stable stems and the chow-novikov t-structure in motivic stable … · 2020. 5. 5. · d2e3 m p3il...

Stable stems and the Chow-Novikov t-structurein motivic stable homotopy category

Zhouli Xu

MIT

May 4, 2020

The Classical Adams Spectral Sequence

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 480

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

The classical Adams spectral sequence

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

36

38

40

42

44

46

48

h0 h1 h2 h3

c0

Ph1 Ph2

h23

d0

h4

Pc0

e0

P2h1

f0

c1

P2h2

g

Pd0

h4c0

i

P2c0

Pe0

P3h1

j

P3h2

d20

k

h24

∆h22

P2d0

h5

n

d0e0+h70h5

d1

∆h1h3

l

P3c0

p

P2e0

P4h1

e20

Pj

m

P4h2

t

Pd20

x

e0g

d0i

h3h5

e1

∆h23

P3d0

h5c0

∆h1d0

Pd0e0

P2i

f1

Ph1h5

g2

d0j

P4c0

c2

∆h20e0

P3e0

P5h1

Ph2h5

∆h1e0

d30

P2j

d0k

P5h2

g2

∆h22d0

P2d20

h23h5

h5d0

∆h1g

d20e0

Pd0i

Mh1

∆h2c1

d0l

i2

P4d0

Ph5c0

∆h22e0

∆h20iP∆h1d0

P2d0e0

h3c2

h5e0

Mh2

d0e20

Pd0j

P5c0

h5f0

d0m

ij

P4e0

P6h1

h5c1

C

∆h22g

P∆h1e0

Pd30

P3j

h3g2

gn

d0e0g

d20i

P6h2

D1

d1g

e0m

P∆h22d0

P3d20

Ph5d0

MP

∆h1d20

Pd20e0

P2d0i

G

h5i

∆2h22

e20g

d20j+P∆c0d0

P5d0

gm

il

P2∆h1d0

P3d0e0

P4i

Ph5e0

gt∆2h1h3

∆h1d0e0

d40

P2d0j

P6c0

Q2

h5j

e0g2

Pd0m

Pij

P5e0

P7h1

D2

∆h22d20

P2∆h1e0

P2d30

P4j

Md0

∆h1e20

d30e0

Pd20i

P7h2

Mh4

B4

g3

d20l

d0i2

P4d20

D3

A′A+A′

∆x

∆h22n

jm

P∆h1d20

P2d20e0

P3d0i

h25

H1

h5n

∆e1

∆e1+C0

Me0∆2h23

∆h1e0g

MP2h1

d20e20

Pd20j+P2∆c0d0

P2i2

P6d0

h6

C′ X2

d20m

d0ij

P3∆h1d0

P4d0e0

A′′

h3Q2

∆2h1h4

∆h22d0g

∆2h21d0

P∆h1d0e0

Pd40

P3d0j

P7c0

h3D2

Mg

Ph5j

∆h22m

∆2h0e0

d0e30

P2d0m

P2ij

P6e0

P8h1

∆1h23

G0

B5

B5 + D′2

nm

d0e0m

P∆h22d20

P3∆h1e0

P3d30

P5j

n1Q3

X3 C′′

∆2c1

lm MPd0

∆h1d30

Pd30e0

P2d20i

P8h2

d2

h3A′

∆g2

∆2h0g

e40

d30j

Pd0i2

P5d20

p′

h3H1

D′3+?h2G0

P (A+A′)

