THE MICROSTABLE ADAMS-NOVIKOV SPECTRAL SEQUENCE
DOUGLAS C. RAVENEL
December 8, 1999
Introduction. For a fixed prime p, recall the spectra T (m) (introduced in
[Rav86 , x6.5]) with
BP*(T (m)) = BP*[t1; : :;:tm ] BP*(BP ):
It is a p-local summand of the Thom spectrum associated with the map
SU(k) ! SU = BU
for any k satisfying pm k < pm+1 . These Thom spectra figure in the proof of t*
*he
nilpotence theorem of [DHS88 ].
Let (A; ) denote the Hopf algebroid (BP*; BP*(BP )); see [Rav86 , A1] for more
information. A change-of-rings isosmorphism identifies the Adams-Novikov E2-
term for for T (m) with Ext(m+1) (A; A) where
(m + 1) = =(t1; : :;:tm ) = BP*[tm+1 ; tm+2 ; : :]:
This Hopf algebroid is cocommutative below the dimension of t2m+2, so its Ext
group (and the homotopy of T (m)) in this range is relatively easy to deal with.
Moreover empirical evidence suggests that ss*(T (m)) for roughly 2pm+1 < * <
2p2m+2 is the same (up to a suitable regrading) as that of ss*(T (m + 1)) for r*
*oughly
2pm+2 < * < 2p2m+3. The purpose of this note is to set up an algebraic framework
that allows us to study the limit of this behavior as m goes to infinity.
This will entail defining a bigraded Hopf algebroid (Ab; b). The grading is o*
*ver
ZZ! where ! becomes pm when we specialize to T (m). We call the corresponding
Ext group the microstable Adams-Novikov E2-term for the following reason. For
each spectrum T (m) one can set up a chromatic spectral sequence as in [Rav86 ,
Chapter 5]. Each Morava stabilizer group Sn gets replaced by a certain open
subgroup which shrinks as m increases. Thus in the limit each Sn gets replaced *
*by
an infinitesimal version of itself. We conjecture that this Ext group is the E2*
*-term
of a trigraded spectral sequence.
The author wishes to thank the Dominique Arlettaz and Kathryn Hess for or-
ganizing a conference in such an inspiring Alpine setting, where the idea for t*
*his
paper originated. I am also grateful to Ippei Ichigi for many useful conversati*
*ons
about this work.
The bigraded Hopf algebroid (Ab; b). Define a Hopf algebroid (Ab; b) over
Z(p), graded over Z Z!, by
bA = BP*[ci;m; ui: 0 i m]=(ci;m- v(p-1)pmici;m+1)
with v0 = p,|ci;m| = (! - pm )|vi|; and |ui| = 2pi! - 2;
b = bA[si: i > 0] with|si| = 2pi! - 2:
____________
The author acknowledges support from NSF grant DMS-9802516.
1
2 DOUGLAS C. RAVENEL
This definition should be compared with the usual (A; ) given by
A = Z(p)[vi: i > 0] with|vi| = 2pi- 2;
= A[ti: i > 0] with|ti| = 2pi- 2:
m
We will denote the element vpici;mby v!ifor 0 i m. Because of the relationsm
in bA, this element is independent of m. Conversely we will often write v!-pi f*
*or
ci;m.
The generators c0;m in dimension 0 are somewhatmproblematic. The defining
relation in bAimplies that c0;m= p!-pm = v!-p0 , so p! is infinitely divisible *
*by p.
It follows that each c0;mmaps to 0 in bA=(p). One could get rid of them by repl*
*acing
(2) below (which is analogous to Araki's definition of the vi) by a formula sim*
*ilar
to Hazewinkel's, namely
X !pj X pj
(1) pki= kjvi-j+ `jui-j;
0 0 is fixed, and we sum over the indicated values of j and k. The pres*
*ence
of the first term on the left would lead to an error (which can be shown to be
divisible by p using [Rav76 , Lemma 2]) in the analog of (10).
We will adopt the following convention: whenever possible an element in b or
a related Ext group will be denoted by the letter of the alphabet preceding the*
* one
usually used for the corresponding element associated with , and algebraic obje*
*cts
assocaied with it will be denoted by a hat over the symbol for the corresponding
object associated with , In particular, the elements ui and si are the microsta*
*ble
analogs of vi and ti.
