Contemporary Mathematics
Commutative Morava homology Hopf algebras
Hal Sadofsky and W. Stephen Wilson
Abstract.We give the Dieudonne module theory for Z=2(pn --1)graded bi-
commutative Hopf algebras over Fp. These objects arise as the Morava K-
theory of homotopy associative, homotopy commutative H-spaces.
1. Introduction
In [DG70 , V], Demazure and Gabriel classify commutative unipotent alge-
braic groups over a perfect field of characteristic p in terms of Dieudonne-mod*
*ules.
Making appropriate translations of terminology, this classifies a category of H*
*opf-
algebras in terms of Dieudonne-modules.
Bousfield, in an appendix to [Bou ], proves many of the results of interest *
*in
an accessible way, adapting and clarifying [DG70 , V] for his specific purpose*
*s. In
particular, he gives much of the structure for Z=(2)-graded bicommutative Hopf
algebras over Fp. He needs these results for his work with mod p K-theory. We
are interested in similar results for Morava K-theory. The Morava K-theory of
homotopy commutative H-spaces gives rise to bicommutative Z=2(pn - 1)-graded
Hopf algebras satisfying certain conditions. Having a Dieudonne module theory f*
*or
these Hopf algebras gives us a great deal of control over their structure. When*
* the
grading is forgotten then Bousfield's results can be applied, so our contributi*
*on is
really to unravel the gradings in our case and make appropriate definitions so *
*that
Bousfield's proofs go through with little or no modification.
Specifically, we want to study the category C(n) of commutative Morava Hopf
algebras. These are bicommutative, biassociative, Hopf algebras over Fp (in our
proofs one could substitute any perfect field of characteristic p) which are gr*
*aded
over Z=2(pn -1) (n > 0) and have an exhaustive primitive filtration. This last *
*con-
dition, an exhaustive primitive filtration, means that some iterate of the redu*
*ced
coproduct is 0. (Bousfield calls this property "irreducible.") Bousfield's work*
* ana-
lyzes these Hopf algebras quite thoroughly by reducing the Z=(2)-graded case to*
* the
ungraded case by splitting off the sub-algebra generated by odd-dimensional pri*
*m-
itives. The study of these Hopf algebras in [HRW97 ] concentrates on important
consequences of the grading. Our goal here is to put the grading into Bousfield*
*'s
results.
____________
1991 Mathematics Subject Classification. Primary 16W30, 55N20.
The first author was partially supported by the National Science Foundation.
Oc0000 (copyright holder)
1
2 HAL SADOFSKY AND W. STEPHEN WILSON
[HRW97 ] shows C(n) splits (for odd primes) as the product of two categori*
*es;
OC(n), whose objects are Hopf algebras generated by primitives in odd degrees,
and hence primitively generated exterior algebras, and EC(n), whose objects are*
* all
concentrated in even degrees. (Note that [Bou ] proves the same result for C(1*
*),
and his proof also holds without alteration for C(n).) So, our real category of*
* study
will be EC(n).
In [Bou96 ], Bousfield shows C(1) is an abelian category, but again, his pro*
*of
works just as well for C(n). Our contribution is to define a functor to Dieudon*
*ne mod-
ules, to which we again appeal to [Bou ] with small alterations.
We define a Morava-Dieudonne module to be a Z=(pn-1)-graded abelian group
M with endomorphisms F (which multiplies degrees by p), V (which divides degrees
by p) : M ! M such that:
(i)F V = p = V F ;
(ii)for each x 2 M, there exists q 1 with V q(x) = 0.
We denote the category of Morava-Dieudonne modules by MD(n)*. Note the
definition implies the modules are all p-torsion, i.e. pq(x) = (F V )q(x) = F q*
*V q(x) =
0 where we have used (i) and (ii).
Our main theorem is:
Theorem 1.1. There is a functor m* such that m* : EC(n) ! MD (n)* is an
equivalence of abelian categories.
Let Z=(n) act on Z=(pn - 1) by k * j = jpk. The orbits are given by fl =
{j; jp; jp2; jp3; : :;:jpn = j}. We can see with no work that MD (n)* splits a*
*s a
product of categories MD fl(n)* where all elements are in the degrees contained
in fl. This translates into a similar theorem, a major result of [HRW97 ], ab*
*out
EC(n).
