Non-nilpotent elements in motivic homotopy theory
Plan
1. The Adams-Novikov spectral sequence
2. The stable motivic homotopy category
3. The motivic Adams-Novikov spectral sequence
4. η −1 π∗,∗ (S 0,0 )
5. A square of spectral sequences
6. Chromatic motivic homotopy theory
The Adams-Novikov spectral sequence
s
H s,u (BP∗ BP) =⇒ πu−s (S 0 ) ⊗ Z(2) .
BP∗ BP = π∗ (BP ∧ BP),
BP = the 2-local Brown-Peterson spectrum.
H(BP∗ BP) = cohomology of the Hopf algebroid BP∗ BP
with coefficents in BP∗ , a Z(2) -algebra.
π∗ (S 0 ) is filtered.
F s πu−s (S 0 )/F s+1 πu−s (S 0 ) approximated by H s,u (BP∗ BP).
The Adams-Novikov spectral sequence
s
Plot H s,u (BP∗ BP) in the (u − s, s)-plane.
8
Nodes will indicate generators for Z(2) -modules.
6
4
2
0
0
2
4
6
8
10
12
14
u−s
The Adams-Novikov spectral sequence
s
αn ∈ H 1,2n (BP∗ BP).
8
6
4
2
α1
0
0
α2
2
α3
4
α4
6
α5
8
α6
10
α7
12
α8
14
u−s
The Adams-Novikov spectral sequence
s
αn ∈ H 1,2n (BP∗ BP).
Generate an algebra.
8
6
Round nodes indicate Z/2.
Square nodes labelled with n 6= ∞ indicate Z/2n .
4
2
α22
2
α1
0
α24
4
α2
α3
3
α4
α5
5
α6
α7
α8
∞
0
2
4
6
8
10
12
14
u−s
The Adams-Novikov spectral sequence
s
8
αn ∈ H 1,2n (BP∗ BP).
Generate an algebra.
6
Lines of slope 1: α1 -multiplication.
4
2
α22
2
α1
0
α24
4
α2
α3
3
α4
α5
5
α6
α7
α8
∞
0
2
4
6
8
10
12
14
u−s
The Adams-Novikov spectral sequence
s
All of H(BP∗ BP) in given range.
8
6
4
2
α22
2
α1
0
α24
4
α2
α3
3
α4
α5
5
α6
α7
α8
∞
0
2
4
6
8
10
12
14
u−s
The Adams-Novikov spectral sequence
s
/ η;
α1
α2
α4
8
6
/ ν;
/ σ.
4
2
α22
2
α1
0
α24
4
α2
α3
3
α4
α5
5
α6
α7
α8
∞
0
2
4
6
8
10
12
14
u−s
The Adams-Novikov spectral sequence
s
/ η.
α1
8
η 4 = 0, but αn1 6= 0 for all n.
6
4
2
α22
2
α1
0
α24
4
α2
α3
3
α4
α5
5
α6
α7
α8
∞
0
2
4
6
8
10
12
14
u−s
The Adams-Novikov spectral sequence
s
/ η.
α1
8
η 4 = 0, but αn1 6= 0 for all n.
6
Even worse: for k 6= 2, αn1 αk 6= 0 for all n.
4
2
α22
2
α1
0
α24
4
α2
α3
3
α4
α5
5
α6
α7
α8
∞
0
2
4
6
8
10
12
14
u−s
The Adams-Novikov spectral sequence
s
Family of differentials (Novikov) resolve this conflict:
8
d3 α4k−1 = α31 α4k−3 ,
d3 α4k+2 = α31 α4k ,
k ≥ 1.
6
4
d3
2
α22
2
α1
0
α24
4
α2
α3
3
α4
α5
5
α6
α7
α8
∞
0
2
4
6
8
10
12
14
u−s
The Adams-Novikov spectral sequence
s
See 2α1 = 0, so 2η = 0.
See 16α4 = 0, so 16σ = 0.
8
/ µ9 = h8σ, 2, ηi.
α5 = h8α4 , 2, α1 i
6
4
d3
2
α22
2
α1
0
α24
4
α2
α3
3
α4
α5
5
α6
α7
α8
∞
0
2
4
6
8
10
12
14
u−s
The (motivic) Adams-Novikov spectral sequence
s
8
6
4
d3
2
α22
2
α1
0
α24
4
α2
α3
3
α4
α5
5
α6
α7
α8
∞
0
2
4
6
8
10
12
14
u−s
The Adams-Novikov spectral sequence
η 4 = 0 but αn1 6= 0 for all n.
=⇒ the ANSS is a poor tool for computing π∗ (S 0 ) ⊗ Z(2) ?
NO!
This arises due to the relationship between the ANSS and the
motivic ANSS.
Motivic homotopy theory
The spaces of topology + the schemes of algebraic geometry
/ motivic homotopy theory.
Have usual topological 1-sphere S 1,0 = S 1
and the algebraic 1-sphere S 1,1 = A1 − 0.
Must specify a ground field. For us it’s C.
New question: what is π∗,∗ (S 0,0 ) ⊗ Z(2) ?
Second grading is called weight.
Realization functor X 7→ X (C) returns classical story.
Motivic η
Recall, classically
η : S 3 ⊂ C2 − 0 → P1 (C) = S 2
so η ∈ π1 (S 0 ).
Motivically,
η : S 3,2 = A2 − 0 → P1 = S 2,1
so η ∈ π1,1 (S 0,0 ).
Remarkably η n 6= 0 for all n in the motivic story.
This is why αn1 6= 0 for all n.
