Towers of M U -algebras and the generalized
Hopkins-Miller theorem
A.Lazarev
Department of Mathematics, Univ. of Bristol, Bristol, BS8 1TW, UK.
email A.Lazarev@bristol.ac.uk
Abstract
Our results are of three types. First we describe a general procedure
of adjoining polynomial variables to A1 -ring spectra whose coefficient
rings satisfy certain restrictions. A host of examples of such spectra
is provided by killing a regular ideal in the coefficient ring of MU,
the complex cobordism spectrum. Second, we show that the algebraic
procedure of adjoining roots of unity carries over in the topological
context for such spectra. Third, we use the developed technology to
compute the homotopy types of spaces of strictly multiplicative maps
between suitable K(n)-localizations of such spectra. This generalizes
the famous Hopkins-Miller theorem and gives strengthened versions of
various splitting theorems.
Key words: S-algebras, topological derivations, Morava K-theories,Witt vectors.
1 Introduction
Our goals in this paper are three-fold. First, we generalize and extend the
methods of [6] of constructing A1 ring spectra by adjoining öp lynomial
variables". Namely, suppose that R is a commutative S-algebra in the sense
of [4] such that its coefficient ring R* has no elements of odd degree and the
graded ideal I* in R* is generated by a regular sequence u1, u2, . ...Then it
is known from work of Strickland, [13] that the R-module R=I obtained by
killing the ideal I has a structure of an R-ring spectrum. We assume that
R=I is actually an R-algebra. Then it turns out that, informally speaking,
all R-modules "between" R=I and R also possess structures of R-algebras.
1
More precisely, there are R-algebras R[uk]=ulkwhose coefficient rings are ob-
tained from R* by killing the sequence (u1, u2, . .,.uk-1, ulk, uk+1, . ...There
are also natural "reduction maps" between these R-algebras.
The basic example is provided by taking R = MU, the complex cobor-
dism S-algebra and MU=I be the Eilenberg-MacLane S-algebra H=p. It
follows by induction that all MU-algebras obtained by killing any sequence
of polynomial generators and/or a prime p have structures of MU-modules.
Second, we consider the question of adjoning roots of unity to an S-
algebra. This problem was also treated in the recent preprint [12] by Schwanzl,
Vogt and Waldhausen. Their definition of a topological extension has bet-
ter formal properties than ours but it applies in a far less general situation.
We show that one can adjoin roots of unity to such spectra as Morava K-
theories K(n), Johnson-Wilson theories E(n) and many other algebras over
the complex cobordism S-algebra MU.
Third, we address the problem of computing S-algebra maps between a
certain completion E^(n) of E(n) and an MU-algebra E which is assumed
to be "strongly K(n)-complete" in some precise sense explained later on in
the paper. Examples of such MU-algebras include ^E(n), K(n) and the Ar-
tinian completion of the vn-localization of the Brown-Peterson spectum BP .
It turns out that the space of S-algebra maps between ^E(n) and a strongly
K(n)-complete MU-algebra is homotopically discrete with the set of con-
nected components being equal to the set of multiplicative cohomology op-
erations ^E(n) ! E. We call the results of this type the generalized Hopkins-
Miller theorem because the original Hopkins-Miller theorem (cf.[10]) asserts
that the space of S-algebra self-maps of the spectrum En is homotopically
equivalent to the (discrete) space of multiplicative operations from En to
itself. Here En is a 2-periodic version of the completed Johnson-Wilson
theory ^E(n) which came to be popularly known as the Morava E-theory.
We chose to work with the 2(pn - 1)-periodic theory E(n) rather than
with En. The relation of E(n) to En is the same as the relation of the p-local
Adams summand of the complex K-theory spectrum KU to KU itself. The
advantage of E(n) is that it is smaller than En, however it does not admit
the action of the full Morava stabilzer group.
One consequence of our generalized Hopkins-Miller theorem is that ^E(n)
admits a unique S-algebra structure, the result previously obtained in [1].
Another consequence is that E^(n) splits off E as an S-algebra for a
certain class of strongly K(n)-complete MU-algebras E. Such splittings
were previously known to be multiplicative only up to homotopy.
Throughout the paper the symbol Fp will denote the prime field with
p elements, L=Fp an arbitrary (but fixed) finite separable extension of Fp.
2
Further W (L) and Wn(L) denote the ring of Witt vectors and the ring of
Witt vectors of length n respectively.
2 Adjoining polynomial variables to S-algebras
In this section we assume that R is a fixed commutative S-algebra such that
R* = ß*R is a graded commutative ring concentrated in even degrees. We
will also assume without loss of generality that R is q-cofibrant in the sense
of [4]. All objects under consideration will be R-modules or R-algebras and
smash products and homotopy classes of maps will be understood to be
taken in the category of R-modules.
We begin by reminding the reader the notions of topological derivations
and topological singular extensions of R-algebras. A detailed account can
be found in [6]. Let A be an R-algebra and M an A-bimodule. Then the
R-module A _ M has the obvious structure of an R-algebra ("square-zero
extensionö f A). Consider the set [A, A _ M]R-alg=A of homotopy classes of
R-algebra maps from A to A _ M which commute with the projection onto
A. Then there exists an A-bimodule A and a natural in M isomorphism
[A, A _ M]R-alg=A ~=[ A , M]A-bimod
where the right hand side denotes the homotopy classes of maps in the
category of A-bimodules.
Definition 2.1 The topological derivations R-module of A with values in
M is the function R-module FA^Aop( A , M). We denote it by Der (A, M)
and its ith homotopy group by Der-i(A, M).
The A-bimodule A is constructed as the homotopy fibre of the multiplica-
tion map A ^ A ! A. There exists the following homotopy fibre sequence
of R-modules:
THH (A, M) ! M ! Der (A, M) (1)
Here THH (A, M) is the topological Hochschild cohomology spectrum of A
with values in M, THH (A, M) := FA^Aop(A, A).
We will frequently use the notion of a primitive operation from A to M.
Denote by m : A ^ A ! A the multiplication map and by ml: A ^ M ! M
and mr : M ^A ! M the left and right actions of A in M respectively. Then
a map p : A ! M is called primitive if p is a "derivation up to homotopy",
3
i.e. the following diagram is homotopy commutative:
A ^ A -m! A
1 ^ p _ p ^ 1 # # p
A ^ M _ M ^ A ml_mr-!M
There is a forgetful map l : Der*(A, M) ! [A, M]* defined as follows.
For any topological derivation d : A ! A _ M let l(d) be the composite map
A ! A _ M ! M where the last map is just the projection onto the wedge
summand. Then it is easy to see that the image of l is contained in the set
of primitive operations from A to M.
Suppose we are given a topological derivation d : A ! A _ M. Consider
the following homotopy pullback diagram
X ! A
# #
A !d A _ M
Here the rightmost downward arrow is the canonical inclusion of a retract.
Then we have the following homotopy fibre sequence of R-modules:
-1M ! X ! A (2)
Definition 2.2 The homotopy fibre sequence (2) is called the topological
singular extension associated with the derivation d : A ! A _ M.
For an element x 2 R we will denote by R=x the cofibre of the map
R !x R Let I* be a graded ideal generated by (possibly infinite) regular
sequence of elements (u1, u2, . .).2 R* of degrees r1, r2, . ... Then we can
form the R-module R=I as the infinite smash product of R=xi. By [13],
Proposition 4.8 there is a structure of an R-ring spectrum on R=I. Clearly
the coefficient ring of R=I is isomorphic to R*=I* where R*=I* is understood
to be the direct limit of R*=(u1, u2, . .,.uk).