∆2h1g

d0gm

∆h20d20e0

MP3

P2∆h1d20

P3d20e0

P4d0i

h3h6

p1+h20h3h6

d1e1

MPe0

m2+∆2h21g

∆h1d20e0

P∆2h0d0

d50

P2d20j+P3∆c0d0

P7d0

h6c0

x71,6

d0Q2

∆2h4c0

∆2h2g

Mj

g2n

e30g

d30k

P2il

P4∆h1d0

P5d0e0

P6i

Ph1h6

h23D2

h4Q2+h23D2

d0D2

e0gm

d0im

P∆2h21d0

P2∆h1d0e0

P2d40

P4d0j

P8c0

h4D2

Md20

P∆2h0e0 ∆h1d0e20

d40e0

P2d20k

P3ij

P7e0

P9h1

h3n1Ph2h6

x74,8

d0B4

e20g2

d30l

P2im

P4∆h1e0

P4d30

P6j

h3d2

x75,7

Mh24

∆h3g2

g2m

d0jm

MP2d0

P∆h1d30

P2d30e0

P3d20i

P9h2

h4D3

x76,6

h23H1

x76,9

Md0e0

g2t

∆2d20

∆h1e30

MPi+∆2h0d20

d30e20

P2d20l

P3ik

P6d20

h23h6

h6d0

x77,7m1

x77,8

gQ2

M∆h1h3

e0B4

∆2h0ke0g

3

d30m

P2jm

P3∆h1d20

P4d20e0

P5d0i

h4h6

t1

x78,9

x78,10

e0A′

MP2e0

d0km

P∆h1d20e0

MP4h1

Pd50

P3d20j

P4i2

P8d0

x1

Ph6c0

∆2n

Me20

∆2d0e0

MPj+∆2h0d0e0

∆h1e20g

d20e30

P2d20m

P3il

P4∆h20i P5∆h1d0

P6d0e0

e2

h6e0

P2h1h6

∆2d1

gB4

∆3h1h3

∆2h0lg4

d20e0m

P2km

P2∆2h21d0

P3∆h1d0e0

P3d40

P5d0j

P9c0

h6f0

c1H1

h3x74,8

gA′

∆2p

∆h22gn

MPd20

∆h22e30

P2∆2h0e0

∆h1d40

Pd40e0

P3d20k

P4ij

P8e0

P10h1

h6c1

h4Q3

e1g2

gH1

P2h2h6

gC0

Me0g

∆2e20

Md0i+∆2h0e

20

∆h1e0g2+∆2h0e

20

d0e40

P2d0e0m

P3im

P5∆h1e0

P5d30

P7j

h6g+h2e2

gC′

∆2m

MP∆h22

d0e20m

MP3d0

d20il

P2∆h1d30

P3d30e0

P4d20i

P10h2

f2

Px76,6

∆2t

∆h20B4

M∆h1d0

MPd0e0

d0m2

∆h1d30e0

P∆2h0d20

d60

P3d20l

P4ik

P7d20

c3

x85,6

Ph6d0

∆2x

Mg2

∆2e0g

Md0j+∆2h0e0g

∆h22gm

e50

P∆2h1d20

d40k

MP5

P3jm

P4∆h1d20

P5d20e0

P6d0i

h4h6c0

h6i

gG0

∆2e1

∆3h23

B4j

gnm

e30m

MP3e0

∆h22d40+P∆

2h31d20

P2∆h1d20e0

∆2P3h0d0

P2d50

P4d20j+h60 ·∆2P3h0d0

P9d0

x87,7

gQ3

∆h1H1

P2h6c0

gC′′

M∆h1e0

∆3h1d0

Md30

∆h22e0g2

P∆2d0e0

MP2j

∆h1d20e

20

d50e0

Pd40i

P3d0ij

P6∆h1d0

P7d0e0

P8i

g22

Ph6e0

∆g2g

∆2f1+?∆g2g

P3h1h6

∆2g2

Md0k

∆2h0e0i

e40g

d40l

d30i2

P3∆2h21d0

P4∆h1d0e0

P4d40

P6d0j

P10c0

h6j

∆2c2

M∆h22d0

∆3h20e0

e20gm+∆3h31d0

MP2d20

∆h22d30e0

P3∆2h0e0

P∆h1d40

P2d40e0

P3d30i

P5ij

P9e0

P11h1

x90,10

M2

P3h2h6

M∆h1g

∆3h1e0

Md20e0

gm2

∆2d30

MPd0i

∆h1d0e30

d40e20

Pd40j

P4im

P6∆h1e0

P6d30

P8j

h24D3

x91,8

h6d20

x91,11

∆h22A′

Md0l+h60x91,11

g3n

∆2h0e0j

d0e0g3

d40m

MP4d0

d30ij

P3∆h1d30

P4d30e0

P5d20i

P11h2

g3

h6k

x92,10

∆h22H1

∆2g2

mQ2

M∆h22e0

∆3h22d0

MP∆h1d0

e0g2m

MP2d0e0

∆h22d20e

20

P2∆2d20

MP3i+P2∆2h0d20

P∆h1d30e0

Pd60

P4d20l

P5ik

P8d20

h24h6

∆h22h6

x93,8

∆h3H1

∆∆1e0

P2h6d0

tQ2

∆3h1g

Md0e20

∆2d20e0

MPd0j

∆h1e40

P∆2h0d0i

d30e30

d50i

P4jm

P5∆h1d20

P6d20e0

P7d0i

h6n

x94,8

h6d0e0

y94,10

∆2Mh1

∆3h2c1

Md0m

∆2d0l

MP∆h20e0

e20g3

∆2i2

d30e0m

MP4e0+P2∆2h21d

20

d30ik+P2∆2h21d

20

P3∆h1d20e0

MP6h1

P3d50

P5d20j+h130 ·∆2i2

