ON THE STRUCTURE OF P (n)*(P (n)) FOR p = 2
CHRISTIAN NASSAU
Abstract.We show that P(n)*(P(n)) for p = 2 with its geometrically in-
duced structure maps is not an Hopf algebroid because neither the augmen*
*ta-
tion ffl nor the coproduct are multiplicative. As a consequence the alg*
*ebra
structure of P(n)*(P(n)) is slightly different from what was supposed to*
* be
the case. We give formulas for ffl(xy) and (xy) and show that the inver-
sion of the formal group of P(n) is induced by an antimultiplicative inv*
*olution
: P(n) ! P(n). Some consequences for multiplicative and antimultiplicat*
*ive
automorphisms of K(n) for p = 2 are also discussed.
1.Introduction and statement of the results
Let BP denote the Brown-Peterson spectrum for p = 2 and recall that BP* =
Z(2)[v1; v2; : :]:with |vn|= 2(2n - 1). As usual we let v0 = 2. BP is a commuta*
*tive
ring spectrum and there are related ring spectra P (n) = BP=(v0; v1; : :;:vn-1),
n 1, which are connected by the Baas-Sullivan cofiber sequences
|vn|P (n)--.vn--!P (n)-jn---!P (n + 1)@n----!|vn|+1P (n):
From the Baas-Sullivan cofiber triangles one obtains Bockstein operations Qn-1 =
@n-1jn-1 : P (n) ! |vn-1|+1P (n).
There are essentially two multiplications m, __mon P (n) that are worth con-
sideration. These are characterised recursively by the fact that they make P (*
*n)
into an P (n - 1) algebra spectrum (cf. [W1 ] with corrections in [N ]). Here *
*we
let P (0) = BP . Both of them are noncommutative: one has __m= mT , where
T : P (n) ^ P (n) ! P (n) ^ P (n) is the switch map. More explicitly their rela*
*tion is
given by (cf. [M ])
__m= m + v
(1) nm(Qn-1 ^ Qn-1):
Furthermore Qn-1 is a derivation with respect to both products. For all this the
reader is referred to [R1 ], [R2 ], [W1 ] and [W2 ].
We prove the following Lemma in section 2 which also contains a formula for
(xy).
Lemma 1. Let ffl : P (n)*(P (n)) ! P (n)* be the augmentation defined by ffl =
ss*(m). Then for all x; y 2 P (n)*(P (n)) we have
(2) ffl(xy) = ffl(x)ffl(y) + vnffl(Qn-1x)ffl(Qn-1y):
Recall from [KW ] the elements ai, 0 i < n, in P (n)*(P (n)) with |ai|= 2i+*
*1-1.
Recall also that BP*(BP ) = BP*[t1; t2; t3; : :]:with |ti|= 2(2i-1). We will de*
*note
the_image_of tiunder the canonical map BP*(BP ) ! P (n)*(P (n)) again by ti. Let
P (n)denote the spectrum P (n) with multiplication __m.
____________
Key words and phrases. Hopf algebroids, Morava K-theory, bordism theory, non*
*commutative
ring spectra.
1
2 CHRISTIAN NASSAU
Lemma 1 leads immediately to the following modification of the main result of
[KW ].
_____
Theorem 2 ([KW ]).P (n)*(P (n)) and P (n)*(P (n)) are polynomial algebras over
P (n)* on the generators
{ a0; a1; : :;:an-1 } [ { ti| i > n }:
There are relations (
a2i= ti+1+ vi+1 in P (n)*(P_(n));_
ti+1 in P (n)*(P (n)):
Here vk is to be interpreted as 0 if k < n.
For commutative ring spectra E and X that satisfy Kronecker duality it is well
known that Kronecker duality sets up a one-to-one correspondence between mor-
phisms of ring spectra X ! E and algebra morphisms E*(X) ! E*. Lemma 1
shows that this is no longer true for the spectra P (n) since the augmentation *
*ffl is
the Kronecker dual of the identity id: P (n) ! P (n). We investigate this probl*
*em
in section 3 and prove the following two Theorems.
Theorem 3. There is an antimultiplicative involution : P (n) ! P (n) which on
Euler classes of complex line bundles L is given by e(L) 7! e(L-1).