The motivic Adams-Novikov spectral sequence
Non-zero elements of H s,u (BP∗ BP) have u even and so we can
assign them a weight w = u/2.
The motivic Adams-Novikov spectral sequence takes the form
s
H s,u (BP∗ BP)[τ ]w =⇒ πu−s,w (S 0,0 ) ⊗ Z(2)
where |τ | = (0, 0, −1).
Classical differentials give motivic differentials:
d2n+1 x = y
/ d2n+1 x = τ n y .
The motivic Adams-Novikov spectral sequence
s
α4 and α5 detect
σ ∈ π7,4 (S 0,0 ) and µ9 ∈ π9,5 (S 0,0 ).
η n σ 6= 0 and η n µ9 6= 0 for all n.
8
6
4
d3
2
α22
2
α1
0
α24
4
α2
α3
3
α4
α5
5
α6
α7
α8
∞
0
2
4
6
8
10
12
14
u−s
The η-local homotopy of the motivic sphere
Theorem (A., Miller)
η −1 π∗,∗ (S 0,0 ) = F2 [η ±1 , σ, µ9 ]/(ησ 2 ),
where η ∈ π1,1 (S 0,0 ) and σ ∈ π7,4 (S 0,0 ) are motivic Hopf invariant
one elements and µ9 = h8σ, 2, ηi ∈ π9,5 (S 0,0 ) is detected by α5 in
the motivic Adams-Novikov spectral sequence.
Key computation: α−1
1 H(BP∗ BP)
Proposition (A., Miller)
H(BP∗ BP) → α−1
1 H(BP∗ BP)
is an isomorphism above a line of slope 1/5 in (u − s, s)
coordinates and
±1
2
α−1
1 H(BP∗ BP) = F2 [α1 , α3 , α4 ]/(α1 α4 ).
The Adams-Novikov E2 -page
s
8
6
4
2
α22
2
α1
0
α24
4
α2
α3
3
α4
α5
5
α6
α7
α8
∞
0
2
4
6
8
10
12
14
u−s
My secret: a square of SSs
Classically, one has a diagram for each prime p:
H(P; Q)
alg-Nov-SS
H(BP∗ BP)
CESS
ANSS
+3 H(A)
ASS
+3 π∗ (S 0 )
The Hopf algebra P and the P-comodule Q
P = Fp [ξ1 , ξ2 , ξ3 , . . .], |ξn | = 2p n − 2, with Milnor diagonal:
ξn 7→
X
j
ξip ⊗ ξj .
i+j=n
Q = Fp [q0 , q1 , q2 , . . .], |qn | = 2p n − 2, with coaction map similar
to the Milnor diagonal:
X pj
qn 7→
ξi ⊗ qj .
i+j=n
The classical chromatic story and Q
The classical chromatic story is loaded up in Q:
qn ∈ Q
/ vn ∈ BP∗ = Z(p) [v1 , v2 , . . .].
Everything one does with BP∗
e.g. BP∗ /(p, v1 , . . . , vn−1 ), vn−1 BP∗ /(p, v1 , . . . , vn−1 ),
∞ )
vn−1 BP∗ /(p ∞ , v1∞ , . . . , vn−1
one can do with Q and one has appropriate algebraic Novikov SSs.
Miller’s proof of the telescope conjecture for height 1
H(P; q1−1 Q/(q0 ))
alg-Nov-SS
H(BP∗ BP; v1−1 BP∗ /p)
CESS
ANSS
+3 q −1 H(A; H∗ (S/p))
1
ASS
+3 v −1 π∗ (S/p)
1
Miller computes this square to obtain:
Theorem (Miller)
v1−1 π∗ (S/p) = Fp [v1±1 ] ⊗ E [h] where
h : S 2p−3
v1
/ Σ−1 S/p
Bockstein
/ S/p.
Uses of the square by Ravenel and myself
Ravenel: TELESCOPE CONJECTURE at height 2.
Tried to DISPROVE by computing the corresponding
square for v2−1 S/(p, v1 ).
My thesis: understanding the square for v1−1 S/p ∞ ,
the analog of Miller’s work for the SPHERE.
Theorem (A.)
For odd primes, there is an ASS for π∗ (v1−1 S/p ∞ ) with E2 -page:
E2-page for the v1-periodic sphere at p=7
500
400
s
300
200
100
0
0
1000
2000
3000
4000
5000
6000
t-s
We understand the differentials in this spectral sequence and above
the red line it coincides with the classical ASS for the sphere.
Motivicifying the square
H(P; Q)[τ ]
alg-Nov-SS[τ ]
H(BP∗ BP)[τ ]
CESS
MANSS
+3 H(Amot )
MASS
+3 π∗,∗ (S 0,0 )
We localized with respect to an element h0 ∈ H(P; Q)[τ ] and
completely described the square.
We did not only compute η −1 π∗,∗ (S 0,0 ) but also the localized
motivic ASS converging to it, confirming a conjecture of
Guillou-Isaksen.
A new chromatic story with P?
h0 ∈ H(P; Q) corresponds to ξ1 ∈ P.
Classically, the chromatic story is governed by p, v1 , v2 , . . . ∈ BP∗
and thus, q0 , q1 , q2 , . . . ∈ Q.
Motivically, there may be other periodicity operators corresponding
to ξ1 , ξ2 , ξ3 , . . . ∈ P.
I want to call them η0 , η1 , η2 , . . .
Mahowald already has the ηj family so I’ll go with w0 , w1 , w2 , . . .