Theorem 1.2. As categories:
Y
EC (n) ' EC(n)fl
fl
where EC(n)flconsists of Hopf algebras in EC(n) with all primitives in degrees *
*2fl.
This result was proven in [HRW97 ] so it could be combined with results of
[RW80 ] to show that there is only the trivial map from K(n)*(K(Z=(pi); s)) to
K(n)*(K(Z=(pj); t)) when s 6= t.
The motivation for us is Morava K-theory, K(n)*(-), which is a generalized
homology theory with a K"unneth isomorphism. The coefficient ring is K(n)* '
Fp[vn; v-1n] where the degree_of_vn is 2(pn - 1). By setting vn = 1 we get a
Z=2(pn - 1)-graded theory,_K(n)_*(-). If a space X is connected and the loop
space of an H-space then K(n) *(X) is in C(n) for odd primes (there are some
complications with commutativity for p = 2).
Some examples from [RW80 ] are
_____
m*(K(n) *(K(Z=(p); n))) ' Z=(p)
in degree 1 + p + . .+.pn-1 with V = 0 and F (1) = (-1)n-1 and
_____
m*(K(n) *(K(Z(p); n + 1))) ' Q=Z(p)
in degree 1 + p + . .+.pn-1 with V = (-1)n-1p and F (x) = (-1)n-1x. In addition,
for all Eilenberg-Mac Lane spaces except S1, the Morava K-theory lies in EC(n).
COMMUTATIVE MORAVA HOMOLOGY HOPF ALGEBRAS 3
There are a couple of additional theorems worth pointing out. The first is
that under very mild conditions, see [Bou , Appendix B, Theorem B.4], Sweedler
proves what we would call a dual Borel theorem (about the coalgebra structure)
in [Swe67a ] and [Swe67b ]. The second is a comment on Theorem 1.2 splitting
EC(n). It is straightforward to modify the Dieudonne theory above to produce ea*
*ch
of these categories. All we have to do is restrict our attention to the functor*
*s mt
where t 2 fl.
Background. A Dieudonne module theory has long been developed for graded
Hopf algebras, [Sch70 ]. The authors had been talking about this project for so*
*me
time when the second author was decimated by the chairmanship of his department.
By the time he had recovered, the two papers, [Bou ] and [Bou96 ], were availab*
*le
and the project was easy to complete by co-opting the proofs from Bousfield, [B*
*ou ].
Pete Bousfield politely but firmly declined the authors' invitation to be a coa*
*uthor.
The authors regret his decision and make no claim to originality. We feel the
theorems and proofs should be made available however. Following a suggestion of
Igor Kriz's we were developing the general Dieudonne module theory for Hopf rin*
*gs
only to find that Paul Goerss had already done it.
2. The Witt algebra, Frobenius and Verschiebung
We review the Witt algebra before constructing the Z=2(pn-1) graded versions
we will use.
Let W = Z[x0; x1; : :]:. We identify
Hom Z-alg(W; A) AN
where N is the set of non-negative integers, and an algebra homomorphism f :
W ! A is identified with the vector (f(x0); f(x1); : :):. Then we define a map
w : AN ! AN
by
2 p 2
(a0; a1; : :):7! (a0; ap0+ pa1; ap0+ pa1 + p a2; : :)::
We also write wn for the nth coordinate of w, and hence
n pn-1 n-1 p n
wn(a0; a1; : :):= ap0 + pa1 + . .+.p an-1 + p an:
Note that the map w is injective if A is p-torsion free.
Theorem 2.1 (Witt). The image of w is a subgroup of AN .
This is proved in [Wit36 , Satz 1], but a more modern treatment following
Lazard can be found on page 40 ff. of [Ser67 ]. The issue in the proof is: gi*
*ven
(a0; a1; : :):and (b0; b1; : :):one needs to find (c0; c1; : :):so that w(c) = *
*w(a)+w(b).
This can clearly be done over p-1A, so the problem is an integrality one.
Corollary 2.2. W has a commutative, associative coproduct determined by
declaring wn(x) to be primitive for all n. This makes W a bicommutative Hopf-
algebra.
Proof. The proof is by Yoneda's lemma. We let A range over p-torsion-free Z-
algebras and give Hom Z-alg(W; A) the group structure derived from Theorem 2.1.