Our standing assumption is that R=I has a structure of an R-algebra
(i.e. strictly associative). This may seem a rather strong condition but, as
we see shortly such a situation is rather typical. In fact P.Goerss proved
in [5] that any spectrum obtained by killing a regular sequence in MU, the
complex cobordism S-algebra, has a structure of an MU-algebra.
The construction we are about to describe allows one to construct new
R-algebras by ä djoining" the indeterminates uk to the R-algebra R=I. The
basic idea is the same as in [6] where Morava K-theories at an odd prime
4
were shown to possess MU-algebra structures. However the arguments we
use here are considerably more general, in particular we make no assumption
that the prime 2 is invertible. Let us introduce the notation R=I[uk]=ulk*
for the R*-algebra
limn!1 R*=(u1, u2, . .,.uk-1, ulk, uk+1, . .,.un).
This notation emphasizes the analogy with the truncated polynomial algebra
even though R=I[uk]=ulk*is not always one. (Take, for instance R* to be the
ring of integers and I* = (u1) = p, a prime number. Then R=I[uk]=ul1*=
Z=plZ.) For each l reduction modulo ul+1kdetermines a map of R*-algebras
R=I[uk]=ul+1k*! R=I[uk]=ulk*.
Now we can formulate our main theorem in this section.
Theorem 2.3 For each k and l there exist R-algebras R=I[uk]=ulkwith co-
efficient rings R=I[uk]=ulk*and R-algebra maps
Ri,k: R=I[uk]=ul+1k! R=I[uk]=ulk
which give the reductions mod ul+1kon the level of coefficient rings.
Proof. Suppose by induction that the R-algebras R=I[uk]=ulkwith the
required properties were constructed for l i. We will show that there
exists an appropriate Bokstein operation from R=I[uk]=uikto |uk|i+1R=I
which allow us to build the next stage.
Consider the cofibre sequence
ulk jl,k l fil,k|u |l+1
|uk|lR ! R ! R=uk ! k R.
According to [13] the R-module R=ulkadmits an associative product OE :
R=ulk^ R=ulk! R=ulkand for any other product OE0 there exists a unique
element u 2 ß2|uk|l+2R=ulkfor which OE0= OE + u O (fil,k^ fil,k).
Lemma 2.4 There exists a map of R-ring spectra ri,k: R=ui+1k! R=uik
realizing the reduction map mod ui+1kon coefficient rings. Moreover in the
cofibre sequence
ri,k i ~fii,k|u |i+1
R=ui+1k! R=uk ! k R=ui (3)
the second map ~fik,i: R=uik! |uk|(i+1)R=uk is a primitive operation.
5
Proof. Use induction on i (in fact we only need the inductive assumption
in order that R=uk be a (bi)module spectrum over R=uik.) Consider the
following diagram of R-modules:
|uk|(i+1)R uk! |uk|iR ! |uk|iR=uk
ui+1k# # uik #
R = R ! pt
# # #
ri,k i u |i+1
R=ui+1k ! R=uk ! k R=uk
# ||
|uk|i+1R ! |uk|i+1R=uk
Here the rows and columns are homotopy cofibre sequences of R-modules.
The map ri,k: R=ui+1k! R=uikis determined uniquely by the requirement
that the diagram commute in the homotopy category of R-modules.
We now show that it is possible to choose a product on R=ui+1kso that
ri,kbecomes an R-ring spectrum map. Take first any associative product
OE : R=ui+1k^ R=ui+1k! R=ui+1kwhich exists by [13], Proposition 3.1. As in
[13], Proposition 3.15 there is an obstruction d(OE) 2 ß2|uk|(i+1)+2(R=uik) for
the map ri,kto be homotopy multiplicative. If the product OE is changed the
obstruction changes according to the formula
d(OE + u O (fii,k^ fii,k)) = d(OE) + ri,k*u.
Since the map ri,k*: R*=ui+1k! R*=uikis just the reduction modulo ui+1k
it is surjective and we conclude that there exists a product on R=ui+1kfor
which the obstruction vanishes.
Further the map ~fii,k: R=uik! |uk|i+1R=uk coincides with the compo-
sition
fii,k|u |i+1ji,k|u |i+1 i |u |i+1
R=uik! k R ! k R=uk ! k R=uk.
Here the last map is (a suspension of) the canonical R-ring map R=uik!
R=uk which exists by the inductive assumption. The composition æi O fii
is the Bokstein operation which is primitive by [13], Proposition 3.14. It
follows that fik,iis also primitive and our lemma is proved.
Now taking the smash product of (3) with R=u1^ R=u2^ . .^.R=uk-1 ^
^R=uk+1 ^. .w.e get the following homotopy cofibre sequence of R-modules:
Ri,k i Qi,k |u |i+1
R=I[uk]=ui+1k-! R=I[uk]=uk -! k R=I.
It follows that Qi,kis a primitive operation and R=I[uk]=ui+1khas an R-ring
spectrum structure such that Ri,kis an R-ring map. (Notice our abuse of
6
notations here in using the symbol Ri,keven though R=I[uk]=ui+1kis not yet
proved to be an R-algebra).
Next we will describe the set of all primitive operations R=I[uk]=uik!
*R=I. Consider the cofibre sequence
|ul|R ul!R jl!R=ul!fil |ul|+1R.
For l 6= k introduce the operation
Ql: R=I[uk]=uik= R=u1^R=u2^. .^.R=uk-1^R=uik^R=uk+1^. .!. |ul|+1R=I
obtained by smashing ælO fil: R=ul! |ul|+1R=ulwith the identity map on
the remaining smash factors. Clearly the operations Ql are primitive. The
next result shows that Ql and Qi,kare essentially all primitive operations
from R=I[uk]=uikinto (suspensions of) R=I.
Lemma 2.5 Any primitive operation R=I[uk]=uik! *R=I can be written
uniquely as an infinite sum akQi,k+ i6=kaiQi for ai2 R=I*.
Proof. We will only give a sketch since the arguments of Strickland ([13],
Proposition 4.17) carry over almost verbatim. Let j : R ! R=I[uk]=uik
be the unit map and J* = (u1, u2 . .,.uk-1, uik, uk+1,...) be the kernel of the
map j* : R* ! R=I[uk]=uik*. Given a primitive operation Q : R=I[uk]=uik!
sR=I define the function d(Q) : J* ! R=I[uk]=uik*as follows. For any
x 2 J* we have a cofire sequence
|x|R x! R jx!R=x fix! |x|+1R.
Then there is a unique map fx : R=x ! R=I[uk]=uiksuch that fx O æx = j.
Further there is a unique map y : |x|+1R ! sR=I such that QOfx = yOfix.
We define d(Q)(x) := y 2 ß|x|+1-sR=I*.
It follows that the function d actually embeds the set of primitive op-
erations from R=I[uk]=uikinto R=I into HomR*(J=J2, R=I*). Further it is
straightforward to check that
d(Qs)(ut) = ffistfor s, t 6= k, d(Qs)(uik) = 0;
d(Qi,k)(uj) = 0 for j 6= k, d(Qi,k)(uik) = 1.
Therefore the elements Qi,k, Qs, s 6= k form a basis in Hom(J*=J2*, R=I*)
dual to the basis (u1, u2, . .,.uk-1, uik, uk+1, . .).in J*=J2*and our lemma is
proved.
7
Now we come to the crucial part of the proof. We are going to show that
the Bokstein operation Qi,kcan be improved to a topological derivation
from which it would follow that R=I[uk]=ui+1kis an R-algebra and Ri,k:
R=I[uk]=ui+1k! R=I[uk]=uiklifts to an R-algebra map.