Slopes
qn / v n
ξn+1 / wn
ASS
1/(2p n − 2)
1/(2p n+1 − 3)
ANSS
0
n+1
1/(2p
− 3)
The chromatic approach to classical homotopy theory
36
18
The E∞ -page of the classical Adams spectral sequence
34
17
P 8 h1
32
P 8 h2
16
P 7 c0
30
15
P 7 h1
28
P 7 h2
14
P 6 c0
26
13
h25
0 h6
P 6 h1
24
P 4 h20 i
P 6 h2
12
P 5 c0
22
11
P 5 h1
20
P 5 h2
10
h70 Q!
P 4 c0
18
9
P 4 h1
16
P 2 h20 i
P 4 h2
d20 u
8
d20 e20
P 3 c0
d0 e0 m
d20 l
14
7
P 3 h1
12
d30
d20
P d0
P h5 c 0
q
B1
P h1 h5
B21
n
P h2
x
B23
B5 + D2!
X3
X2
A!
h5 d 0
q1
E1 + C 0
B3
C
h3 d 1
R
h1 X1
h1 Q2
B2
P h2 h5
h30 G21
h5 n
h3 (E1 + C0 )
D3!
h3 Q2
C!
h22 H1
h 3 A!
k1
h1 H1
d1 e1
h1 h3 H1
h2 Q3
r1
Q3
h3 g2
2
d0
g
h30 h4
h23
h1
h2
h4 c 0
d1
c1
1
h0
0
t
h0 h4 x !
P h5 j
h0 h5 i
d1 g
N
g2
c1 g
c0
2
x!
gn
w
e20
3
P h1
4
g3
C11
e0 r
z
u
h20 i
P 2 h2
4
P c0
6
e20 g
e0 m
d0 l
5
P 2 h1
8
gw
d0 u
d0 e20
h10
0 h5
P 2 c0
10
d0 w
Pu
P 3 h2
6
h1 W1
d0 x! e0 gr
d0 e0 r
h1 h4
p
h20 h3 h5
h22 h5
h0 h2 h5
h24
h2 h4
h5 c 0
g2
f1
D3
h5 c 1
d2
h23 h5
h1 h3 h5
p!
p1
h22 h6
h25
h1 h5
h1 h6
h3 h6
h3
0
0
0
1
2
2
4
3
6
4
8
5
10
6
12
7
14
8
16
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13
26
14
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15
30
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32
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30
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31
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32
64
33
66
34
68
35
70
The chromatic approach to classical homotopy theory
36
18
The E∞ -page of the classical Adams spectral sequence
34
17
P 8 h1
32
P 8 h2
16
P 7 c0
30
15
P 7 h1
28
P 7 h2
14
P 6 c0
26
13
h25
0 h6
P 6 h1
24
22
v0
11
P 5 c0
P 5 h1
20
P 4 h20 i
P 6 h2
12
P 5 h2
10
h70 Q!
P 4 c0
18
9
P 4 h1
16
P 2 h20 i
P 4 h2
d20 u
8
d20 e20
P 3 c0
d0 e0 m
d20 l
14
7
P 3 h1
12
d30
d20
P d0
P h5 c 0
q
B1
P h1 h5
B21
n
P h2
x
B23
B5 + D2!
X3
X2
A!
h5 d 0
q1
E1 + C 0
B3
C
h3 d 1
R
h1 X1
h1 Q2
B2
P h2 h5
h30 G21
h5 n
h3 (E1 + C0 )
D3!
h3 Q2
C!
h22 H1
h 3 A!
k1
h1 H1
d1 e1
h1 h3 H1
h2 Q3
r1
Q3
h3 g2
2
d0
g
h30 h4
h23
h1
h2
h4 c 0
d1
c1
1
h0
0
t
h0 h4 x !
P h5 j
h0 h5 i
d1 g
N
g2
c1 g
c0
2
x!
gn
w
e20
3
P h1
4
g3
C11
e0 r
z
u
h20 i
P 2 h2
4
P c0
6
e20 g
e0 m
d0 l
5
P 2 h1
8
gw
d0 u
d0 e20
h10
0 h5
P 2 c0
10
d0 w
Pu
P 3 h2
6
h1 W1
d0 x! e0 gr
d0 e0 r
h1 h4
p
h20 h3 h5
h22 h5
h0 h2 h5
h24
h2 h4
h5 c 0
g2
f1
D3
h5 c 1
d2
h23 h5
h1 h3 h5
p!
p1
h22 h6
h25
h1 h5
h1 h6
h3 h6
h3
0
0
0
1
2
2
4
3
6
4
8
5
10
6
12
7
14
8
16
1. 2 : S 0 → S 0 ,
9
18
10
20
11
22
12
24
13
26
14
28
15
30
16
32
17
34
18
36
19
38
20
40
21
42
22
44
23
46
24
48
25
50
26
52
27
54
28
56
π∗ (2−1 S 0 );
29
58
30
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31
62
32
64
33
66
34
68
35
70
The chromatic approach to classical homotopy theory
36
18
The E∞ -page of the classical Adams spectral sequence
34
17
P 8 h1
32
P 8 h2
16
P 7 c0
30
15
P 7 h1
28
v1
P 7 h2
14
P 6 c0
26
13
h25
0 h6
P 6 h1
24
22
v0
11
P 5 c0
P 5 h1
20
P 4 h20 i
P 6 h2
12
P 5 h2
10
h70 Q!
P 4 c0
18
9
P 4 h1
16
P 2 h20 i
P 4 h2
d20 u
8
d20 e20
P 3 c0
d0 e0 m
d20 l
14
7
P 3 h1
12
d30
d20
P d0
P h5 c 0
q
B1
P h1 h5
B21
n
P h2
x
B23
B5 + D2!