Precisely, given f; g 2 Hom Z-alg(W; A), f is determined by the vector f(x) = a,
4 HAL SADOFSKY AND W. STEPHEN WILSON
g is determined by the vector g(x) = b. Then f + g is determined by the vector c
where w(c) = w(a) + w(b), so (f + g)(xi) = ci.
This gives a product
(2.3) Hom Z-alg(W Z W; A) '
Hom Z-alg(W; A) x Hom Z-alg(W; A)! Hom Z-alg(W; A):
By Yoneda's lemma, we get a map of Z-algebras
: W ! W Z W:
Since Hom Z-alg(W; A) is an abelian group, this makes W a cocommutative Hopf-
algebra.
To see the wn(x) are primitive, we take A = W Z W in (2.3) and want that
(iL + iR )(wn(x)) = wn(x) 1 + 1 wn(x)
where iL : W ! W Z W takes a to a1, and iR is defined similarly. So if we write
xLifor xi1 and xRi= 1xi, iL is represented by (xL0; xL1; : :):and iR is represe*
*nted
by (xR0; xR1; : :):. Hence (iL + iR ) is represented by polynomials (c0; c1; : *
*:):where
wn(c) = wn(xL ) + wn(xR ). In other words,
(iR + iL)(wn(x)) = wn(c) = wn(xL) + wn(xR) = wn(x) 1 + 1 wn(x);
as was to be shown.
To note that wn(x) primitive determines the co-algebra structure, observe th*
*at
the wn(x) are polynomial generators of W Z Q, so since W is torsion free, there_
is at most one Hopf algebra structure with the wn(x) primitive. |_*
*_|
Remark 2.4. If we grade W by setting |x0| = 2t and |xi| = pi2t, then the
wn(x) are homogeneous, so we get a graded Hopf-algebra. We call the resulting
Hopf algebra Wt. Another reference for the above is [Kra72 ].
Definition 2.5.We give a Hopf algebra graded over Z=2(pn - 1) by defining
W (s; t) Fp[x0; x1; : :;:xs] Wt Fp:
Frobenius and Verschiebung. Bicommutative Hopf algebras over fields of
characteristic p come with two self maps, the Frobenius F and the Verschiebung,*
* V .
The Frobenius is the pth power map, i.e. F (x) = xp, which is a map of Hopf alg*
*ebras
since we are in characteristic p, but not of graded Hopf algebras. The Verschie*
*bung
can be defined as the dual of the Frobenius map on the dual Hopf algebra, as in
[HRW97 ], or in terms of the coproduct, as in [Bou ]. One construction of the
Verschiebung is as follows: choose a basis for the bicommutative Hopf algebra of
characteristicPp, A. Write p-1(x) as a sum of tensor products of basis elements,
k
i=1ai1 . . .aip. Then
X
V (x) = ai1:
{i|ai1=...=aip}
It is an easy exercise to check that this definition does not depend on the bas*
*is
chosen, and another exercise to check that it gives a map of Hopf algebras. It
doesn't give a map of graded Hopf algebras since |V (x)| = |x|=p.
When x is primitive, V (x) = 0, so V is 0 on primitively generated bicommuta-
tive Hopf algebras. Clearly F V = V F since V is a map of algebras. Also, a dir*
*ect
consequence of the addition as defined above in Hom Z-alg(A; B) (where A is some
COMMUTATIVE MORAVA HOMOLOGY HOPF ALGEBRAS 5
bicommutative Hopf algebra over Fp and B runs through Fp-algebras) is that F V
on A induces multiplication by p in the abelian group Hom Z-alg(A; B). It follo*
*ws
that if A is primitively generated, Hom Z-alg(A; B) is an Fp-vector space.