Recall the the log coefficients `i2 A Q are related to the viby Araki's form*
*ula
X pj
p`i= `jvi-j;
0ji
and the right unit is defined by
X pj
jR (`i) = `jti-j:
0ji
The microstable analogs of these formulas are
X !pj X pj
(2) pki= kjvi-j+ `jui-j
0 0 by
bA(m) = Z(p)[v1; : :;:vm ; v!-pm0; v!-pm1; : :;:v!-pmm; u1; : :;:um+1 ]
bG(1; m)= bA(m)[s1; : :;:sm+1 ]
Ab(m; n) = Z(p)[v1; : :;:vm+n ; v!-pm0; v!-pm1; : :;:v!-pmm+n; u1; : :;:um+*
*n+1 ]=In
bG(1; m; n)= bA(m; n)[s1; : :;:sm+1 ]:
Let
A(k) = Z(p)[v1; : :;:vk];
G(m + 1; m) = A(2m + 1)[tm+1 ; : :;:t2m+1] as in [Rav86 , x7.1];
and G(m + 1; k; n)= A(m + 1 + k + n)=In[tm+1 ; : :;:tm+1+k ]:
There are maps
(5) bG(1; m)____wG(mm+ 1; m) (m + 1)
and
(6) bG(1; m; n)____G(mw+m1; m; n) (m + 1)=In
given by
vi 7! vi
ui 7! vi+m
m
v!i 7! vpi
si 7! ti+m:
In addition, we have
_
m (ki) = `m+i
_
where `m+i is obtained from `m+i by removing all terms which are monomials in
the vj for 0 < j m. The analog of Araki's formula for these elements is
_ X pj X _ pm+j
p`m+i = `jvm+i-j + `m+j vi-j :
0j* 0:
One can derive a similar formula for Ext0b(Ab; bA=In) from (10). Let
bV= Z(p)[vi; ci;m: 0 i m] bA:
Then we have (
Vb ifn = 0
(12) Ext0b(Ab; bA=In) = b
V =In[u1; : :u:n]ifn > 0:
The microchromatic spectral sequence. The chromatic spectral sequence
converging to Ext (A; A) is obtained from the resolution
0 ! BP* ! M0 ! M1 ! M2 ! : : :
where
Mn = v-1nBP*=(p1 ; v11; : :;:v1n-1):
More details can be found in [Rav86 , Chapter 5]. This can be tensored over A w*
*ith
bA, leading in the same way to a spectral sequence converging to Extb(Ab; bA) w*
*hich
we call the microchromatic spectral sequence.
We also define
Mn-ii= v-1nBP*=(p; : :;:vi-1; v1i; : :;:v1n-1):
so for each i > 0 there is a resolution
0 ! BP*=Ii! M0i! M1i! M2i! : :;:
and there are short exact sequences
vi
0 _________wMn-i-1i+1___w|vi|Mn-ii ______wMn-ii__________w0
which lead to Bockstein spectral sequences. In particular there is a chain of n
Bockstein spectral sequences leading from Ext (A; v-1nBP*=In) to Ext (A; Mn ).
The former group can be identified with
Ext(n)(K(n)*; K(n)*)
where
(n) = K(n)* BP* BP*(BP ) BP* K(n)*
n pi
= K(n)*[ti: i > 0]=(vntpi - vn ti)
6 DOUGLAS C. RAVENEL
as an algebra, with coproduct inherited from BP*(BP ). The formula (11) is pivo*
*tal
in the proof of this result.
The microstable analog, which can proved in a similar way using (10) is
(13) Extb(Ab; v-1nbA=In) = Extb(n)(Kb(n)*; bK(n)*)
where
bK(n)* = v-1nbV=In[u1; : :;:un]
n !pi
and b(n) = Kb(n)*[si: i > 0]=(vnspi - vn si):
The methods of [Rav86 , Chapter 6] can be used to compute this Ext group explic*
*itly,
and the result is
(14) Extb(Ab; v-1nbA=In) = bK(n)* E(gi+1;j: 0 i; j < n)
j
where gi;jcorresponds to spi. It is also true that
Ext(m) (A; v-1nA=In) = K(n)*[vn+1; : :;:v2n] E(hi+m+1;j: 0 i; j < n) form n;
j
where hi+m;jcorresponds to tpi+m. Thus the microchromatic spectral sequence is
simpler than the usual one.
The microstable 1-line. We can use the microchromatic spectral sequence to
compute Ext1b(Ab; bA) in the same way that we use the chromatic spectral sequen*
*ce
to compute Ext1(A; A). We need to analyze the Bockstein spectral sequence going
from
Extb(Ab; cM01) = bK(1)* E(g1;0)
to Extb(Ab; cM10). This behaves in much the same way as the stable analog, i.e.*
*, the
one going from
Extb(A; cM01) = K(1)* E(h1;0)
to Ext (A; M10).