The following lemma (interesting in its own right) is preparatory for
computing the topological Hochschild cohomology of R=I[uk]=ulkwith coef-
ficients in R=I.
Lemma 2.6 For any regular ideal J* = (x1, x2, . .).in R* and any product
on R=J there is a multiplicative isomorphism
ß*(R=I ^R R=Jop) ~= R*=J*(ø1, ø2, . .).
where |øi| = |xi| + 1.
Proof. The standard spectral sequence arguments show that ß*(R=J ^R
R=Jop) is a filtered R*=J*-algebra whose associated graded is isomorphic
to R*=J*(ø1, ø2, . .).so the problem is to show that the possible multi-
plicative extensions are trivial. More precisely one needs to show that all
skew-commutators of the elements øi in ß*(R=J ^R R=Jop) as well as their
squares are zero. Consider the map of R-ring spectra f : R=J ^R R=Jop !
FR (R=J, R=J) which is induced by the structure of a left R=J ^R R=Jop-
module spectrum on R=J. Then f induces a map of R*=J*-algebras f* :
ß*R=J ^R R=Jop ! ß*FR (R=J, R=J). By [13], Proposition 4.15 the ring
ß*FR (R=J, R=J) is a completed exterior algebra over R*=J*, in particular it
is (graded) commutative. Therefore all possible nontrivial relations between
the elements øi map to zero in the R*=J*-algebra ß*FR (R=J, R=J). Since
these relations belong to R*=I* and f* is R*=I*-linear we conclude that they
are actually zero and ß*(R=J ^R R=Jop) is in fact a (graded) commutative
R*=J*-algebra. This finishes the proof of lemma 2.6.
Remark 2.7 The R*=J*-algebra ß*(R=J ^R R=J) need not be commutative
in general. For example take R=J = MU=2, the reduction of MU modulo
2. Then it can be shown using the methods of [8] that MU=2 ^MU MU=2
is an MU=2*-algebra on one generator in degree 1 whose square is equal to
x1 2 MU=2*. We will discuss this and related phenomena elsewhere.
Lemma 2.8 There is the following isomorphism of graded R*-modules:
grT HH*(R=I[uk]=uik, R=I) = R=I*[~u1, ~u2, . .,.~uk, . .].;
grDer*-1(R=I[uk]=uik, R=I) = R=I*[~u1, ~u2, . .,.~uk, . .].=(R=I*).
8
Here gr(?) denotes the associated graded module. Moreover the image of the
forgetful map
Der*(R=I[uk]=uik, R=I) ! [R=I[uk]=uik, R=I]*
is the set of all primitive operations from R=I[uk]=uikto *R=I.
Proof. Denote by J* the ideal in R generated by (u1, u2, . .,.uk-1, uik, uk+1, *
*. .)..
Then we have the following spectral sequence
Ext**i*R=I[uk]=ui (R=iopI[uk]=uik*, R=I*)
k^R=I[uk]=uk
= Ext**i*R=J^R=Jop(R=J*, R=I*) ) T HH*(R=I[uk]=uik, R=I).
By Lemma 2.6 ß*R=J ^ R=Jop = R=J*(ø1, ø2 . .).. Therefore
Ext**i*R=J^R=Jop(R=J*, R=I) = R=I*[~u1, ~u2, . .,.~ui, . .].
where |~ul| = -|ul| + 2 for l 6= k and |~uk| = -|uk|i + 2. Our spectral sequence
collapses and we obtain the desired isomorphism
grT HH*(R=I[uk]=uik, R=I) = R=I*[~u1, ~u2, . .,.~ui, . .]..
The isomorphism involving Der*(R=I[uk]=uik, R=I) is obtained similarly.
Now consider the spectral sequence
Ext**R*(R=I[uk]=uik*, R=I*) = Hom( (z1, z2, . .)., R=I*) ) [R=I[uk]=uik, R=I]*.
Here |zl| = |ul| + 1 for l 6= k and |zk| = |uk|i + 1. This spectral sequence is
easily seen to collapse and denoting by ~zithe elements in the dual basis in
(z1, z2, . .).we identify its E2 = E1 -term with R=I*(~z1, ~z2, . .).. (Of co*
*urse
this is an identification only as R=I*-modules, not as rings.) It is clear that
the elements zl correspond to the primitive operations Ql : R=I[uk]=uik!
|ul|+1R=I for l 6= k whilst zk corresponds to Qi,k: R=I[uk]=uik! |ul|i+1R=I.
Furthermore the forgetful map l (operating on the level of E2-terms)
sends ~ul2 grDer*(R=I[uk]=uik, R=I) to ~zl. In other words all primitive
operations R=I[uk]=uik! *R=I are covered by l up to higher filtration
terms. Since the image of l : Der*(R=I[uk]=uik, R=I) ! [R=I[uk]=uik, R=I]*
is contained in the subspace of the primitive operations no higher filtration
terms are present and we conclude that all Bokstein operations Ql and Qi,k
are in the image of l. With this Lemma 2.8 is proved.
So we proved that there exists a topological derivation ~Qi,k: R=I[uk]=uik!
R=I[uk]=uik_ |uk|i+1R=I such that its composition with the projection onto
9
the wedge summand R=I[uk]=uik_ |uk|i+1R=I ! |uk|i+1R=I is the Bok-
stein operation Qi,k. Associated with ~Qi,kis a topological singular extension
|uk|iR=I ! X ! R=I[uk]=uik
such that X is weakly equivalent to R=I[uk]=ui+1k. In other words the R-
module R=I[uk]=ui+1kadmits a structure of an R-algebra so that the reduc-
tion map R=I[uk]=ui+1k! R=I[uk]=uikis an R-algebra map. The inductive
step is completed and Theorem 2.3 is proved.
The singular extension of R-algebras
Ri,k i Qi,k |u |i+1
R=I[uk]=ui+1k-! R=I[uk]=uk -! k R=I
is of fundamental importance to us. We will call the R-algebra R=I[uk]=ui+1k
an elementary extension of R=I[uk]=uik. Sometimes we will use the notation
R=I[uk] for holimi!1 R=I[uk]=uik. Of course the coefficient ring of R=I[uk]
is not always a polynomial algebra on R=I*.
Now let R= MU, the complex cobordism spectrum. It is well-known that
MU is a commutative S-algebra and MU* = Z[x1, x2, . .]., the polynomial
algebra on infinitely many generators in even degrees.
Corollary 2.9 Let J* be a regular ideal in MU* = Z[x1, x2, . .].generated
by any subsequence of the regular sequence p 0, x11, x22, . .w.here l 1 for
all l. Then MU=J admits a structure of an MU-algebra.
Proof. First assume that the element p0 does not belong to J*. Denote by
I* be the ideal in MU* generated by all polynomial generators (x1, x2, . .)..
Then MU=I is the integral Eilenberg-MacLane spectrum HZ which pos-
sesses a canonical structure of an MU-algebra (even a commutative MU-
algebra).
Taking successive elementary extensions corresponding to elements xiikk
with ik> 1 we construct an MU-algebra
MUg=J = ((HZ[xi1]=x i1 ik
i1 )[xi2]=x ) . ...
Similarly taking elementary extensions corresponding to those polynomial
generators {u1, u2, . .}.whose powers are not in J* and passing to the (in-
verse) limit we construct the MU-algebra MUg=J [u1, u2, . .].whose underly-
ing MU-module is MU=J.
If the element p 0 does belong to J* the only difference is that we start
our induction with the Eilenberg-MacLane MU-algebra HZ=p 0 instead of
HZ. With this Corollary 2.9 is proved.