Let K(n) denote the nth Morava K-theory at p = 2. It inherits two mul-
tiplications from P (n) of which each is the opposite of the other. Let Mult+
(resp. Mult-) denote the set of all multiplicative (resp. antimultiplicative) a*
*utomor-
phisms of K(n). Then Mult := Mult+ [ Mult- is a group. Recall from [Y ] that
the map Z 3 k 7! [k]Fn(x) 2 End(Fn) extends to an isomorphism Z2 ~=End(Fn),
where Fn is the formal group law associated to K(n), End(Fn) is the endomorphism
ring of Fn (considered as a formal group over the graded ring K(n)*) and Z2 is *
*the
ring of 2-adic integers.
Theorem 4. The canonical map Mult ! Aut(Fn) is an isomorphism of groups.
Mult+ (resp. Mult-) corresponds to those x 2 Z*2that are congruent to 1 (resp. *
*-1)
modulo 4.
This paper is a condensed version of my Diplomarbeit [N ] which I wrote under
the supervision of Prof. R. Kultze. I want to take this opportunity to thank hi*
*m for
paying close attention to certain parts of that work. I also thank the Studiens*
*tiftung
des Deutschen Volkes for support during my studies.
2.Proofs
Regard n as fixed and write v instead of vn, Q instead of Qn-1. Choose once a*
*nd
for all an admissible multiplication m : P (n) ^ P (n) ! P (n). We will repeate*
*dly
make use of the following fact, mostly without explicitly mentioning it.
Useful fact.Left and right multiplication by v, denoted Lv and Rv, agree, that *
*is
we have m(v ^ id) = m(id^v) as maps P (n) ! P (n).
Proof.Suppressing the coherent chain of identifications S ^ P (n) ~=P (n) ~=P (*
*n) ^
S ~=S ^ P (n) from the notation we can write m(v ^ id) = mT (id^v) = m(id^v) +
vm(Q^Qv). But Qv = 0 since v is in the image of the canonical map BP* ! P (n)*_
and Q = jn-1@n-1 annihilates this (@n-1 already does). |__|
ON THE STRUCTURE OF P(n)*(P(n)) FOR p = 2 3
The benefit of this observation is that we don't have to worry about the nonc*
*o-
mutativity of the multiplication as long as one factor is vn.
We can now prove Lemma 1. The proof is straightforward: just draw the relevant
diagrams and check that they commute. This diagramatic reasoning is, however,
not very enlightening. (Such proofs were given in [N ], and anybody who had a
glimpse of that will agree instantly. This is especially true for the proof of *
*Lemma
5 which would require at least Din A3 paper.) The simple facts behind it become
much clearer when written out using fictitous elements of our spectra as placeh*
*olders
as in the following
Proof of Lemma 1.Recall that the Pontrjagin product of x; y 2 P (n)*(P (n)) is *
*the
composite of x ^ y : S ~=S ^ S ! P (n) ^ P (n) ^ P (n) ^ P (n) with
P (n) ^ P (n) ^ P (n) ^ Pi(n)d^T^id-----!P (n) ^ P (n) ^ P (n) ^ P (n)
and
P (n) ^ P (n) ^ P (n) ^ Pm(n)^m----!P (n) ^ P (n):
So ffl(xy) = ss*(m)(xy) is naturally given as a composite of x ^ y : S ! P (n)^4
and a certain map OE from P (n)^4 to P (n) which is made up of multiplications *
*and
transpositions only. Fictitous elements a; b; c; d 2 P (n) can be used to keep *
*track of
these multiplications and transpositions; then, for example, ffl is given by (a*
*; b) 7! ab
and the Pontrjagin product of x = (a; b) and y = (c; d) is (ac; bd). In this no*
*tation
OE becomes
P (n)^4 3 (a; b; c; d) 7! (a; c; b; d) 7! (ac; bd) 7! acbd 2 P (n):
We compare this with ffl(x)ffl(y). This can be decomposed similarly as (x ^ y)*
* with
: P (n)^4 ! P (n) given by
(a; b; c; d) 7! (ab; cd) 7! abcd:
Writing (1) as cb = bc + vQ(b)Q(c) we obtain
OE(a; b; c; d) = (a; b; c; d) + v (a; Q(b); Q(c); d):
If one observes that (a; Q(b)) = Q*(x) and (Q(c); d) = Q(y) this can be rephras*
*ed
as saying that
(3) ffl(xy) = ffl(x)ffl(y) + vnffl(Q*(x))ffl(Q(y)):
This is (2) except that we have Q*(x) instead of Q(x). But we shall see later
that Q* : P (n)*(P (n) ! P (n)*(P (n)) and Q : P (n)*(P (n) ! P (n)*(P (n)) are_
identical, so this finishes the proof. |_*
*_|
The categorically minded reader will probably and rightly frown on these "el-
ements" since obviously their sole purpose is to hide some general nonsense fac*
*ts
about monoidal categories. I could not find a convenient reference, however, an*
*d it
wouldn't really have improved the exposition.