For odd primes the relevant fact about the right unit is that for all t > 0,
jR (ut1) ut1+ ptut-11s1 mod (p2t):
From this we deduce that Ext1b(Ab; bA) is the bV-module generated by the set
ae oe
ut1_: t > 0:
pt
For p = 2 let
w1;1= u21+ 2v2!-11u1 + 4v-11u2:
Then for all s > 0 we have
jR (u2s-11) u2s-11+ 2u2s-21s1mod(4)
and jR (ws1;1) ws1;1+ 4su2s-11s1mod(8s):
From this we deduce that Ext1b(Ab; bA) is the bV-module generated by the set
ae2s-1 s oe
u1___; w1;1_: s > 0:
2 4s
THE MICROSTABLE ADAMS-NOVIKOV SPECTRAL SEQUENCE 7
For all primes it follows that
Ext1b(Ab; cM10) = 0;
unlike the stable case where Ext1(A; M10) Q=Z.
Note that for odd primes each element in Ext1can be pulled back to
i j
ExtbG(1;0)bA(0); bA(0);
so we can map them via the map m of (5) to
Ext1(m)(A; A)
for m 0. For p = 2 we can only do this for m 1. This is to be expected since
the structure of Ext1(1)(A; A) for p = 2 differs from that of Ext1b(Ab; bA) in *
*that for
2s u2s
s > 1, v1_2is divisible by 4s while _1_2is only divisible by 2s.
The Miller-Wilson computation. In [MW76 ] Miller-Wilson computed
Ext0(A; M1n-1)
in nearly all cases by studying the relvant Bockstein spectral sequence. Their
result is restated as [Rav86 , 5.2.13], which says that the group is a direct s*
*um of
K(n - 1)*=k(n - 1)* generated by {__1_vj: j > 0}, and cyclic k(n - 1)*-modules
n-1
generated by the set
( )
xsn;i_
(15) : i 0; s > 0; s 6= 0 mod;p
van;in-1
where xn;iis a certain expression of the form vpinmodulo decomposables, and an;i
is a certain integer not less than pi. For n = 2 we have
x2;0 = v2;
x2;1 = vp2- vp1v-12v3;
2-1 p2-p+1 p2+p-1 p2-2p
x2;2 = xp2;1- vp1 v2 - v1 v2 v3; and
(
x22;i-1 ifp = 2
x2;i = p b2;i(p-1)pi-1+1 fori 3;
x2;i-1- 2v1 v2 ifp > 2
where b2;i= (p + 1)(pi-1- 1). The exponents a2;iare given by
a2;0 = 1;
a2;1 = p;ae
(16) i+ 2i-1 ifp = 2
and a2;i= 2pi+ pi-1- 1 ifp > 2 fori 2:
A microstable analog of this computation should be feasible, and we give some
partial results here. Analyzing the Bockstein spectral sequenceifor a fixed n >*
* 1
amounts to finding elements wn;i2 v-1nbA=In-1 congruent to upn modulo In and
integers en;iso that Ext0b(Ab; cM1n-1) is the bk(n - 1)*-module generated by th*
*e set
( )
wsn;i_
: i 0; s > 0; s 6= 0 mod;p
ven;in-1
8 DOUGLAS C. RAVENEL
where bk(n - 1)* = bV=In-1[u1; : :;:un-1]. To prove such a result one needs to
compute
jR (wn;i) - wn;i mod (v1+en;in-1)
and show that these represent vn-1-multiples of linearly independent (over bK(n*
*)*)
elements in the group Ext1b(Ab; cM0n) given in (14).
Here is the relevant information for n = 2; details will appear in a forthcom*
*ing
paper with Nakai. Let
w2;0 = u2;
w2;1 = wp2;0- vp1r2;3;
2-1 a+1 p2+p -p -p! !p
w2;2 = wp2;1- vp1 v2 w2;0+ v1 v2 (r2;4- v2 v3 r2;3);
w2;3 = wp2;2;
2 p3(p+1)p2a -p(p+1)p -p! p!
w2;4 = wp2;2+ v1 v2 (r2;4- v1 v2w2;2- v2 v3 r2;3)
(17) 3(p+1)+pp2a p-1
+vp1 v2 r2;3w2;1
2 p3(p+1)-pp2a p -p2 p p2 -1 !p2
= wp2;2+ v1 v2 (u3 - v1 v2u2 + v1 v2 u2)
3(p+1)p2a p-1 p
-vp1 v2 w2;1(w2;1- w2;0); and
i-1(p+1)pi-2ap-1 p
w2;i = wp2;i-1- vp1 v2 w2;i-3(w2;i-3- w2;i-4)
fori > 4;
where
a = p2! - p - 1;
r2;3 = v-12u3;
and r2;4 = v-12(u4 - v3rp3) + v-p1v-p!2vp!3w2;1- v-11va2v3w2;0:
The elements rn;n+i= v-1nun+i+ : :a:re chosen so that
n pi!-1
d(rn;n+i) spi - vn si mod In:
where d(x) = jR (x) - x.