X3
X2
A!
h5 d 0
q1
E1 + C 0
B3
C
h3 d 1
R
h1 X1
h1 Q2
B2
P h2 h5
h30 G21
h5 n
h3 (E1 + C0 )
D3!
h3 Q2
C!
h22 H1
h 3 A!
k1
h1 H1
d1 e1
h1 h3 H1
h2 Q3
r1
Q3
h3 g2
2
d0
g
h30 h4
h23
h1
h2
h4 c 0
d1
c1
1
h0
0
t
h0 h4 x !
P h5 j
h0 h5 i
d1 g
N
g2
c1 g
c0
2
x!
gn
w
e20
3
P h1
4
g3
C11
e0 r
z
u
h20 i
P 2 h2
4
P c0
6
e20 g
e0 m
d0 l
5
P 2 h1
8
gw
d0 u
d0 e20
h10
0 h5
P 2 c0
10
d0 w
Pu
P 3 h2
6
h1 W1
d0 x! e0 gr
d0 e0 r
h1 h4
p
h20 h3 h5
h22 h5
h0 h2 h5
h24
h2 h4
h5 c 0
g2
f1
D3
h5 c 1
d2
h23 h5
h1 h3 h5
p!
p1
h22 h6
h25
h1 h5
h1 h6
h3 h6
h3
0
0
0
1
2
2
4
3
6
4
8
5
10
6
12
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14
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13
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14
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32
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38
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40
21
42
22
44
23
46
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48
25
50
26
52
27
54
28
56
1. 2 : S 0 → S 0 ,
π∗ (2−1 S 0 );
2. v14 : S/2 → Σ−8 S/2,
π∗ (v1−1 S/2);
29
58
30
60
31
62
32
64
33
66
34
68
35
70
The chromatic approach to classical homotopy theory
36
18
The E∞ -page of the classical Adams spectral sequence
34
17
P 8 h1
32
P 8 h2
16
P 7 c0
30
15
P 7 h1
28
v1
P 7 h2
14
P 6 c0
26
13
h25
0 h6
P 6 h1
24
22
v0
11
P 5 c0
P 5 h1
20
P 4 h20 i
P 6 h2
12
P 5 h2
10
h70 Q!
P 4 c0
18
9
P 4 h1
16
P 2 h20 i
P 4 h2
d20 u
8
d20 e20
P 3 c0
d0 e0 m
d20 l
14
7
P 3 h1
12
d30
d20
P d0
P h5 c 0
q
t
P h1 h5
n
P h2
x
B21
v2
R
h1 X1
h30 G21
B23
B5 + D2!
X3
B3
X2
A!
C
h5 d 0
q1
E1 + C 0
h1 Q2
B2
P h2 h5
h3 d 1
h0 h4 x !
P h5 j
h5 n
h3 (E1 + C0 )
D3!
h3 Q2
C!
h22 H1
h 3 A!
k1
h1 H1
d1 e1
h1 h3 H1
h2 Q3
r1
Q3
h3 g2
2
d0
g
h30 h4
h23
h1
h2
h4 c 0
d1
c1
1
h0
0
B1
h0 h5 i
d1 g
N
g2
c1 g
c0
2
x!
gn
w
e20
3
P h1
4
g3
C11
e0 r
z
u
h20 i
P 2 h2
4
P c0
6
e20 g
e0 m
d0 l
5
P 2 h1
8
gw
d0 u
d0 e20
h10
0 h5
P 2 c0
10
d0 w
Pu
P 3 h2
6
h1 W1
d0 x! e0 gr
d0 e0 r
h1 h4
p
h20 h3 h5
h22 h5
h0 h2 h5
h24
h2 h4
h5 c 0
g2
f1
D3
h5 c 1
d2
h23 h5
h1 h3 h5
p!
p1
h22 h6
h25
h1 h5
h1 h6
h3 h6
h3
0
0
0
1
2
2
4
3
6
4
8
5
10
6
12
7
14
8
16
9
18
10
20
11
22
12
24
13
26
14
28
15
30
16
32
17
34
18
36
19
38
20
40
21
42
22
44
23
46
24
48
25
50
26
52
27
54
28
56
29
58
30
60
31
62
1. 2 : S 0 → S 0 ,
π∗ (2−1 S 0 );
2. v14 : S/2 → Σ−8 S/2,
π∗ (v1−1 S/2);
3. v232 : S/(2, v14 ) → Σ−192 S/(2, v14 ),
π∗ (v2−1 S/(2, v14 )).
32
64
33
66
34
68
35
70
The chromatic approach to motivic homotopy theory
8
36
18
The E∞ -page of the motivic Adams spectral sequence
34
17
P 8 h1
32
P 8 h2
16
P 7 c0
30
15
P 7 h1
28
v1
P 7 h2
14
P 6 c0
26
13
h25
0 h6
P 6 h1
24
22
v0
11
P 5 c0
P 5 h1
20
P 4 h20 i
P 6 h2
12
P 5 h2
10
h70 Q!
P 4 c0
18
9
P 4 h1
16
P 2 h20 i
P 4 h2
d20 u
8
e40
d20 e20
P 3 c0
τ 2 d0 e0 m
d0 u! + τ d20 l
14
7
d0 z
P 3 h1
12
6
d30
h10
0 h5
P 2 c0
10
d20
P d0
τ g3 + h41 h5 c0 e0
h61 h5 e0
h1 G 3
gn
P h5 c 0
q
h1 h3 g
x!
