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 MoravaK -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 NilpotenceII 2 5 Endomorphisms, up to nilpotents 37 5.1 N-endomorphisms : :: : : : : : : : : : : : : : : : : 37 5.2 Classification ofN -endomorphisms : : : : : : : : : : 39 5.3 Some technicalto ols : : : : : : : : : : : : : : : : : : 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 familyof "higher periodicities" generalizing Bott periodicity, and dependedon 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: Definition1. i)A map of spectra f : F ! X is smashnilpotent if for n AE 0 the map f (n): F(n)! X(n) isnull. ii)A self map f : kF ! F is nilpotent if for n AE 0 the map fn : knF ! F NilpotenceII 3 iii)A map f : Sm ! R; from the sphere spectrum to aring 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 isp-lo cal, the condition MU f = 0 can be replaced with the condition B Pf = 0. The purpose of this paper is to refine thiscriterion and to pro- duce some interestingnon-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 andonly 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 amap 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. Definition4. A ring spectrum E is said to detect nilpotence if, equivalently, i)for any ring spectrum R, the kernel of the Hurewicz homomor- NilpotenceII 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. Corollary5. 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 thehomotopy category of p-local finite spectra and let Cnae C0 be the full subcategoryof K (n 1)-acyclics. The Cn fit into a sequence ae Cn+1 ae Cn ae ae C0: This is a nontrivialfact. That there are inclusions Cn+1 ae Cn is essentially the Invariant Prime Ideal Theorem. See [15]. That the inclusions are proper is a result of Steve Mitchell [12]. Definition6. 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 ob ject of C is in C. ii)If X ! Y ! Zis a cofibration, and two of fX; Y; Zg are in C then so is the third. iii)A retract of an object of C is inC . 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 atthe end of Section 4). It is often used in the followingmanner. Call a property of p-local finite spectra generic if the full sub- category of C0consisting of the objects with P is closed under cofi-