Next we show how Lemma 1 afflicts the algebra structure of P (n)*(P (n)).
Proof of Theorem 2.Following the proof of Theorem 1.4 in [KW ] up to the middle
of page 199 one finds that the additive structure of P (n)*(P (n)) and enough of
4 CHRISTIAN NASSAU
the multiplicative structure have been determined to leave open only the questi*
*on
whether a2n-1= tn or a2n-1= tn + vn. It is shown that
( 2
a2n-1= tn if ffl(an-1);= 0
tn + vn if ffl(a2n-1):= vn
In Lemma 2.2 the authors also observed that Qn-1(an-1) = 1, and a similar argu-
ment shows that Qn-1*(an-1) = 1, too. From (3) it follows that
ffl(a2n-1) = ffl(an-1)2 + vnffl(Qn-1 *(an-1))ffl(Qn-1(an-1)) = vn;
so that a2n-1= tn + vn in P (n)*(P (n)). _____
To prove the claim about the structure of P (n)*(P (n)) we will first prove t*
*hat
(4) x _?y = x ? y + vnQ*(x) ? Q*(y)
for x; y 2 P (n)*(P (n)). Here Q = Qn-1 and (just for this proof)_?_(resp.__?)
denotes Pontrjagin multiplication in P (n)*(P (n)) (resp. P (n)*(P (n))). This*
* is
easily accomplished using fictitous elements again. With x = (a; b), y = (c; d)*
* we
have
x _?y = (a; b) _?(c; d) = (ac; db); x ? y = (a; b) ? (c; d) = (ac; bd):
Using (1) we get
(ac; db) = (ac; bd) + vn(ac; Q(b)Q(d)) = x ? y + vnQ*(x) ? Q*(y):
From this we get an-1 _?an-1 = tn + vn + vn = tn as claimed. (4) shows also
that x ? y = x _?y whenever x or y lies in the subalgebra generated by a0; : :;*
*:an-2
and the ti; since this subalgebra comes from P (n - 1)*(P (n - 1)) it is annihi*
*lated_
by Q and Q*. So the rest of the multiplicative structure is not afflicted. *
* |__|
Note that both Q : P (n)*(P (n)) ! P (n)*(P (n)) and Q* : P (n)*(P (n)) !
P (n)*(P (n)) are derivations. In the proof just given we noted that they agree*
* on
the algebra generators of Theorem 2, so we obtain another
Useful fact.Q; Q* : P (n)*(P (n)) ! P (n)*(P (n)) are equal.
This has been used in the proof of Lemma 1 above and will be used henceforth
without explicit reference.
To give a formula for (xy) we first have to recall the definition of the copr*
*oduct
: P (n)*(P (n)) ! P (n)*(P (n)) P(n)*P (n)*(P (n)):
There are two ingredients: firstly the map
__ ss*(id^i^id)
: P (n)*(P (n)) ~=ss*(P (n) ^ S ^-P-(n))------!ss*(P (n) ^ P (n) ^ P (n));
where i : S ! P (n) is the unit. Secondly
O : P (n)*(P (n)) P(n)*P (n)*(P (n)) ! ss*(P (n) ^ P (n) ^ P (n))
which is given by x y 7! (id^ m ^ id)(x ^ y) and which_is an isomorphism since
P (n)*(P (n)) is P (n)*-flat. is defined to be O-1 .
Recall that P (n)*(P (n)) is a bilateral P (n)*-module, courtesy of the left *
*and
right unit maps jL; jR : P (n)* ! P (n)*(P (n)). The ""s above are to be under-
stood with respect to this bimodule structure. Luckily, vn is invariant in P (n*
*)*, so
we don't need to discriminate between vnx y, xvn y, x vny and x yvn.
ON THE STRUCTURE OF P(n)*(P(n)) FOR p = 2 5
Lemma 5. For all x; y 2 P (n)*(P (n)) we have
(xy) = (x)(y) + vn ([(idQn-1)(x)][(Qn-1 id)(y)]) :
Proof.First note that there is an obvious algebra structure on ss*(P (n) ^ P (n*
*) ^
P (n)). With elements (a; b; c) and (d; e; f) this is given by
(a; b; c) . (d; e; f) = (ad; be; cf):
__
is given by (a; b) 7! (a; 1; b) so it is obviously multiplicative.