B8
gt
h0 h5 i
d1 g
N
B1
B21
D11
v2
h3 d 1 g
h0 g
P h2
t
n
h2 g
P h1 h5
x
h3 d 1
h5 d 0
h2 C !!
q1 τ B23 c0 Q2
τ 2 B5 + D2!
h21 X2
B3
τ X2
A!
h23 g2
C
P h2 h5
R
τ h1 X1
h5 n
h22 C !
h3 (E1 + C0 )
X3
τ D3!
d2
h3 Q2 1
E1 + C 0
j1
C11
τ h30 G21
h2 B23
h2 B22
h31 D4
h1 Q2
i1
B2
h0 h4 x !
P h5 j
τ h21 X1
c1 g 2
C!
h22 H1
h 3 A!
k1
τ h1 H1
r1
d1 e1
τ h1 h3 H1
h2 Q3
h0 Q3
τ Q3
h3 g2
2
d0
τg
h30 h4
c0
h23
h1
h2
h4 c 0
d1
c1
1
h0
0
h1 h3 g 2
τw
h2 g 2
τ g2
c1 g
τ h1 W1
h1 h3 g 3
B8 d0
τ gw + h41 X1
h2 g 3
3
P h1
2
h0 g 2
e20
c0 d 0
4
τ e20 g
τ 2 e0 m
e0 r
z
u
h20 i
P 2 h2
4
P c0
6
h0 g 3
P h31 h5 e0
d0 e20
τ d0 l + u!
c0 e20
h2 e20 g
d0 u
5
P 2 h1
8
τ d0 w
Pu
P 3 h2
d0 x! e0 gr
d0 e0 r
h1 h4
p
h20 h3 h5
h22 h5
h0 h2 h5
h24
h2 h4
h5 c 0
g2
f1
D3
h5 c 1
d2
h23 h5
h1 h3 h5
p!
p1
h22 h6
h25
h1 h5
h1 h6
h3 h6
h3
0
0
0
1
2
2
4
3
6
4
8
5
10
6
12
7
14
8
16
9
18
10
20
11
22
12
24
1. π∗,∗ (2−1 S 0,0 );
2. π∗,∗ (v1−1 S/2);
3. π∗,∗ (v2−1 S/(2, v14 )).
13
26
14
28
15
30
16
32
17
34
18
36
19
38
20
40
21
42
22
44
23
46
24
48
25
50
26
52
27
54
28
56
29
58
30
60
31
62
32
64
33
66
34
68
35
70
But is there more?
8
36
18
The E∞ -page of the motivic Adams spectral sequence
34
17
P 8 h1
32
P 8 h2
16
P 7 c0
30
15
P 7 h1
28
v1
P 7 h2
14
P 6 c0
26
13
h25
0 h6
P 6 h1
24
22
v0
11
w0
P 5 c0
P 5 h1
20
P 4 h20 i
P 6 h2
12
P 5 h2
10
h70 Q!
P 4 c0
18
9
P 4 h1
16
P 2 h20 i
P 4 h2
d20 u
8
e40
d20 e20
P 3 c0
τ 2 d0 e0 m
d0 u! + τ d20 l
14
7
d0 z
P 3 h1
12
6
d30
h10
0 h5
P 2 c0
10
d20
P d0
τ g3 + h41 h5 c0 e0
h61 h5 e0
h1 G 3
gn
P h5 c 0
q
h1 h3 g
x!
B8
gt
h0 h5 i
d1 g
N
B1
B21
D11
v2
h3 d 1 g
h0 g
P h2
t
n
h2 g
P h1 h5
x
h3 d 1
h5 d 0
h2 C !!
q1 τ B23 c0 Q2
τ 2 B5 + D2!
h21 X2
B3
τ X2
A!
h23 g2
C
P h2 h5
R
τ h1 X1
h5 n
h22 C !
h3 (E1 + C0 )
X3
τ D3!
d2
h3 Q2 1
E1 + C 0
j1
C11
τ h30 G21
h2 B23
h2 B22
h31 D4
h1 Q2
i1
B2
h0 h4 x !
P h5 j
τ h21 X1
c1 g 2
C
!
h22 H1
h 3 A!
k1
τ h1 H1
r1
d1 e1
τ h1 h3 H1
h2 Q3
h0 Q3
τ Q3
h3 g2
2
d0
τg
h30 h4
c0
h23
h1
h2
h4 c 0
d1
c1
1
h0
0
h1 h3 g 2
τw
h2 g 2
τ g2
c1 g
τ h1 W1
h1 h3 g 3
B8 d0
τ gw + h41 X1
h2 g 3
3
P h1
2
h0 g 2
e20
c0 d 0
4
τ e20 g
τ 2 e0 m
e0 r
z
u
h20 i
P 2 h2
4
P c0
6
h0 g 3
P h31 h5 e0
d0 e20
τ d0 l + u!
c0 e20
h2 e20 g
d0 u
5
P 2 h1
8
τ d0 w
Pu
P 3 h2
d0 x! e0 gr
d0 e0 r
h1 h4
p
h20 h3 h5
h22 h5
h0 h2 h5
h24
h2 h4
h5 c 0
g2
f1
D3
h5 c 1
d2
h23 h5
h1 h3 h5
p!
p1
h22 h6
h25
h1 h5
h1 h6
h3 h6
h3
0
0
0
1
2
2
4
3
6
4
8
5
10
6
12
7
14
8
16
9
18
10
20
11
22
12
24
13
26
14
28
15
30
16
32
1. w0 = η : S 0,0 → Σ−1,−1 S 0,0 .
17
34
18
36
19
38
20
40
21
42
22
44
23
46
24
48
25
50
26
52
27
54
28
56
29
58
30
60
31
62
32
64
33
66
34
68
35
70
But is there more?