We would like to calculate the value of V on W Fp. We begin with the
observation that since F is injective, we can deduce V from F V . Since F V ind*
*uces
multiplication by p in Hom Fp-algebra(W Fp; A), we'll begin by analyzing the s*
*elf-
map of W that induces multiplication by p in Hom Z-alg(W; A). Recall that f :
W ! A is represented by a vector (a0; a1; : :):where ai= f(xi) and that the p-f*
*old
sum of f with itself, which we'll write [p](f) is represented by a vector (c0; *
*c1; : :):
where ci= ([p]f)(xi) and pwn(a) = wn(c). Now observe that
(2.6) pw(a) =
n 2 pn-1 n p n+1
(pa0; pap0+ p2a1; : :;:pap0 + p a1 + . .+.p an-1 + p an; : :)::
So c0 = pa0; c1 = ap0+ p(a1 - pp-2ap0). If one assumes inductively that cj =
apj-1+ pbj, for j i one sees that
i+1 pi i p i+1
wi+1(c) = cp0 + pc1 + . .+.p ci + p ci+1
i+1 p pi i p p i+1
= (pb0)p + p(a0 + pb1) + . .+.p (ai-1+ pbi) + p ci+1
i+1 i p2 i+1 i+2
pap0 + . .+.p ai-1+ p ci+1 mod p :
and hence by equating with (2.6 ) that ci+1 = api+ pbi+1. On reduction mod p,
it follows that multiplication by p on Hom Fp-algebra(Wp; A) is induced by the *
*map
that sends xi to xpi-1. Since F (xi-1) = xpi-1, it follows then that V (xi) = x*
*i-1.
3. The functor to Dieudonne-modules
Although, as we have remarked, neither F nor V give maps of graded Hopf
algebras, we have maps F nand V nin our cyclicly graded category EC(n). The V n
induce surjections by:
Fp[x0; : :;:xns] = W (ns; t) -! W (n(s - 1); t) = Fp[x0; : :;:xn(s-1)]:
We use the sequence:
n V n
W (0; t) -V- W (n; t) . . .W (n(s - 1); t) -- W (ns; t) . . .
to define
mt: EC(n) -! MD(n)t
by
mt(A) = lim!Hom(W (ns; t); A):
Remark 3.1. To get the equivalence of EC(n)fland MD (n)flwe can modify
the definition slightly. Instead of using V nwe can replace n with the order of*
* the
orbit fl. In particular, when fl = {0} we are precisely in the ungraded case of*
* [Bou ]
except with slightly modified, but completely equivalent, definitions for F and*
* V
on the image of our functors.
6 HAL SADOFSKY AND W. STEPHEN WILSON
To complete our definition of m* : EC(n) ! MD(n)* we need to define the
action of F and V on m*(A). We require a minor departure from the standard
definition in order to preserve grading. In our case we use the fact that the i*
*mage
of F on W (s; t) is canonically isomorphic to W (s; pt), i.e. the composite
W (s; t) '-!W (s; pt) ,! W (s; t)
that sends xi7! xi7! xpiis F . The second map here preserves degree, so is in o*
*ur
category.
Because F and V commute we get a map of sequences
W(0; t) W(n; t) . . . W(n(s - 1); t) W(ns; t) . . .
" " " "
W(0; pt) W(n; pt) . . . W(n(s - 1); pt) W(ns; pt). . .
which induces the map F
mt(A) = lim!Hom(W (ns; t); A) -! mpt(A) = lim!Hom(W (ns; pt); A):
We now need the map V . The image of V acting on W (s; t) is the sub-Hopf algeb*
*ra
W (s - 1; t). There is a canonical map of W (s; t=p) which surjects to this ima*
*ge. In
particular, this gives us maps
W (ns; t=p) ! W (ns - 1; t) W (ns; t):
The corresponding map of sequences
W(0; t) W(n; t) . . . W(n(s - 1); t) W(ns; t) . . .
" " " "
W(0; t=p) W(n; t=p) . . . W(n(s - 1); t=p) W(ns; t=p). . .
induces V
mt(A) = lim!Hom(W(ns; t); A) -! mt=p(A) = lim!Hom(W(ns; t=p); A):
4. The equivalence of categories
We are now set to prove our main Theorem 1.1.
Proof. We follow Bousfield's analogous proof in [Bou , Appendix A] very
closely. As in his proof and the similar result in [DG70 , V,x1,n.4], we veri*
*fy
directly that m* is an isomorphism of categories rather than constructing an in*
*verse
functor from abstract considerations. We remind the reader as we remarked in the
introduction that the proof for the Z=(2)-graded case in [Bou96 ] implies that *
*EC(n)
is an abelian category. The structure of the rest of the proof is as follows:
(i)Show m* is exact (for which we'll need to prove that W (s; t) is projecti*
*ve in
the subcategory EC(n; s) defined below).