The exponents e2;iare
e2;0 = 1;
e2;1 = p;
ae
and for i 2 e2;i = pe2;i-1+ 0p ifiot0hmode3rwise.
Note that these microstable exponents e2;iare are in general greater than the
corresponding stable exponents a2;idefined in (16).
Then we have
d(w2;0) v1sp1 mod (v21);
d(w2;1) vp1vp!-12s1 mod (v1+p1);
d(w2;2) -vp(p+1)1va2s22 mod (v1+p(p+1)1);2
d(w2;3) -vp1(p+1)vpa2sp23 mod (v1+p1(p+1));
2a p-1 1+p2(p+1)+p
d(w2;4) -vp1(p+1)vp2 w2;1d(w2;1) mod (v1 );
i-2ap-1 1+e2;i
and d(w2;i) -ve2;i-e2;i-31vp2 w2;i-3d(w2;i-3)mod(v1 )
fori > 4:
THE MICROSTABLE ADAMS-NOVIKOV SPECTRAL SEQUENCE 9
Alternatively we can get some simpler formulas by defining
w2;0 = u2;
w2;1 = wp2;0;
2 p p2-1 a+1
w2;2 = wp2;1- vp1r2;3- v1 v2 w2;0
w2;3 = wp2;2;
3(p+1)-p3 p2 -p(p+1)!p4+p-p2 -p3! !p3 p2
w2;4 = wp2;3+ vp1 v2 (r2;4- v1 v2 w2;2- v2 v3 r2;3);
i-1(p+1)pi-2ap-1 p
and w2;i = wp2;i-1- vp1 v2 w2;i-3(w2;i-3- w2;i-4)
fori > 4;
which leads to
d(w2;0) v1sp12 mod (v21);
d(w2;1) vp1sp1 2 mod (v1+p1);
d(w2;2) -vp(p+1)1v-p2sp22 mod (v1+p(p+1)1);
2 p3 1+p2(p+1)
d(w2;3) -vp1(p+1)v-p2s23 mod (v1 );
2a p-1 1+p2(p+1)+p
d(w2;4) -vp1(p+1)vp2 w2;1d(w2;1) mod (v1 );
i-2ap-1 1+e2;i
and d(w2;i) -ve2;i-e2;i-31vp2 w2;i-3d(w2;i-3)mod(v1 )
fori > 4:
These elements all pull back to an Ext group over bG(1; 2; 1), so we can map *
*them
via the map m of (6) to
Ext(m+1)(A; M11)
for m 2.
The Thom reduction. One can ask about the image of Extb(Ab; bA) in Extb(Ab; b*
*A=I),
where I = (p; v1; v2; : :):, since the latter can be computed explicitly. Each*
* si is
primitive mod I, so we have
Extb(Ab; bA=I)= Ab=I E(gi;j: i > 0; j 0) E(ai;j: i > 0; j 0)
j(pi!-1) pj 2;2pj+1(pi!-1)
where gi;j2 Ext1;2p corresponds to si , and ai;j2 Ext is its
transpotent, not to be confused with the exponents ai;jof (15).
10 DOUGLAS C. RAVENEL
Let ae denote the mod I reduction in Ext. Then we have
t ae t-1
ae u1_pt= u1utg1;0-1 t-2 for p odd
1 g1;0+ (t - 1)u1 g1;1for p:= 2
sut
ae u1_2_pv= stus-11ut-12g1;1g1;0+ tus1ut-12a1;0
1
+ t(t - 1)us1ut-22g1;1g2;0
s pjt!
j
ae u1u2__pj= stus-11u(t-1)p2g1;j+1g1;0
pv1
j(t-1)
+ tus1up2 a1;j forj > 0
t
ae __u3_pv= t(t - 1)ut-23(g1;2a2;0- g2;1a1;1)
1v2
+ t(t - 1)(t - 2)ut-33g1;2g2;1g3;0
Hence the image appears to be rather complicated.
On the other hand, it appears likely that all of the ai;jare in the image. Gi*
*ven
x 2 BP*[s1; : :]: Q, let x(j)denote the expression obtained from x by replacing
each vk and sk by its pjth power. Using chromatic notation, we conjecture that
X (pi-1`kspk )(j+1)
Ai;j= ________i-k____i
0k*