O, however, is not multiplicative. Let x = (a; b) (c; d), y = (e; f) (g; h)*
* be
elements of P (n)*(P (n)) P(n)*P (n)*(P (n)). Then xy = (ae; bf) (cg; dh) and
O(xy) = (ae; bfcg; dh):
On the other hand O(x) = (a; bc; d), O(y) = (e; fg; h) so that
O(x)O(y) = (ae; bcfg; dh):
Using fc = cf + vnQ(c)Q(f) we get
O(xy) = (ae; bcfg; dh) + (ae; bQ(c)vnQ(f)g; dh)
(5)
= O(x)O(y) + MvO((idQ)x)O((Q* id)y):
Here we used that (idQ)x = (a; b) (Q(c); d) and (Q* id)y = (e; Q(f)) (g; h).
Mv denotes multiplication by vn in the middle.
Using (5) we can verify the formula for (xy) given in the Lemma: we have to
show that
__
O ((x)(y) + vn ([(idQ)(x)][(Q* id)(y)]) ) = (xy):
We compute
O ((x)(y) + vn ([(idQ)(x)][(Q* id)(y)]) )
= O((x)(y)) + MvO ([(idQ)(x)][(Q* id)(y)])
= O((x))O((y))
+ MvO ([(idQ)(x)]) O ([(Q* id)(y)])
+ MvO ([(idQ)(x)]) O ([(Q* id)(y)])
2 2
+ M2vO [(idQ) (x)] O [(Q* id) (y)]
= O((x))O((y)) (since Q2 = Q2*= 0)
__ __
= (x) (y)
__
= (xy):
|___|
Using the identities (Q id)(x) = (Qx) and (idQ*)(x) = (Q*(x)) one
can check that Lemma 5 says that is an algebra homomorphism
_____
P (n)*(P (n)) ! P (n)*(P (n)) P(n)*P (n)*(P (n)):
This observation might at least mnemonically be useful.
6 CHRISTIAN NASSAU
3.Multiplicative and antimultiplicative maps
To get at multiplicative or antimultiplicative maps P (n) ! P (n) or K(n) !
K(n) we need to be able to characterise them in terms of their Kronecker duals.
The following Lemma is a first step.
Lemma 6. Let E and X be ring spectra and suppose that the Kronecker homo-
morphism
E*(X ^ X) ----! Hom E*(E*(X ^ X); E*)
is an isomorphism. Assume that E*(X) is a flat E*-module. Then a : X ! E is
multiplicative iff
(6) ffl(*(xy)) = ffl(*(x)*(y))
holds for all x; y 2 E*(X).
Proof. is multiplicative iff mX = mE ( ^ ). Both are maps from X ^ X to E,
so by Kronecker duality this is equivalent to
ffl((mX )*(z)) = ffl((mE ( ^ ))*(z))
for all z 2 E*(X ^ X). Since E*(X) is a flat E*-module, the exterior homology
product ^_ : E*(X) E* E*(X) ! E*(X ^ X) is an isomorphism. Hence we can
assume that z = x ^_y for x; y 2 E*(X). The lemma follows because *(x ^_y) =_
*(x) ^_*(y), xy = (mX )*(x ^_y) and *(x)*(y) = (mE )*(*(x) ^_*(y)). |__|
For the P (n) this gives
Lemma 7. Let : P (n) ! P (n) be any map and denote its Kronecker dual by .
Then is multiplicative iff
: P (n)*(_____P)(n)! P (n)*
is an algebra homomorphism. It is antimultiplicative iff
: P (n)*(P (n)) ! P (n)*:
is an algebra homomorphism.
Proof.We prove the assertion about multiplicative maps only, the antimultiplica-
tive case being completely analogous.
According to Lemma 6 is multiplicative iff (6) holds. Using (2) this may be
rewritten
(7) ffl(*(xy)) = ffl(*(x))ffl(*(y)) + vnffl(Q*(x))ffl(Q*(y)):
_____
Let _?denote the multiplication in P (n)*(P (n))._We_recall_from the proof of T*
*heo-
rem 2 that x _?y = xy + vnQ(x)Q(y). So : P (n)*(P (n)) ! P (n)* is multiplicat*
*ive
iff
(8) ffl(*(xy + vnQ(x)Q(y))) = ffl(*(x))ffl(*(y)):
From both (7) and (8) we can conclude that
(9) ffl(*(x))ffl(*(y)) = ffl(*(xy)) if Qx = 0 or Qy = 0.