10
Remark 2.10 Notice that our method also shows that the MU-algebra struc-
tures on MU=J for different J are compatible in the sense that various re-
duction maps given by killing elements xii are actually MU-algebra maps.
Recall that the Brown-Peterson spectrum BP is obtained from MU by
killing all polynomial generators in MU* except for vi = x2(pi-1)and local-
izing at the prime p. Therefore BP* = Z(p)[v1, v2, . .].. It follows that BP
possesses an MU-algebra structure. We define
BP < n >= BP=(vi, i > n) = HZ(p)[v1, v2, . .,.vn]
P (n) = BP=(vi, i < n) = HFp[vn, vn+1, . .].
B(n) = v-1nP (n) = v-1nHFp[vn, vn+1, . .].
k(n) = BP=(vi, i 6= n) = HFp[vn]
K(n) = v-1nBP=(vi, i 6= n) = HFp[vn, v-1n]
E(n) = v-1nBP=(vi, i > n) = HZ(p)[v1, v2, . .,.vn, v-1n]
Since inverting an element in the coefficient ring is an instance of Bous-
field localization and Bousfield localization preserves algebra structures we
conclude that all spectra listed above admit structures of MU-algebras.
We conclude this section with a few remarks about the commutativity
of the products on the MU-algebras considered. First in the case of the
odd prime p BP and other spectra derived from it have coefficient rings
concentrated in degrees congruent to 0 mod 4 and therefore all products
are unique up to homotopy and automatically commutative. If p=2 then
Strickland shows that BP still admits a structure of a commutative MU-
ring spectrum, but this does not follow directly from our construction. We
conjecture that in the context of Corollary 2.9 one can choose an MU-algebra
structure on MU=J compatible with any given structure of an MU-ring
spectrum on MU=J.
The situation with E(n) is different. The spectrum E(n) is Landweber
exact regardless of the prime p. Therefore it possesses a unique homotopy
associative and commutative multiplication compatible with its structure
(in the classical sense) of an MU-algebra spectrum. Our results show that
this multiplication can be improved to an MU-algebra structure which in
turn gives an A1 -structure. On the other hand Strickland shows in [13] that
at the prime 2 there is no homotopy commutative multiplication on E(n)
in the category of MU-modules (to get a homotopy commutative product
one has to choose a different set of generators). Therefore, quite curiously,
E(n) is a homotopy commutative A1 -ring spectrum and simaltaneously a
MU-algebra that is not commutative even up to homotopy.
11
3 Separable extensions of R-algebras
We keep our convention that R is a commutative S-algebra with coefficient
ring R* concentrated_in even degrees. Let A be an R-algebra with coefficient
ring A* and A* A *a ring extension in the usual algebraic_sense. In this
section we consider the_problem of finding an R-algebra A together_with
an R-algebra map A ! A wich realizes_the given extension A* A * in
homotopy._If such an algebra A exists we say that the algebraic extension
A* A* admits a topological lifting.
We now describe a general framework in which_it_is possible to prove that
topological liftings exist. Suppose that R* ! R *is a separable extension_
of R._ That means that R* is a subring of a graded_commutative ring R *
and_R *is_a separable algebra over R*,_i.e._R * is a projective module over
R* R* R*. In addition we assume that R*_is_a projective R*-module._Then
for any ideal I* in R* the_R*-algebra_R*=I *:= R*=I* R* R* is a separable
R*=I*-algebra so R*=I* ! R*=I *is also a separable extension.
Further assume that the ideal I* is generated by a regular sequence
(u1, u2, . .).(possibly infinite), the R-module R=I is supplied_with a struc-
ture of an R-algebra and the algebraic extension R*=I* ! R*=I * admits_
a topological lifting._In_other words there exists an R-algebra R=I and an
R-algebra map R=I ! R=I realizing the given separable extension on the
level of coefficient rings.
Recall that in the previous section we constructed an R-algebra structure
on the R-module
R=I[uk]=ulk:= R=u1 ^ R=u2 . .^.R=uk-1 ^ R=ulk^ R=uk+1 ^ . ...
Then we have the following
___________
Theorem 3.1___There_exist_R-algebras R=I[uk]=ulkrealizing in homotopy the
__
R*-module R=I[uk]=ulk*:= R=I[uk]=ulk* R* R and reduction maps
____ _____________l+1__________
Ri,k: R=I[uk]=uk ! R=I[uk]=ulk.
___________
Moreover the algebraic extension of rings R=I[uk]=ulk*! R=I[uk]=ulk*ad-
mits a topological lifting so that the following diagram of R-algebras is ho-
motopy commutative:
Ri,k l
R=I[uk]=ul+1k -! R=I[uk]=uk
# #
_____________ ____Ri,k________
R=I[uk]=ul+1k -! R=I[uk]=ulk
12
___________
Proof. Proceeding by induction suppose that the R-algebras R=I[uk]=ulk
with required properties were constructed for l i Consider the spectral
sequence ___________ ___________op
T orR***(R=I[uk]=uik*, R=I[uk]=uik*)
__ __ R i iop
= R* R* R * R* T or***(R=I[uk]=uk *, R=I[uk]=uk* )
__ __
= R* R* R * R* R=I[uk]=uik*(ø1, ø2, . .).
___________ ___________
= R=I[uk]=uik* R=I[uk]=uik*R=I[uk]=uik* (ø1, ø2, . .).
___________ ___________op
) ß*R=I[uk]=uik^R R=I[uk]=uik .
Since by Lemma 2.6
ß*R=I[uk]=uik^R R=I[uk]=uikop= R=I[uk]=uik*(ø1, ø2, . .).
we see that the exterior generators øiare permanent cycles and it_follows_that
our_spectral_sequence collapses multiplicatively so the ring ß*R=I[uk]=uik^R
op
R=I[uk]=uik__is_isomorphic_to the_exterior_algebra_on ø1, ø2, . .w.ith coeffi-
cients in R=I[uk]=uik* R=I[uk]=uik*R=I[uk]=uik*.
___________
Further since R=I[uk]=uik*is a separable R=I[uk]=uik*-algebra we see that
___________ ____
grDer*-1(R=I[uk]=uik, R=I)
___________ ____ ____
= R=I[uk]=uik* R=I[uk]=uik* R=I *[u~1, ~u2, . .].=(R=I *).
Similarly ____
grDer*-1(R=I[uk]=uik, R=I)
___________ ____ ____
= R=I[uk]=uik* R=I[uk]=uik* R=I *[u~1, ~u2, . .].=(R=I *).
___________ ____ ____
Therefore the map Der (R=I[uk]=uik,_R=I)_!_Der_(R=I[uk]=uik, R=I) induced
by the R-algebra map R=I[uk]=uik! R=I[uk]=uikis an isomorphism in ho-
motopy.
Consider the following homotopy commutative diagram of R-algebras:
___________ d ____ ____
R=I[uk]=uik -! R=I _ |uk|i+1R=I
" " ____
R=I[uk]=uik -! R=I _ |uk|i+1R=I (4)
|| "
Qi,k |u |i+1
R=I[uk]=uik -! R=I _ k R=I
13
Here the horizontal arrows are topological derivation. The lower arrow is the
Bokstein operation Qi,kconstructed in the previous section,the middle arrow
is the_only_one_that makes the_lower_square commute. Since derivations
of R=I[uk]=uik with values in R=I are in_one-to-one_correspondence with
derivations of R=I[uk]=uikwith values in R=I the upper horizontal arrow d
exists making the upper square commute.