8
36
18
The E∞ -page of the motivic Adams spectral sequence
34
17
P 8 h1
32
P 8 h2
16
P 7 c0
30
15
P 7 h1
28
v1
P 7 h2
14
P 6 c0
26
13
h25
0 h6
P 6 h1
24
22
v0
11
µ9
w0
P 5 c0
P 5 h1
20
P 4 h20 i
P 6 h2
12
P 5 h2
10
h70 Q!
P 4 c0
18
9
P 4 h1
16
P 2 h20 i
P 4 h2
d20 u
8
e40
d20 e20
P 3 c0
τ 2 d0 e0 m
d0 u! + τ d20 l
14
7
d0 z
P 3 h1
12
6
d30
h10
0 h5
P 2 c0
10
d20
P d0
τ g3 + h41 h5 c0 e0
h61 h5 e0
h1 G 3
gn
P h5 c 0
q
h1 h3 g
x!
B8
gt
h0 h5 i
d1 g
N
B1
B21
D11
v2
h3 d 1 g
h0 g
P h2
t
n
h2 g
P h1 h5
x
h3 d 1
h5 d 0
h2 C !!
q1 τ B23 c0 Q2
τ 2 B5 + D2!
h21 X2
B3
τ X2
A!
h23 g2
C
P h2 h5
R
τ h1 X1
h5 n
h22 C !
h3 (E1 + C0 )
X3
τ D3!
d2
h3 Q2 1
E1 + C 0
j1
C11
τ h30 G21
h2 B23
h2 B22
h31 D4
h1 Q2
i1
B2
h0 h4 x !
P h5 j
τ h21 X1
c1 g 2
C
!
h22 H1
h 3 A!
k1
τ h1 H1
r1
d1 e1
τ h1 h3 H1
h2 Q3
h0 Q3
τ Q3
h3 g2
2
d0
τg
h30 h4
c0
h23
h1
h2
h4 c 0
d1
c1
1
h0
0
h1 h3 g 2
τw
h2 g 2
τ g2
c1 g
τ h1 W1
h1 h3 g 3
B8 d0
τ gw + h41 X1
h2 g 3
3
P h1
2
h0 g 2
e20
c0 d 0
4
τ e20 g
τ 2 e0 m
e0 r
z
u
h20 i
P 2 h2
4
P c0
6
h0 g 3
P h31 h5 e0
d0 e20
τ d0 l + u!
c0 e20
h2 e20 g
d0 u
5
P 2 h1
8
τ d0 w
Pu
P 3 h2
d0 x! e0 gr
d0 e0 r
h1 h4
p
h20 h3 h5
h22 h5
h0 h2 h5
h24
h2 h4
h5 c 0
g2
f1
D3
h5 c 1
d2
h23 h5
h1 h3 h5
p!
p1
h22 h6
h25
h1 h5
h1 h6
h3 h6
h3
0
0
0
1
2
2
4
3
6
4
8
5
10
6
12
7
14
8
16
9
18
10
20
11
22
12
24
13
26
14
28
15
30
16
32
17
34
18
36
19
38
20
40
21
42
22
44
23
46
24
48
25
50
26
52
27
54
28
56
29
58
30
60
1. w0 = η : S 0,0 → Σ−1,−1 S 0,0 .
2. µ9 ∈ π9,5 (S 0,0 ) is v1 and w0 -periodic. It fills the gap.
31
62
32
64
33
66
34
68
35
70
But is there more?
8
36
18
The E∞ -page of the motivic Adams spectral sequence
34
17
P 8 h1
32
P 8 h2
16
P 7 c0
30
15
P 7 h1
28
v1
P 7 h2
14
P 6 c0
26
13
h25
0 h6
P 6 h1
24
22
v0
11
µ9
w0
P 5 c0
P 5 h1
20
P 4 h20 i
P 6 h2
12
P 5 h2
10
h70 Q!
P 4 c0
18
9
P 4 h1
16
P 2 h20 i
P 4 h2
d20 u
8
e40
d20 e20
P 3 c0
τ 2 d0 e0 m
d0 u! + τ d20 l
14
7
d0 z
P 3 h1
12
6
d30
h10
0 h5
P 2 c0
10
d20
P d0
τ 2 e0 m
h1 h3 g 2
τ g3 + h41 h5 c0 e0
h61 h5 e0
h1 G 3
gn
P h5 c 0
q
h1 h3 g
x!
B8
gt
h0 h5 i
d1 g
N
B1
B21
D11
v2
h3 d 1 g
h0 g
P h2
t
n
h2 g
P h1 h5
x
h3 d 1
h5 d 0
h2 C !!
q1 τ B23 c0 Q2
τ 2 B5 + D2!
h21 X2
B3
τ X2
A!
h23 g2
C
P h2 h5
R
τ h1 X1
h5 n
h22 C !
h3 (E1 + C0 )
X3
τ D3!
d2
h3 Q2 1
E1 + C 0
j1
C11
τ h30 G21
h2 B23
h2 B22
h31 D4
h1 Q2
i1
B2
h0 h4 x !