So to show the equivalence of (7) and (8) we may assume (9). But from (9) it
follows easily that
vnffl(Q*(x))ffl(Q*(y)) = vnffl(*(Q(x)Q(y)));
which is the difference between (7) and (8). (Use that QQ = 0 and Q* = *Q.) |_*
*__|
ON THE STRUCTURE OF P(n)*(P(n)) FOR p = 2 7
To prove Theorems 3 and 4 we have to recall the relation of the P (n) to for-
mal group laws. [R1 ], especially Appendix 2, is a good reference for most of t*
*his
material.
(P (n)*; P (n)*(BP )) inherits from (BP*; BP*(BP )) an Hopf algebroid structu*
*re.
Recall that P (n)*(BP ) is the P (n)*-subalgebra of P (n)*(P (n)) generated by *
*the
ti. For a ring R of characteristic 2 the set of ring homomorphisms P (n)* ! R
may naturally be identified with the set FG n(R) of all 2-typical formal group *
*laws
of height n over R. Similarly, the ring P (n)*(BP ) corepresents the set SIn(R)
of triples (F; f; G) with F; G 2 FG n(R) and f : G ! F a strict isomorphism.
Given OE : P (n)*(BP ) ! R the triple (F; f; G) is obtained as follows: F (resp*
*. G) is
the formal group law classified by OEjL (resp. OEjR ). The isomorphism f(x) is *
*then
defined by X
F 2i
f-1 (x) = x +F OE(ti)x :
i1
Proof of Theorem 3.We here are particularly interested in the natural transfor-
mation FG n(R) 3 F 7! (F; [-1]F (x); F ) 2 SIn (R). This is induced by a ring
homomorphism ffin : P (n)*(BP ) ! P (n)*. These ffin are obviously compatible as
n varies and, obviously again, ffinjL = ffinjR = id, so ffin is P (n)*-linear. *
*If we can
extend this to an algebra map
__
ffin: P (n)*(P (n)) ! P (n)*
its Kronecker dual : P (n) ! P (n) will be antimultiplicative by Lemma 7 and
will by construction have the stated effect on Euler classes.
We don't have much choice in extending ffin: since |ai|is odd for every_i and
P (n)* does not have nonzero odd dimensional elements we have to let ffin(ai) =*
* 0,
so the required multiplicativity implies that there is at most one such extensi*
*on.
All we have to do is to check that this is consistent with the relations a2i= t*
*i+1,
0 i < n - 1 and a2n-1= tn + vn. So we have to show that ffin(ti) = 0 for
1 i n - 1 and that ffin(tn) = vn.
The first requirement follows easily from dimensional considerations. To see *
*that
ffin(tn) = vn recall that with Araki's vk we have
k
[2]F (x) = 2x +F v1x2 +F v2x4 +F . .+.Fvkx2 +F . . .
for the formal groupnlaw F of BP . For the formal group Pn of P (n) this gives
[2]Pn(x) = vnx2 + higher terms. Comparing coefficients in [-1]Pn(x)+Pn[2]Pn(x) =
x then gives the result.
That is idempotent is quite clear: since_2_is_multiplicative its Kronecker
dual _fflis an algebra homomorphism P (n)*(P (n)) ! P (n)*. Since it is the ide*
*ntity
on Euler classes of complex line bundles its restriction to P (n)*(BP ) agrees *
*with
ffl. Since _ffland ffl both are multiplicative extensions of this restriction,_*
*uniqueness_
implies _ffl= ffl, i.e. 2 = id. |_*
*_|
To prove Theorem 4 we first have to carry over some of the results on P (n) to
K(n). Let K(n)* = F2[vn; v-1n] as a module over P (n)*. Then X 7! K(n)* P(n)*
P (n)*(X) =: K(n)*(X) is a homology theory on the stable homotopy category and
X 7! Hom K(n)*(K(n)*(X); K(n)*) a cohomology theory, and both are represented
by the same spectrum K(n). The two multiplications m and __mon P (n) give simil*
*ar
ring spectrum structures on K(n) that will be denoted by the same symbols. The
Bockstein Qn-1, too, carries over to K(n) and (1) still holds. If we denote the
8 CHRISTIAN NASSAU
_*
*____
augmentation of K(n)*(K(n)) again by ffl then (2) continues to hold, too. Let K*
*(n)
have the obvious meaning. _____
To get at the structure of K(n)*(K(n)) and K(n)*(K(n) ) let
n = K(n)* P(n)*P (n)*(BP ) P(n)*K(n)*
be the nth Morava stabiliser algebra (cf. [R1 ], ch. 6). The following Lemma fo*
*llows
easily from Theorem 2 and the definitions.