Now the derivation d gives rise to a topological singular extension
____ ___________
|uk|iR=I!? ! R=I[uk]=uik
which on the level of coefficient rings reduces to the algebraic singular ex-
tension ____ _____________ ___________
|uk|iR=I*! R=I[uk]=ui+1k*! R=I[uk]=uik*
_____________
Therefore we can denote ? by R=I[uk]=ui+1kand because of the diagram (4)
we have a map of topological singular extensions
|uk|iR=I ! R=I[uk]=ui+1k ! R=I[uk]=uik
#____ _____#_______ ____#______
|uk|iR=I ! R=I[uk]=ui+1k ! R=I[uk]=uik
With this the inductive step is completed and Theorem 3.1 is proved.
We now show that Theorem 3.1 gives a way of adjoining roots of unity
to various MU-algebras. Let Fp ,! L be a finite separable extension of
the field Fp (typically obtained by adjoining roots of an irreducible factor
of the cyclotomic polynomial xpn-1 - 1 for some n). Let_R = MUp, the p-
completion of the complex cobordism spectrum MU and R* = W (L) MU*
where W (L) is the ring of Witt vectors of L. Since L is a separable extension
of Fp the algebra W (L) is separable over the p-adic integers ^Zpand it follows
that W (L) MU* is a separable algebra over MUp* = ^Zp MU*.
Corollary 3.2 Let J* be the ideal in MU* = Z[x1, x2, . .].generated by
(p, xi1, xi2, . .).where xik are polynomial generators in MU*. Then for any
finite separable extension Fp ,! L there exist MUp-algebras MU=J, MU=JL,
MU=JW(Fp) and MU=JW(L) together with the commutative diagram in the
homotopy category of MUp-algebras
MU=JW(Fp) ! MU=JW(L)
# # (5)
MU=J ! MU=JL
14
which reduces in homotopy to the following diagram of MU*-algebras:
^Zp MU*=(xi1, xi2. .).! W (L) MU*=(xi1, xi2. .).
modp # # modp
MU*=J* ! L MU*=J*
Proof. Denote by u1, u2, . .t.he collection of polynomial generators of
MU* which are not in J*. There is an isomorphism of rings MU*=J* ~=
Fp[u1, u2, . .].. Let I be the maximal ideal (p, x1, x2, . .).in MU* = Z[x1, x2*
*, . .]..
Then MU=I is the Eilenberg-MacLane spectrum HFp and the canonical
map MUp ! MU=I is a commutative S-algebra map. Therefore MU=I is
an MUp-algebra (even a commutative MUp-algebra). Further the extension
Fp ,! L determines a map of commutative S-algebras HFp ! HL which
is also a map of commutative MUp-algebras. Using Theorem 3.1 we can
adjoin the variable ui to the topological extension HFp ! HL and obtain
the tower of topological extensions
{HFp[ui]=[uki] ! HL[ui]=[uki]}.
Passing to the limit we get the topological extension HFp[ui] ! HL[ui].
This procedure could be repeated so we can adjoin another polynomial
generator to the extension HFp[ui] ! HL[ui] (or any number of polynomial
generators). In this way we construct the extension
HFp[u1, u2, . .].! HL[u1, u2, . .].
which is the same as the extension MU=J ! MU=JL for the ideal J* =
(p, xi1, xi2, . .).in MU*. At this point we adjoin the indeterminate corre-
sponding to the prime p. We then get the following diagram of MU-algebras:
MU=J MU=JW2(Fp) . . . MU=JWn(Fp)
# # #
MU=JL MU=JW2(L) . . . MU=JWn(L)
where the MUp-algebras MU=JWn(Fp) and MU=JWn(L) realize the MUp*-
algebras Wn(Fp) MU*=(xi1, xi2, . .).and Wn(L) MU*=(xi1, xi2, . .).re-
spectively.
Finally taking the (homotopy) inverse limit we get an MUp-algebra map
MU=JW(Fp) ! MU=JW(L) which realizes in homotopy the extension of rings
^Zp MU=J* ! W (L) MU=J*. Corollary 3.2 is then proved.
Remark 3.3 It is not hard to prove that the diagram (5) is equivariant
with respect to Gal(L=Fp) in a suitable sense but we save this observation
for future work.
15
4 Localized towers of M U -algebras
In this section we investigate the behaviour of the towers of MU-algebras
constructed in the previous section under Bousfield localization. We start
with an almost trivial
Theorem 4.1 Let
I ! A ! B (6)
be a singular extension of R-algebras for a commutative S-algebra R. Then
for any R-module M the cofibre sequence
IM ! AM ! BM (7)
is also a singular extension of R-algebras (here ?M denotes Bousfield local-
ization with respect to M)
Proof. The singular extension (6) is associated with a certain topological
derivation d : B ! B _ M so that there is a homotopy pullback square of
R-algebras
A ! B
# #
B !d B _ I
where the vertical map B ! B _ M is the canonical inclusion of the wedge
summand. Localizing this diagram with respect to M we get the homotopy
pullback diagram
AM ! BM
# #
BM dM! BM _ IM
Therefore (7) is a singular extension associated with the topological deriva-
tion BM dM!BM _ IM and the theorem is proved.
We now specialize to the case R = MU. Let I* be a regular ideal
(ui1, ui1, . .).in MU* such that the elements uik have positive degrees and
MU=I is an MU-algebra. Consider the MU-algebra MU=I[ui1, ui1, . .,.uik]
obtained by adjoining the indeterminates ui1, ui1, . .,.uik to MU=I as ex-
plained in Section 2. Set
MU=I[[ui2, ui3, . .,.uik]][ui1, u-1i1] = MU=I[ui2, ui3, . .,.uik]MU=I[ui1,u-1*
*i],
*
* 1
the Bousfield localization of the MU-algebra MU=I[ui2, ui3, . .,.uik] with
respect to MU=I[ui1, u-1i1] := u-1i1MU=I[ui1]. The reason for this notation
is the following
16
Proposition 4.2 The coefficient ring of MU=I[[ui2, ui3, . .,.uik]][ui1, u-1i1]
as an MU*-algebra is MU=I*[[ui2, ui3, . .,.uik]][ui1, u-1i1].
Proof. We have the following tower of MU-algebras:
MU=I[ui1] MU=I[ui1][ui2]=u2i2 . . .MU=I[ui1][ui2]=uli2 . . .
Bousfield-localizing it with respect to MU=I[ui1, u-1i1] we get the tower
MU=I[ui1, u-1i1] MU=I[ui1, u-1i1][ui2]=u2i2 . . .
Notice that the first term of the localized tower as well as its successive sub-
quotients (isomorphic to suspensions of MU=I[ui1, u-1i1]) are MU=I[ui1, u-1i1]-
local and therefore so is its homotopy inverse limit. Denote this limit by
MU=I[ui1, u-1i1][[ui2]], clearly
MU=I[ui1, u-1i1][[ui2]]* = MU=I*[ui1, u-1i1][[ui2]]
Further the canonical map
holim(MU=I[ui1][ui2]=uli2) ' MU=I[ui1, ui2] ! MU=I[ui1, u-1i1][[ui2]]
is MU=I[ui1, u-1i1]-equivalence and we conclude that
MU=I[ui1, u-1i1][[ui2]] ~=MU=I[ui1, ui2]MU=I[ui1,u-1i] (8)
1
We then proceed by adjoining ui3. There is a tower of MU-algebras
MU=I[ui1, ui2] MU=I[ui1, ui2][ui3]=u2i3 . . .
where the successive stages
|ui3|sMU=I[ui1, ui2][ui3] ! MU=I[ui1, ui2][ui3]=us+1i3! MU=I[ui1, ui2][ui3]=us*
*i3
are singular extensions of MU-algebras. Localizing this tower with respect
to MU=I[ui1, u-1i1] and using (8) and Theorem 4.1 we get the tower
MU=I[[ui2]][ui1, u-1i1] MU=I[[ui2]][ui1, u-1i1][ui3]=u2i3 . ...
whose homotopy limit
holim(MU=I[[ui2]][ui1, u-1i1][ui3]=uki3) = MU=I[[ui2, ui3]][ui1, u-1i1]
17
is clearly weakly equivalent to the localization of MU=I[ui1, ui2, ui3] with
respect to MU=I[ui1, u-1i1. Moreover
MU=I[[ui2, ui3]][ui1, u-1i1]* = MU=I*[[ui2, ui3]][ui1, u-1i1]
Repeating this process for ui4, ui4, . .,.uikwe obtain the desired isomorphism
MU=I[[ui2, ui3, . .,.uik]][ui1, u-1i1]* = MU=I*[[ui2, ui3. .,.uik]][ui1, u-1i*
*1].