P h5 j
τ h21 X1
c1 g 2
C
!
h22 H1
h 3 A!
k1
τ h1 H1
r1
d1 e1
τ h1 h3 H1
h2 Q3
h0 Q3
τ Q3
h3 g2
2
d0
τg
h30 h4
c0
h23
h1
h2
h4 c 0
d1
c1
1
h0
0
τ e20 g
w1
τw
h2 g 2
τ g2
c1 g
τ h1 W1
h1 h3 g 3
B8 d0
τ gw + h41 X1
h2 g 3
3
P h1
2
h0 g 2
e20
c0 d 0
4
e0 r
z
u
h20 i
P 2 h2
4
P c0
6
h0 g 3
P h31 h5 e0
d0 e20
τ d0 l + u!
c0 e20
h2 e20 g
d0 u
5
P 2 h1
8
τ d0 w
Pu
P 3 h2
d0 x! e0 gr
d0 e0 r
h1 h4
p
h20 h3 h5
h22 h5
h0 h2 h5
h24
h2 h4
h5 c 0
g2
f1
D3
h5 c 1
d2
h23 h5
h1 h3 h5
p!
p1
h22 h6
h25
h1 h5
h1 h6
h3 h6
h3
0
0
0
1
2
2
4
3
6
4
8
5
10
6
12
7
14
8
16
9
18
10
20
11
22
12
24
13
26
14
28
15
30
16
32
17
34
18
36
19
38
20
40
21
42
22
44
23
46
24
48
25
50
26
52
27
54
28
56
29
58
30
60
1. w0 = η : S 0,0 → Σ−1,−1 S 0,0 .
2. µ9 ∈ π9,5 (S 0,0 ) is v1 and w0 -periodic. It fills the gap.
3. w14 : S/η → Σ−20,−12 S/η?
31
62
32
64
33
66
34
68
35
70
But is there more? A new constellation
The α family of the Adams-Novikov spectral sequence
6
4
2
3
w0
2
1
α1
0
0
0
12
4
α4/4
α3
α2/2
α6/3
α5
α6/2
5
α8/5
α7
α10/3
α9
α10/2
4
α12/4
α11
α14/3
α13
α14/2
6
α16/6
α15
α18/3
α17
α18/2
4
α20/4
α19
α22/3
α21
α22/2
5
α24/5
α23
α26/3
α25
α26/2
4
α28/4
α27
α30/3
α29
α30/2
0
1
2
2
4
3
6
4
8
5
10
6
7
12
14
8
16
9
10
18
20
11
12
22
24
13
26
14
15
28
16
30
17
32
34
18
19
36
20
38
21
40
22
42
23
44
24
46
25
48
26
50
27
52
28
54
29
56
30
58
6
z59,11
10
5
z54,10
z48
z43
8
4
z40,8
w1
6
z59,9
z52,8
z39,7
3
z51,7
z34,6
z28
z40,6
a59,7
z53,7
z52,6
b50
!
z59,7
z59,7
z58,6
b58
a54
a60,6
z47,5
z23,5
4
0
z53,5
2
z20,4
1
z6
z14
z8
!
z14
z16
z18
z36
z32,4
z19
2
z39,5
z37
z31
z23,3
z30
b30
z20,2
z40,4
z32,2
!
z32,2
z34,2
a34
a36
z39,3
!
z37
z38
b38
!
z39,3
!!
z39,3
b44
z44
a41
z45
b43
z40,2
a42
z42
z50
z46
a47
!!
z45
!
z45
a55
z53,5
!
z50
z59,5
a!55
z47,3
a59,3
b57
c44
a50
z54,2
e56
60,4
ca60
z58,4
b56
a58
a60,2
0
0
0
1
2
2
4
3
6
4
8
5
10
6
12
7
14
8
16
9
18
10
20
11
22
12
24
13
26
14
28
15
30
16
32
17
34
18
36
19
38
20
40
21
42
22
44
23
46
24
48
25
50
26
52
27
54
28
56
29
58
30
60
But is there more? A new constellation
The α family of the Adams-Novikov spectral sequence
6
4
2
3
w0
2
1
α1
0
0
0
12
4
α4/4
α3
α2/2
α6/3
α5
α6/2
5
α8/5
α7
α10/3
α9
α10/2
4
α12/4
α11
α14/3
α13
α14/2
6
α16/6
α15
α18/3
α17
α18/2
4
α20/4
α19
α22/3
α21
α22/2
5
α24/5
α23
α26/3
α25
α26/2
4
α28/4
α27
α30/3
α29
α30/2
0
1
2
2
4
3
6
4
8
5
10
6
7
12
14
8
16
9
10
18
20
11
12
22
24
13
26
14
15
28
16
30
17
32
34
18
19
36
20
38
21
40
22
42
23
44
24
46
25
48
26
50
27
52
28
54
29
56
30
58
6
z59,11
10
5
z54,10
z48
z43
8
z59,9
4
z40,8
z52,8
z39,7
6
3
z51,7
z34,6
z28
z40,6
a59,7
z53,7
z52,6
b50
!
z59,7
z59,7
z58,6
b58
a54
a60,6
z47,5
z23,5
4
0
z53,5
2
z20,4
1
z6
z14
z8
!
z14
z16
z18
z36
z32,4
z19
2
z39,5
z37
z31
z23,3
z30
b30
z20,2
z40,4
z32,2
!
z32,2
z34,2
a34
a36
z39,3
!
z37
z38
b38
!
z39,3
!!
z39,3
b44
z44
a41
z45
b43
z40,2
a42
z42
z50
z46
a47
!!
z45
!
z45
a55
z53,5
!
