the kernel of f consists of nilpotent elements and. 6 Hopkins and Smith Nilpotence II 7, ii given b B bp is in the image of f for some n Proof of Theorem 13 Since hXi hY i if and only if hX p i.

hY p i for all primes p we may localize everything at a prime p. Two rings A and B are F isomorphic A F B if there is an For a fixed Y the property of X. F isomorphism between them, Theorem 11 Let X Cn Cn 1 The K n Hurewicz homomor is a generic property It follows that the class. phism gives rise to an F isomorphism,X hXi hY i, Z p n 0 is equal to Cm for some m Suppose that Y Cn Cn 1 We need. 11 1 center X X F, Fp vn n 6 0 to show that m n Since hY i hY i m n But if X. hXi 6 hY i since K n 1,hXi so m 6 n, The description of spectra as cell complexes encourages the in. tuition that the endomorphism rings of finite spectra approximate. matrix algebras over the ring S 0 This would suggest that the. centers of these rings are generated by the maps obtained by smash Acknowledgements. ing the identity map with a map between spheres an impossibil Most of the results of this paper date from 1985 and there have. ity by Theorem 11 A more accurate description might be that been many people who helped shaped the course of the results. the Morita equivalence classes of these rings are determined by Special thanks are due to Emmanuel Dror Farjoun whose prodding. the integer n of Theorem 11 This integer invariant can also be eventually led to the formulation of Theorem 7 and to Clarence. thought of as determining the birational equivalence classes of Wilkerson for helpful conversations concerning the proof of Theo. finite spectra For more on this analogy see 9 rem 4 12 Even deeper debts are owed to Doug Ravenel for formu. There is a less metaphorical interpretation of the integer which lating such a beautiful body of conjectures and to Mark Mahowald. occurs in Theorems 9 and 11 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, Definition 12 Let X be a spectrum The Bousfield class of X.

denoted hXi is the collection of spectra Z for which X Z is not. contractible,Notation and conventions, The Bousfield classes of spectra are naturally ordered by inclu. For the most part we will work in the homotopy category of spec. sion though the relation is indicated with rather than. tra Of course to form things like the induced map of cofibers. For a finite spectrum X let Cl X N P denote the set. requires choosing a diagram in a model for the category of spectra. of pairs n p for which K n X 6 0 at p Here N is the set of. introducing a certain ambiguity into the resulting map This am. nonnegative integers and P is the set of primes, biguity plays only a very small role in this paper and is dealt with. each time it comes up, Theorem 13 Let X and Y be finite spectra Then hXi hY i if The cofiber of a map X Y will be written with the cone. and only if Cl X Cl Y coordinate on the right, Theorem 13 affirms Ravenel s Class Invariance conjecture 15 Y f CX Y X I. 8 Hopkins and Smith Nilpotence II 9, With this convention the cofiber of 1 1 Construction.

Z X Z Y The study of a ring is often simplified by passage to its quotients. and localizations The same is true of ring spectra though con. is Z Y f CX modulo associativity of the smash product structing quotients and localizations can be difficult In good cases. not just isomorphic to it With this convention the cofiber of the following constructions can be made. Y Y f CX is X S 1 which is isomorphic but not equal to. X This avoids encountering the troublesome sign that can crop Quotients. up when trying to relate the connecting homomorphism in a cofi Suppose that E is a ring spectrum and that E R is commu. bration with the connecting homomorphism in some suspension of tative Given x R define the spectrum E x by the cofibration. 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 If x is a non zero divisor then E x is isomorphic to the ring. spectra this term is used to refer to a spectrum which is weakly R x In good cases E x will still be a ring spectrum and the. equivalent to the p localization of a finite spectrum The only prop map. erty of finite spectra that is used is that the set of homotopy classes E E x. of maps from a finite spectrum to a directed colimit is the colimit. of the maps will be a map of ring spectra Given a regular sequence. X Y x1 xn R, In general and object of a category with this property is said to be one can hope to iterate the above construction and form a ring. small It can be shown that the small objects of the category of p spectrum E x1 xn with. local spectra are precisely the objects which are weakly equivalent E x1 xn R x1 xn. to the p localizations of a finite spectrum, A spectrum X is connective if k X 0 for k 0 It is and such that the natural map. connected if k X 0 for k 0 Thus connected and 1 E E x1 xn. connected are synonymous Similarly a graded abelian is con. nective if the homogeneous part of degree k is zero for k 0 A is a map of ring spectra. graded abelian group is connected if the homogeneous component. of degree k is zero for k 0 Localizations, The Eilenberg MacLane spectrum with coefficients in an abe Let E and R be as above and suppose that S R is a multiplica. lian group A will be denoted HA To be consistent with this the tively closed subset Since S 1 R is a flat R module the functor. homology of a spectrum X with coefficients in A will be denoted. HA X S 1 R E, Finally the suspension of a map will always be labeled with the.

same symbol as the map 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. 1 Morava K theories,10 Hopkins and Smith Nilpotence II 11. 1 2 Spectra related to BP respectively Then there is an equality of Bousfield classes. When the ring spectrum in question is BP the above constructions hXi hX vXi hv 1 Xi. 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 vn with vn 2pn 2 To. fix notation take the set vn to be the Hazewinkle generators 8 Proposition 1 3 There is an equality of Bousfield classes. For 0 n the ring spectra K n and P n are defined by the. isomorphisms hvn 1 P n i hK n i,K n Fp vn vn 1,P n Z p vn vn 1. 1 3 Fields in the category of spectra, with the understanding that they are constructed from BP using. a combination of the above methods It is also useful to set 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. K 0 HQ of the Morava K theories, There are maps P n P n 1 and the limit Proposition 1 4 For any spectrum X K n X has the homo.

topy type of a wedge of suspensions of K n,is the Eilenberg MacLane spectrum HFp. Proof Choose a basis ei i I of the free K n module K n X. and represent it as a map, Proposition 1 1 The Bousfield classes of K n and P n are re. lated by S ei K n X,hP n i hK n i hP n 1 i i I,Consequently The composition. hBP i hK 0 i hK n i hP n 1 i,K n S ei K n K n X K n X. Proof The proposition follows from the next two results of. Ravenel 15 is then a weak equivalence, Proposition 1 2 Let v k X X be a self map of a spectrum Proposition 1 5 For any two spectra X and Y the natural map.

X Let X vX and v 1 X denote the cofiber of v and the colimit of. the sequence 1 5 1 K n X K n K n Y K n X Y,is an equivalence. 12 Hopkins and Smith Nilpotence II 13, Proof Consider the map 1 5 1 as a transformation of func Proof Since 1 E is non nilpotent for some prime p and. tors of Y The left side satisfies the Eilenberg Steenrod axioms for some n. since K n Y is a flat in fact free K n module The right side K n E 6 0. satisfies the Eilenberg Steenrod axioms by definition The trans. Since K n and E are both fields it follows from Lemma 1 7 that. formation is an isomorphism when Y is the sphere hence for all. 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. Proof of Corollary 5 If for some n K n E 0 then E does proposition. not detect the nonnilpotent map, S 0 K n 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. If K n E 6 0 then by Proposition 1 4 has the homotopy type of a wedge of suspensions of K n. 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 De nition 1 i A map of

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