Our main application of the developed localization techniques is to Johnson-
Wilson theories E(n).
Recall that the MU-algebra spectrum E(n) has the coefficient ring E(n)* =
Z(p)[v1, v2, . .,.vn-1][vn, v-1n]. The MU*-algebra structure on E(n)* is de-
fined by the correspondence xi ! 0 for i 6= 2(pn - 1) and x2(pn-1)! vn
where xi are polynomial generators of MU*. We have the following
Corollary 4.3 There exist MU-algebras E^(n), E^W(L)(n), K(n), K(n)L,
and the homotopy commutative diagram of MU-algebras
^E(n) ! ^EW(L)(n)
# # (9)
K(n) ! K(n)L
which realizes in homotopy the diagram of MU*-algebras
Z^p[[v1, v2, . .,.vn-1]][vn,!v-1n]W (L)[v1, v2, . .,.vn-1][[vn, v-1n]]
# #
Z=pZ[vn, v-1n] ! L[vn, v-1n]
Proof. We first construct the map of MU-algebras k(n) ! k(n)L by adjoin-
ing vn to the extension HFp ! HL. Here k(n) and k(n)L are the connective
Morava K-theories with coefficient rings k(n)* = Fp[vn] and k(n)* = L[vn].
Adjoining indeterminates p, v1, v2, . .,.vn-1 we construct the homotopy com-
mutative diagram of MU-algebras
dBP < n > ! BdP < n >W(L)
# # (10)
k(n) ! k(n)L
which realizes in homotopy the diagram
Z^p[v1, v2, . .,.vn-1][vn]!W (L)[v1, v2, . .,.vn-1][vn]
# #
Z=pZ[vn] ! L[vn]
18
According to Proposition 4.2 Bousfield localization of dBP < n > with re-
spect to K(n) on the level of coefficient rings amounts to inverting vn
and completing at the ideal (v1, v2, . .,.vn-1). Further notice that since
ß*dBP < n > W(L) is a free ß*dBP < n > -module of finite rank (which is equal
to the degree of the field extension L=Fp) the MU-algebra dBP < n > W(L)
is a finite cell BdP < n >-module and therefore its K(n)-localization is
equivalent to
BdP < n > K(n)^ dBP < n >
BcP W(L)
and its coefficient ring is
W (L) ß*dBP < n > K(n)= W (L)[[v1, v2, . .,.vn-1]][vn, v-1n].
Therefore we obtain the diagram (9) by localizing (10) with respect to K(n)
in the category of MU-algebras. Corollary 4.3 is proved.
5 Computing spaces of S-algebra maps
In this section we investigate spaces of strictly multiplicative maps from
^E(n)W(L) to certain MU-algebras which we call called "strongly K(n)L-
complete.". As a consequence we obtain versions of the Hopkins-Miller
theorem as well as splitting theorems for such spectra. Such theorems (in
a weaker, up to homotopy form) were previously obtained by methods of
formal group theory, cf. [2].
Let A, B be S-algebras, which we will assume without loss of generality
to be q-cofibrant in the sense of [4]. That means in particular, that the
topological space of S-algebra maps B ! A has the öc rrect" homotopy
type (i.e. the one that depends only on the homotopy type of A and B as
S-algebras). Denote this topological space by FS-alg(B, A). Also denote
the set of multiplicative up to homotopy maps from A to B by Mult(A, B).
We recall one result from [6] which will be needed later on.
Theorem 5.1 Let -1M ! X ! A be a singular extension of S-algebras
associated with a derivation d : A ! A _ M and f : B ! A a map of
R-algebras. Then f lifts to an R-algebra map B ! X iff a certain element
in Der0(B, M) is zero. Assuming that a lifting exists the map
FR-alg(B, X) ! FR-alg(B, A)
has homotopy fibre over the point f 2 FR-alg(B, A) weakly equivalent to
1 Der (B, -1M) (the 0th space of the spectrum Der (X, -1M)).
19
Recall from the previous section that the spectrum ^E(n)W(L) with coef-
ficient rings W (L)[[v1, v2, . .,.vn-1]][vn, v-1n] has a structure of an S-alge*
*bra.
We say that the generalized Hopkins-Miller theorem holds for an S-
algebra E if ß0FS-alg(E^(n)W(L) , E) = Mult(E^(n)W(L) , E) whilst
ßiFS-alg(E^(n)W(L) , E) = 0 for i > 0.
Proposition 5.2 The generalized Hopkins-Miller theorem holds for the S-
algebra K(n)L.
Proof. Consider the Bousfield-Kan spectral sequence (cf.[3]) for the map-
ping space FS-alg(A, B). The identification of the E2-term is standard and
we refer the reader to [10] for necessary details. Here we only mention
that the key ingredient in this identification is the existence of the Kunneth
formula
K(n)L*(E^(n)W(L) ^ ^E(n)W(L) )
= K(n)L*(E^(n)W(L) ) K(n)L*K(n)L*(E^(n)W(L) ).
This is the result we need:
Est2= DerstK(n)L*(E^(n)W(L)*K(n)L, K(n)L*) for s, t 6= 0;
E002= Mult(E^(n)W(L) , K(n)L).
Here Der**k(?, ?) is defined for a graded k-algebra R* and a graded R*-
bimodule M* as the shifted Hochschild cohomology:
Der**k(A*, M*) := HH*+1*k(A*, M*)
(the second grading reflects the fact that A* and M* are graded objects).
Further
E^(n)W(L)*K(n)L = ^E(n)*K(n) W (L) L = E(n)*K(n) L L
and computations in [9], Chapter VI show that
n pk-1
E(n)*K(n) = * = Fp[vn, v-1n][tk|k > 0]=(tpk - vn tk)
where the degree of tk is 2(pk - 1). We have:
HH**K(n)L*( * L L, K(n)L*) = HH**K(n)*( * L, K(n)* L)
= HH**K(n)*( *, K(n)*) HH*Fp(L, L) = HH**K(n)*( *, K(n)*) L.
20
An easy cohomological calculation (due to A.Robinson, cf. [11]) shows that
HH**K(n)*( *, K(n)*) = K(n)* and therefore
HH**K(n)L*(E^(n)W(L)*K(n)L) = K(n) L = K(n)L* (11)
Thus the Bousfield-Kan spectral sequence reduces to its corner term and
Proposition 5.2 is proved.
Remark 5.3 Notice that even though in Proposition 5.2 and we used the
results in the previous section that spectra K(n)L, and E^(n)W(L) admit S-
algebra structures we did not specify which structures are used. In other
words Proposition 5.2 is valid with any choice of S-algebra structures on the
spectra in question.