Lemma 8. We have
K(n)*(K(n)) = n[a0; : :;:an-1]
with relations a2i= ti+1+ vi+1 for 0 i n - 1 and
_____
K(n)*(K(n) ) = n[a0; : :;:an-1]
with relations a2i= ti+1 for 0 i n - 1. |___|
Lemma 7 carries over, too, as does its proof.
Lemma 9. Let : K(n) ! K(n) be any map and denote its Kronecker dual by .
Then is multiplicative iff
: K(n)*(_____K(n)) ! K(n)*
is an algebra homomorphism. It is antimultiplicative iff
: K(n)*(K(n)) ! K(n)*:
is an algebra homomorphism. |___|
Finally recall (and there will be no more recollections, I promise) that grad*
*ed ring
homomorphisms n ! K(n)* classify strict graded automorphisms of the canonical
formal group law Fn over K(n)* and that the group Aut(Fn) of such automorphisms
is isomorphic to Z*2, the isomorphism being given by Z*23 k 7! [k]Fn(x) 2 Aut(F*
*n).
Proof of Theorem 4.For each OE 2 Aut(Fn), OE : n ! K(n)*, we have to_provide_
a multiplicative extension to either K(n)*(K(n)) ! K(n)* or K(n)*(K(n) ) !
K(n)*. As in the proof of Theorem 3, an extension to the former will exist (and
be unique) iff OE(ti) = 0 for 1 i n - 1 and OE(tn) = vn. The first condition
is automatically satisfied for dimensional reasons. Similarly, an extension to*
* the
latter exists iff OE(tn) = 0. Since 0 and vn are the only elements in K(n)* of *
*degree
|tn|exactly one of these conditions is fulfilled. (We leave it to the intereste*
*d reader
to verify that OE(tn) depends on the congruence class of OE modulo 4 in the way
claimed.) Thus we obtain a well-defined map
Aut(Fn) ! Mult
which is an inverse to the geometrically defined map
Mult ! Aut(Fn)
that gives the effect on Euler classes. Since the latter is obviously a_group_h*
*omo-
morphism, we are done. |__|
ON THE STRUCTURE OF P(n)*(P(n)) FOR p = 2 9
References
[KW] R. Kultze and U. W"urgler, A note on the algebra P(n)*(P(n)) for the prime*
* 2, manuscripta
math. 57 (1987), 195-203
[M] O. K. Mironov, Multiplications in cobordism theories with singularities an*
*d Steenrod - tom
Dieck Operations, Math. USSR-IZV 13, (1979), 89-106
[N] C. Nassau, Eine nichtgeometrische Konstruktion der Spektren P(n), Multipli*
*kative und
antimultiplikative Automorphismen von K(n), Diplomarbeit, Johann Wolfgang *
*Goethe-
Universit"at Frankfurt, October 1995
[R1] D. Ravenel, Complex cobordism and stable homotopy groups of spheres, Acade*
*mic Press,
Inc., Orlando 1986
[R2] D. Ravenel, Nilpotence and periodicity in stable homotopy theory, Annals o*
*f Mathematics
Studies 128, Princeton University Press, 1992
[W1] U. W"urgler, Commutative ring spectra of characteristic 2, Comment. Math. *
*Helvetici 61
(1986), 33-45
[W2] U. W"urgler, Morava K-theories: A survey in Algebraic Topology, Poznan 198*
*9 Proceed-
ings, Springer Lecture Notes 1474 (1991), 111-138
[Y] N. Yagita, On the Steenrod algebra of Morava K-theory, J. London Math. Soc*
*. (2), 22
(1980), 423-438
Johann Wolfgang Goethe-Universit"at Frankfurt, Fachbereich Mathematik, Robert
Mayer Strasse 6-8, 60054 Frankfurt, Germany
E-mail address: nassau@math.uni-frankfurt.de