(ii)Show every element of MD (n) is m*(B) for some B 2 EC(n).
(iii)Show m* is an isomorphism on Hom sets.
(iv)Show m* is an isomorphism on the class of objects, which is implied by the
previous two points.
We begin by defining EC(n; k) to be the objects in EC(n) on which V k+1is 0,
and similarly for MD (n; k).
We observe that EC(n; 0) is the category of primitively generated objects in
EC(n). This follows from the same fact in the ungraded case, [Swe67a , Lemma 3]
(see the corollary immediately after the proof of Lemma 3), because the primiti*
*ves
COMMUTATIVE MORAVA HOMOLOGY HOPF ALGEBRAS 7
of the ungraded Hopf algebra are generators, but since the diagonal map on an
object of EC(n) preserves the grading, the primitives of such an object are the
direct sum of the homogeneous primitives.
On the other hand MD (n; 0) can be thought of as restricted Fp-Lie algebras
with trivial bracket (V = 0 implies p = 0 on objects of MD (n; k)). For A 2
EC(n; 0) we check that m*(A) = P A, the primitives of A. To see this note that
Hom (W (ns; t); A) = PtA because any map from W (ns; t) to A is 0 on the image *
*of
V , so factors through W (ns; t) ! W (0; tpns) = W (0; t). Since this factoriza*
*tion is
preserved under the maps in the system:
n
W (ns; t)-V---!W (n(s - 1); t)
?? ?
y ?y
W (0; t)--=--! W (0; t)
we get that mt(A) = lim! Hom (W (ns; t); A) = PtA. The functor back from
MD (n; 0) is of course the restricted enveloping algebra, which gives an equiv*
*a-
lence of categories EC(n; 0) and MD (n; 0). For this we apply [MM65 , Theorem
6.11] in the ungraded case to see that as an ungraded Hopf-algebra, A 2 EC(n; 0)
implies A = V (P A); the restricted enveloping algebra on the primitives. Again
since P (A) is just the direct sum of its homogeneous pieces we can consider P *
*(A)
as a cyclically graded Fp-Lie algebra with trivial bracket (A = V (P A) is comm*
*uta-
tive). Again, since M = P V (M) as an ungraded Lie algebra, the same fact about
the diagonal (now on V (M)) tells us that this formula holds as a graded Lie al*
*gebra.
Next one wants to see that W (s; t) is projective in EC(n; s). We've establi*
*shed
this above when s = 0. One can follow the proof of [Bou , Lemma A.12] precisely:
it suffices to prove that for each B 2 EC(n; s) that Ext1EC(n;s)(B; W (s; t)) =*
* 0. Since
B is filtered by the kernels of V i, with filtration quotients in EC(n; 0) it s*
*uffices to
check this when B 2 EC(n; 0). So we look at such an extension
__
(4.7) B ! B ! W (s; t):
Taking both the kernel and cokernel of V of this short exact sequence we can use
the snake lemma to get a six term exact_sequence relating them. In fact this is*
* two
short exact sequences because the map B ! W (s; t) is still onto when restricted
to the kernel of V because the kernel of V on W (s; t) is also the image of V s*
*and
the image of V sis a subset of the kernel of V on B since B 2 EC(n; s). Hence t*
*he
sequence
__ s
(4.8) B ! B==V ! W (s; t)==V = W (0; p t)
in EC(n; 0) is exact, and therefore split as W (0; pst) is_projective_in EC(n; *
*0). It
follows that we can split the inclusion in (4.8) by a map B==V ! B, and hence c*
*an
split the inclusion in (4.7).
Now we need to show that m*(-) is exact. It is obviously left exact, so the
issue is showing that if A g-!B is onto, and z 2 mt(B), then z = g*(y) = g O y *
*for
y 2 mt(A). Choose s so z is given by a map f : W (ns; t) ! B. Choose lifts of
f(xi), i = 0; : :;:ns in A. Choose m so V m annihilates all those lifts. Let Bm*
* B
be the kernel of V m and similarly Am A. Then f factors through Bm , and let
8 HAL SADOFSKY AND W. STEPHEN WILSON
f*Am be the pullback:
f*Am ----! Am
? ?
p?y ?y
W (ns; t)----! Bm :
If n(k + s) m, then fV nk: W (n(k + s); t) ! Bm ! B also represents z, and
since p is surjective the projectivity of W (n(k +s); t) allows us to lift V nk*
*to f*Am ,
and hence fV nkto A as we require.