We can now regard K(n)L as a bimodule over E^(n)W(L) by choosing an
S-algebra map ^E(n)W(L) ! K(n)L (which is supplied by Propositions 5.2).
Therefore it makes sense to consider spectra of topological Hochschild co-
homology and topological derivations of ^E(n)W(L) with values in K(n)L. It
turns out that these spectra do not depend up to weak equivalence which
bimodule structure we choose.
Proposition 5.4 The canonical map of S-modules
THH (E^(n)W(L) , K(n)L) ! K(n)L;
is a weak equivalences. Furthermore the S-module Der (E^(n)W(L) , K(n)L)
is contractible.
Proof. Since there is a cofibre sequences of S-modules
-1Der (E^(n)W(L) , K(n)L) ! THH (E^(n)W(L) , K(n)L) ! K(n)L
it suffices to prove the statement about THH. We have the following spec-
tral sequence
Ext**^E(n)W(L) ^ (E^(n)W(L)*, K(n)L*) ) T HH*(E^(n)W(L) , K(n)L)
*E(n)W(L)
(12)
Since E^(n)W(L) is a Landweber exact theory the ß*E^(n)W(L) - module
^E(n)W(L)*E^(n)W(L) is flat and by flat base change we obtain
Ext**^E(n)W(L) ^ (E^(n)W(L)*, K(n)L*)
*E(n)W(L)
21
= Ext**^E(n)W(L) (K(n)L*, K(n)L*)
*K(n)L
= HH*K(n)L*(E^(n)W(L)*K(n)L, K(n)L*).
The isomorphism (11) shows that the spectral sequence (12) collapses giving
the weak equivalence
THH (E^(n)W(L) , K(n)L) ' K(n)L
and our theorem is proved.
We now describe a particular class of MU-algebras E which will be
called strongly K(n)L - complete and for which the set of multiplicative
maps ^E(n)W(L) ! E has an especially simple form.
Let = { 0, 1, . .,.^ 2(pn-1), . .}.(i.e.the 2(pn - 1)th spot is missed) be
a sequence of positive integers. Associated to is the regular sequence
p 0, x11, x22, . .i.n MU*. Consider the MU-module
MU( ) := x-12(pn-1)MUW(L) =(p 0, x11, x22, . .).
= v-1nMUW(L) =(p 0, x11, x22, . .)..
Since MU( ) can be constructed from K(n)L by a sequence (possi-
bly infinite) of elementary extensions we see that there is a structure of
an MU-algebra on MU( ) compatible with the MU*-algebra structure on
v-1nMU( )* = W (L) v-1nMU*=(p 0, x11, x22, . .).. Notice that there is a
reduction MU-algebra map MU( ) ! K(n)L. Furthermore the filtration
on MU( ) by powers of the maximal ideal has subquotients isomorphic to
products of copies of K(n)L* = L[vn, v-1n].
Let E and F be two MU-algebras supplied with an MU-algebra map
E ! K(n)L and F ! K(n)L. We say that an MU-algebra map E ! F is
compatible if it is a map over K(n)L.
Definition 5.5 The class SC(K(n)L) of strongly K(n)L-complete MU-
algebras is the smallest class of MU-algebras over K(n)L that satisfies the
following two axioms:
(i) SC(K(n)L) contains MU( ) for any = { 1, 2, . .,.^ 2(pn-1), . .}.as
above;
(ii) For any sequential inverse system of compatible maps in SC(K(n)L) its
homotopy inverse limit is also in SC(K(n)L).
In other words an MU-algebra is K(n)L-complete if it can be built from
K(n)L by taking elementary extensions and (homotopy) inverse limits. If
22
E1 E2 . .i.s an inverse system in SC(K(n)L) then ß*holim(En) =
limn!1 ß*En since coefficients rings of En are all in even degrees. Therefore
we never have to worry about lim1-problems.
Examples of strongly K(n)L - complete MU-algebras include K(n)L,
^E(n)W(L) and also v-1ndBPW(L) and v-1nbP(n)W(L) , the Artinian completions
of v-1nBPW(L) and v-1nP (n)W(L)
Let E be a strongly K(n)L-complete MU-algebra. The canonical S-
algebra map (which is also an MU-algebra map) E ! K(n)L determines
via composition the morphism of topological spaces (unbased)
i : FS-alg(E(n)W(L) , E) ! FS-alg(E(n)W(L) , K(n)L) (13)
Proposition 5.6 The map (13) is a weak equivalence. In particular
FS-alg(E(n)W(L) , E) is a homotopically discrete space. Moreover the MU-
module Der (E(n)W(L) , E) is contractible.
Proof. First assume that E = MU( ) where all l except for i are equal
to one, i.e. that E = K(n)L[xk]=xki. The we have the tower
K(n)L K(n)L[xk]=x2k . . .K(n)L[xk]=xki= E.
Applying to this tower the functor FS-alg(EW(L) , ?) we get the tower of
topological spaces
FS-alg(EW(L) , K(n)L) FS-alg(EW(L) , K(n)L[xk]=x2k) . . .
FS-alg(EW(L) , K(n)L[xk]=xki) = FS-alg(EW(L) , E) (14)
Using Theorem 5.1 and Proposition 5.4 we see that the homotopy fibre of
each map
FS-alg(EW(L) , K(n)L[xk]=xsk) FS-alg(EW(L) , K(n)L[xk]=xs+1k)
is weakly equivalent to the space 1 Der (EW(L) , |xk|sK(n)L) which is con-
tractible. Hence all maps in (14) are weak equivalences of spaces and in par-
ticular FS-alg(EW(L) , K(n)L[xk]=xki) is weakly equivalent to FS-alg(EW(L) , K(*
*n)L.
Similarly Der (EW(L) , K(n)L[xk]=xki) is weakly equivalent to Der (EW(L) , K(n)L
and is contractible.
Further obvious induction shows that the statement of Proposition 5.6
holds for an MU-algebra of the form MU( ) where all l except for for a
finite number are equal to one. Passing to the limit we obtain the result for
an arbitrary MU-algebra MU( ).
23
Now using transfinite induction suppose that the conclusion of Proposi-
tion 5.6 holds for an inverse system {El} of strongly K(n)L-complete MU-
algebras consisting of compatible maps and let E = holim El. Apply-
ing the functor MapS-alg(EW(L) , ?) to {El} we get an inverse system of
topological spaces {FS-alg(EW(L) , El)}. By inductive assumption all spaces
FS-alg(EW(L) , El) are weakly equivalent to FS-alg(EW(L) , K(n)L) and the
maps in the inverse system {FS-alg(EW(L) , El)} respect this equivalence.
We conclude that the space FS-alg(EW(L) , E) is also weakly equivalent to
FS-alg(EW(L) , K(n)L). Similar arguments show that Der (EW(L) , K(n)L) is
contractible and Proposition 5.6 is proved.
Remark 5.7 We see that for any strongly K(n)L-complete MU-algebra E
the canonical S-algebra map E^(n)W(L) ! K(n)L lifts uniquely to a map
^E(n)W(L) ! E. Therefore E is naturally a (bi)module over ^E(n)W(L) .
The following lemma descibes the structure of cohomology operations from
^E(n)W(L) to a strongly K(n)L-complete MU-algebra E.
Lemma 5.8 Let E be a strongly K(n)L-complete MU-algebra. Then the
evaluation map
E*E^(n)W(L) ! HomE^(n)W(L)*(E^(n)W(L)*E^(n)W(L) , E*)
is an isomorphism.
