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-