Now we need that m*W (s; t) is projective in MD (n; s). We do as we did for
W (s; t). It suffices to check that Ext1MD(n;s)(B; m*W (s; t)) = 0 for B 2 MD *
*(n; 0).
So if we have an extension of Dieudonne-modules,
__
(4.9) B ! B ! m*W (s; t)
we can observe by exactness of m* that the kernel of V is the image of V son
m*W (s; t), so that we have an extension
(4.10) B ! B=V B ! m*W (s; t)=V m*W (s; t)
in MD (n; 0). But (4.10) splits by the equivalence between MD (n; 0) and EC(n*
*; 0),
and it follows that (4.9) splits.
Now we wish to prove that
(4.11) m* : Hom EC(n)(W (s; t); A) ~=Hom MD (n)(m*W (s; t); m*A):
It clearly suffices to show this when A 2 EC(n; s) since the kernels of V s+1on*
* A
and m*A is where morphisms from W (s; t) and m*W (s; t) land, and the functor
m* commutes with this kernel.
By exactness, it also suffices to assume A 2 EC(n; 0). But then
Hom EC(n)(W (s; t); A) = Hom EC(n)(W (s; t)==V; A)
= Hom EC(n)(W (0; pst); A) = PpstA
and
Hom MD (n)(m*W (s; t); m*A) = Hom MD (n)(m*W (s; t)=V; m*A)
= Hom MD (n)(m*(W (s; t)==V ); m*A) =
Hom MD (n)(m*W (0; pst); m*A) = PpstA:
Next we need to show that every M 2 MD (n) is m*B for some B. To do this,
we take M and realize it as a cokernel (see below):
M M
(4.12) m*W (si; ti) ! m*W (sj; tj) ! M:
I J
Since m* is a direct limit, it commutes with sums, and we get M = m*B where B
is the cokernel in
O O
(4.13) W (si; ti) ! W (sj; tj) ! B:
I J
So it remains to prove that M can be written as in (4.12). Let x 2 Mt. Then
V rx = 0 for some r. m*W (r - 1; t) is
Z=(p) Z=(p2) . . .Z=(pr-1) Z=(pr) Z=(pr) : : :
COMMUTATIVE MORAVA HOMOLOGY HOPF ALGEBRAS 9
where if we denote the generator of the ith summand by ei, then
V (ei) = ei-1 ifi r; V (ei) = pei-1 ifi > r
F (ei) = ei+1ifi r; F (ei) = pei+1ifi < r;
and the degrees are determined by |er| = t. Then m*W (r - 1; t) is the free
Dieudonne-module on a class er in dimension t with V r(er) = 0, and there is a
map of Dieudonne-modules m*W (r - 1; t) ! M defined by taking er to x. It fol-
lows there is a surjective map from a sum of m*W (sj; tj) to M, and that we can
realize M as a cokernel as in (4.12). Hence M = m*B for some B.
Finally, to prove m* is an equivalence of categories, it suffices to show th*
*at
m* : Hom EC(n)(B; A) ! Hom MD (n)(m*B; m*A) is an isomorphism. We do this by
showing any B 2 EC(n) can be realized as a cokernel as in (4.13). Once we know
this, we get a map of exact sequences
0 - ---! 0
?? ?
y ?y
Hom EC(n)(B; A) - ---! Hom MD (n)(m*B; m*A)
?? ?
y ?y
Hom EC(n)(JW (sj; tj); A)----!Hom MD (n)(Jm*W (sj; tj); m*A)
?? ?
y ?y
Hom EC(n)(IW (si; ti);-A)---!Hom MD (n)(Im*W (si; ti); m*A):
The third and fourth horizontal maps are isomorphism since in both categories
Hom out of a sum is the product of Hom 's and we've already checked (4.11). So_*
*by
the five lemma, we get the desired isomorphism. |__|
References
[Bou] A. K. Bousfield. On p-adic -rings and the K- theory of H-spaces. Mathema*
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University of Oregon, Eugene, OR, 97403
E-mail address: sadofsky@math.uoregon.edu
Johns Hopkins University, Baltimore, Maryland 21218
E-mail address: wsw@math.jhu.edu