z50
z59,5
a!55
z47,3
a59,3
b57
c44
a50
z54,2
e56
60,4
ca60
z58,4
b56
a58
a60,2
0
0
0
1
2
2
4
3
6
4
8
5
10
6
12
7
14
8
16
9
18
10
20
11
22
12
24
13
26
14
28
15
30
16
32
17
34
18
36
19
38
20
40
21
42
22
44
23
46
24
48
25
50
26
52
27
54
28
56
29
58
30
60
But is there more? A new constellation
The α family of the Adams-Novikov spectral sequence
6
4
2
3
w0
2
1
α1
0
0
0
12
4
α4/4
α3
α2/2
α6/3
α5
α6/2
5
α8/5
α7
α10/3
α9
α10/2
4
α12/4
α11
α14/3
α13
α14/2
6
α16/6
α15
α18/3
α17
α18/2
4
α20/4
α19
α22/3
α21
α22/2
5
α24/5
α23
α26/3
α25
α26/2
4
α28/4
α27
α30/3
α29
α30/2
0
1
2
2
4
3
6
4
8
5
10
6
7
12
14
8
16
9
10
18
20
11
12
22
24
13
26
14
15
28
16
30
17
32
34
18
19
36
20
38
21
40
22
42
23
44
24
46
25
48
26
50
27
52
28
54
29
56
30
58
6
z59,11
10
5
z54,10
z48
z43
8
4
z40,8
w1
6
z59,9
z52,8
z39,7
3
z51,7
z34,6
z28
z40,6
a59,7
z53,7
z52,6
b50
!
z59,7
z59,7
z58,6
b58
a54
a60,6
z47,5
z23,5
4
0
z53,5
2
z20,4
1
z6
z14
z8
!
z14
z16
z18
z36
z32,4
z19
2
z39,5
z37
z31
z23,3
z30
b30
z20,2
z40,4
z32,2
!
z32,2
z34,2
a34
a36
z39,3
!
z37
z38
b38
!
z39,3
!!
z39,3
b44
z44
a41
z45
b43
z40,2
a42
z42
z50
z46
a47
!!
z45
!
z45
a55
z53,5
!
z50
z59,5
a!55
z47,3
a59,3
b57
c44
a50
z54,2
e56
60,4
ca60
z58,4
b56
a58
a60,2
0
0
0
1
2
2
4
3
6
4
8
5
10
6
12
7
14
8
16
9
18
10
20
11
22
12
24
13
26
14
28
15
30
16
32
17
34
18
36
19
38
20
40
21
42
22
44
23
46
24
48
25
50
26
52
27
54
28
56
29
58
30
60
What powers of w1 and between what?
Classically, we have non-nilpotent self maps
v14 : Σ8 S/2 → S/2,
Motivically, we have...
Theorem (A.)
There is a non-nilpotent self map
w14 : Σ20,12 S/η → S/η.
What’s the map?
Σ20,12 S/η
S/η
O
S 20,12
η 2 η4
/ S 2,1
What’s the map?
Σ20,12 S/η
w14
O
S 20,12
η 2 η4
/ S/η
/ S 2,1
Orthogonality of the vn ’s and the wn ’s
Originally I expected a self map
Σ20,12 (S/2 ∧ S/η)/v1 → (S/2 ∧ S/η)/v1 .
I thought one would have to kill everything with higher slope than
w1 in the motivic ASS before defining a w1 -self map.
i
n−1
However, it is enough to kill p i0 , v1i1 , . . . , vn−1
to construct a vn -self
map.
That we only have to kill w0 to get w14 suggests we would only
in−1
to get a wn -self map.
have to kill w0i0 , . . . , wn−1
The two families of self maps are orthogonal in a sense.
The nilpotence theorem
Classically the nilpotence theorem says the following.
Theorem (Devinatz, Hopkins, Smith)
A non-nilpotent self map on a finite p-local spectrum induces a
non-zero homomorphism on BP-homology.
To have a nilpotence theorem motivically we’d need a spectrum N
such that N∗,∗ is a Z(p) [v1 , v2 , v3 , . . . , w0 , w1 , w2 , . . .] module and
each of 2, v1 , v2 , . . . , w0 , w1 , . . . acts nilpotently on N∗,∗ .
How would N see our elements?
Since 2η = 0 on S 0,0 but 2 is non-nilpotent on S/η,
we must have 2w0 · N∗,∗ = 0 and 2 · N∗ (S/w0 ) 6= 0.
Since µ9 is non-nilpotent v14 w0 should act non-nilpotently on N∗,∗ .
This second observation prevents us having
N∗,∗ = BP∗,∗ ⊕ Z/2[w0 , w1 , . . .]
so the two families are not completely orthogonal. Maybe
N∗,∗ = BP∗,∗ [w0 , w1 , . . .]/(2w0 and other relations)?
Since S 0,0 has maps inducing multiplication by v0 , w0 and v14 w0 ,
and S/w0 has maps inducing multiplication by v0 and w14 , even if
one had a nilpotence theorem, it is hard to imagine what the thick
subcategories of the stable motivic homotopy category are.
Odd primes: β
The first nontrivial element in the cokernel of J is β.
H s,u (BP∗ BP)
(
b=
s
(−1)i i p−i
i=1
i [t1 |t1 ]
Pp−1
β ∈ π2(p2 −p−1) (S 0 ).
β is nilpotent but b n 6= 0 for all n.
)
+3 πu−s (S 0 ) ⊗ Z(p)
/β
Odd primes: β motivically
Can use the classical ANSS to deduce the motivic ANSS.
Motivically, b detects a non-nilpotent element
β ∈ π2(p2 −p−1),p2 −p (S 0,0 )
and τ p
2 −2p+1
βp
2 −p+1
= 0.
Conjecture
2
τ p −2p β n 6= 0 for all n.
2
τ n β p −p 6= 0 for all n and p > 5.
Moreover, this remains true in S/(p, v1 ).
Try to compute β −1 π∗,∗ (S 0,0 ) or β −1 π∗,∗ (S/(p, v1 )).
First step: b −1 H(P; Q/(q0 , q1 )).
The end: thank you for listening