Proof. We have the following natural isomorphism of S-modules
FS(E^(n)W(L) , E) ~=FE^(n)W(L)(E^(n)W(L) ^ ^E(n)W(L) , E)
Consider further the spectral sequence
Ext**^E(n)W(L)*(E^(n)W(L)*E^(n)W(L) , E*)
) [E^(n)W(L) ^ ^E(n)W(L) , E]*(E^(n)W(L))= [E^(n)W(L) , E]*(15)
We claim that all higher Ext groups vanish so that our spectral sequence
reduces to Ext0*^E(n) (E^(n)W(L)*E^(n)W(L) , E*). (This would clearly give
W(L)*
us the statement of the lemma). To see this first assume that E is of the
form MU( ) for some sequence .
Then since for the maximal ideal M of E* the E*-module Mn=Mn+1 is
isomorphic to direct product of copies of K(n)L* it is enough to prove that
Exti^E(n)W(L)*(E^(n)W(L)*E^(n)W(L) , K(n)L) = 0
24
for i > 0 (here we suppressed the second, internal grading on Ext from the
notations). Since the ^E(n)W(L)*-module ^E(n)W(L)*E^(n)W(L) is flat we have
by flat base change:
Exti^E(n)W(L)*(E^(n)W(L)*E^(n)W(L) , K(n)L*)
= ExtiK(n)L*(E^(n)W(L)*E^(n)W(L) E^(n)W(L)*K(n)L*, K(n)L*)
and the last Ext-group is zero for i > 0 since K(n)L* is injective as a graded
module over itself.
Therefore our claim about the vanishing of higher Ext-groups in (15)
is proved for E = MU( ). To get the general case suppose that E is
the homotopy inverse limit of strongly K(n)L-complete MU-algebras Elfor
which higher Ext's do vanish. Then
Exti*^E(n)W(L)*(E^(n)W(L)*E^(n)W(L) , E*)
= Exti*^E(n)W(L)*(E^(n)W(L)*E^(n)W(L) , liml!1 El*)
liml!1 Exti*^E(n)W(L)*(E^(n)W(L)*E^(n)W(L) , El*) = 0
for i > 0. With this Lemma 5.8 is proved.
We can now formulate our main
Theorem 5.9 The generalized Hopkins-Miller theorem holds for any strongly
K(n)L-complete MU-algebra E.
Proof. By Proposition 5.6 we know that the space MapS-alg(E^(n)W(L) , E)
is homotopically discrete with
ß0(MapS-alg(E^(n)W(L) , E)) = HomK(n)L*-alg( * L, K(n)L*).
So it remains to show that an arbitrary multiplicative operation ^E(n)W(L) !
E lifts to an S-algebra map. Since any such operation determines an
^E(n)W(L)*-algebra map ^E(n)W(L)*E(n)W(L) ! E* we conclude by Lemma
5.8 that there is an injective map
Mult(E^(n)W(L) , E) ! HomE^(n)W(L)*-alg(E^(n)W(L)*E^(n)W(L) , E*)(16)
We claim that any ^E(n)W(L)*-algebra map ^E(n)W(L)*E^(n)W(L) ! K(n)L*
lifts uniquely to a map E^(n)W(L)*E^(n)W(L) ! E* so that there is an iso-
morphism
HomE^(n)W(L)*-alg(E^(n)W(L)*E^(n)W(L) , E*
~=HomE^(n) (E^(n) E^(n) , K(n) ) (17)
W(L)*-alg W(L)* W(L) L*
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To see this assume first that E is of the form MU( ). Induction up the
filtration of the E^(n)W(L)*-module E* by the powers of the maximal ideal
shows that our claim is equivalent to the vanishing of certain Hochschild co-
homology classes in HH*^E(n) (E^(n)W(L)*E^(n)W(L) , K(n)L*). Flat base
W(L)*
change gives an isomorphism
HH*^E(n)W(L)*(E^(n)W(L)*E^(n)W(L) , K(n)L*)
= HH*K(n)L*(E^(n)W(L)*E^(n)W(L) E^(n)W(L)*K(n)L*, K(n)L*)
= HH*K(n)L*(K(n)L*E^(n)W(L) , K(n)L*) = K(n)L*.
We see that all higher Hochschild cohomology groups are zero and our claim
is proved (in the special case E = MU( )). Then the standard by now
arguments (which we omit) show that (17) holds for any E.
Further we have an obvious change of rings isomorphism
HomE^(n)W(L)*-alg(E^(n)W(L)*E^(n)W(L) , K(n)L*)
= HomK(n)L*-alg(K(n)L*E^(n)W(L) , K(n)L*)
The last term corresponds bijectively by Proposition 5.6 to homotopy classes
of S-algebra maps from ^E(n)W(L) to E. This shows that any multiplicative
operation E^(n)W(L) ! E lifts to an S-algebra map and Theorem 5.9 is
proved.
Remark 5.10 The class of strongly K(n)L-complete MU-algebras is prob-
ably not the most general one for which the conclusion of Theorem 5.9 holds.
It is plausible that one has the generalized Hopkins-Miller theorem for any
K(n)L-local MU-algebra but such a result seems out of reach at present.
Tracing the proofs of Proposition 5.2 and Theorem 5.9 we see that they
don't depend on the choice of the S-algebra structure on ^E(n)W(L) as long
as the ring spectrum structure is fixed. (But they do depend on the choice
of the S-algebra structure on E.) This gives the following
Corollary 5.11 The spectrum E^(n)W(L) has a unique (up to a noncanon-
ical isomorphism) structure of an S-algebra compatible with its structure of
a ring spectrum.
Proof. Let E^0(n)W(L) denote the S-algebra whose underlying ring spec-
trum is equivalent to ^E(n)W(L) but the S-algebra structure is possibly dif-
ferent. Then the given multiplicative up to homotopy weak equivalence
26
^E0(n)W(L) ! ^E(n)W(L) can be lifted to an S-algebra map which shows that
the would-be exotic S-algebra structure on ^E0(n)W(L) is actually isomorphic
to the standard one.
Remark 5.12 A similar statement about the uniqueness of the A1 struc-
ture on E^(n) was proved by A.Baker in [1]. However the proof in the cited
reference was based on the obstruction theory of A.Robinson and it is unclear
at this time to what extent this theory carries over in the present context.
Theorem 5.9 allows one to obtain splittings of various strongly K(n)L-
complete MU-algebras. For example consider v-1ndBPW(L), the Artinian
completion of the vn-localization of the Brown-Peterson spectrum BPW(L)
(cf.[2] concerning Artinian completions). Then v-1ndBPW(L) is a strongly
K(n)L-complete MU-algebra obtained from K(n)W(L) by adjoining the col-
lection of indeterminates {p, v1, v2, . .}.:
v-1ndBPW(L) = K(n)W(L) [[v1, v2, . .].].
There is a canonical map v-1ndBPW(L) ! ^E(n)W(L) which on the level of coef-
ficient ring reduces to killing the ideal (vn+1, vn+2, . .).in v-1ndBPW(L)*. Fr*
*om
Theorem 5.9 we conclude that any S-algebra map E^(n)W(L) ! E^(n)W(L)
lifts uniquely to an S-algebra map ^E(n)W(L) ! v-1ndBPW(L). In particular
the identity map id : ^E(n)W(L) ! ^E(n)W(L) can be so lifted and we obtain
Theorem 5.13 The S-algebra map v-1ndBPW(L) ! ^E(n)W(L) admits a unique
S-algebra splitting ^E(n)W(L) ! v-1ndBPW(L)
An analogous theorem was proved (in a weaker, up to homotopy form) in
[2].
Acknowledgement. The author is indebted to A. Baker for many
useful discussions on the subject of the paper.
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27
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