P6i2

P10d0

h6d1

x95,7

∆h1h3h6

h6l

∆x71,6

P3h6c0

M∆h22g

∆3h22e0

MP∆h1e0

g3m

∆3h20i

P∆3h1d0

MPd30

∆h22d20e0g

P2∆2d0e0

MP3j+P2∆2h0d0e0

∆h1d50

Pd50e0

P4d20m

P5il

P7∆h1d0

P8d0e0+P6∆h50i

7

The Classical Adams E8-page

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 480

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

The E∞-page of the classical Adams spectral sequence

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

36

38

40

42

44

46

48

h0 h1 h2 h3

c0

Ph1 Ph2

h23

d0 h30h4

h1h4

Pc0

P2h1

h2h4

c1

P2h2

g

Pd0

h4c0

h20i

P2c0

P3h1 P3h2

d20

h24

n

h100 h5

h1h5

d1

∆h1h3

P3c0

p

P4h1

h0h2h5

d0g

P4h2

t

h22h5

x

h20h3h5

h1h3h5

h5c0

h3d1

∆h1d0

P2h20i

f1

Ph1h5

g2

P4c0

∆h20e0

P5h1

Ph2h5

d30

P5h2

g2

h23h5

h5d0

∆h1g

Mh1

∆h2c1

d0l

Ph5c0

∆h22e0

P∆h1d0

h70Q′

Mh2

d20g

P5c0

P6h1

h5c1

C

h3g2

gn

P6h2

d1g

e0m

MP

∆h1d20

h0h5i

e20g

P4h20i

P6c0

P7h1

h1Q2

Md0

∆h1d0g

P7h2

g3

d20l

h25

h5n

∆e1+C0

∆2h23

h1H1

X2+C′

h250 h6

h1h6

h2D3

h2A′

h3Q2

∆2h1h4

P7c0

Mg

Ph5j

∆h1g2

P8h1

r1

B5+D′2

d0e0m

Q3+n1

∆2c1

P8h2

h3A′

h22h6

p′

h3(∆e1+C0)

h30h3h6

h1h3H1

h1D′3

m2+∆2h21g

h1h3h6

h6c0

h1p1

h3A′′

h23Q2

d0Q2

∆2h4c0

∆2h2g

P6h20i

Ph1h6

h4Q2+h23D2

h0d0D2

e0gm

P8c0

h0h4D2

Md20

∆h1d0e20

P9h1

h3(Q3+n1)

Ph0h2h6

x74,8

e20g2

h0h3d2

P9h2

h4A

d1g2

x76,9

h23h6

h6d0

h1h4D3

m1

h1x76,6

h0x77,7

M∆h1h3

h60h4h6

x78,9

e0A′

h1h4h6

h2x76,6

Ph6c0

h1x78,10

∆2n

Me20

P4h70Q′

h1x1

c1A′

P2h1h6

∆2d1

g4

P9c0

h2h4h6

h23n1

gD3

∆2p

P10h1

h6c1

h25g

e1g2

P2h2h6

h6g+h2e2

gC′

P10h2

h2gD3

Px76,6

∆2t

M∆h1d0

h1f2

x85,6

h2h4Q3

Ph6d0

∆2x

Mg2

∆h1g3

h4h6c0

h30h6i

M∆h20e0

h21c3

x87,7

gQ3

∆h1H1

P2h6c0

∆2h3d1

P8h20i

h1x87,7 + g22

∆2f1+?∆g2g

P3h1h6

P10c0

P11h1

M2

P3h2h6

M∆h1g

gm2

h24D3

∆h22A′

Md0l+h60x91,11

P11h2

h0g3

h1x91,8

x92,10

∆2g2

M∆h22e0

MP∆h1d0

h30h24h6

h30 · ∆h22h6

P2h6d0

e0x76,9

Md0e20

h1h24h6

h21g3

h6n

x94,8

d1H1

y94,10

∆h1h3H1

∆2Mh1

∆3h2c1

h2g3

h6d1

∆h1h3h6

e1Q2

∆2Mh21+?M∆h22g

P3h6c0

g3m

P6h60Q′

8

Adams differentials

Theorem (Isaksen - Wang - X.)

Up to 6 differentials, we have complete info on E8 through 90.

§ 71-stem, d5 on h1p1,

§ 77-stem, d2 on x77,7,

§ 83-stem, d9 on h6g ` h2e2,§ 85-stem, d9 on x85,6,

§ 87-stem, d6 on ∆h1H1,

§ 87-stem, d9 on x87,7.

Theorem (Chua)

d2px77,7q “ h30x76,6.

Theorem (Burklund - Isaksen - X.)

d5ph1p1q “ 0.

Hopf classes and Kervaire classes

§ hj : Hopf classes

§ h0, h1, h2, h3 survive and detect 2, η, ν, σ.

§ Adams: d2phjq “ h0h2j´1, for n ě 4.

§ h2j : Kervaire classes

§ h2j , 0 ď j ď 5 survive and detect θj .

§ Hill-Hopkins-Ravenel: For j ě 7, h2j supports a nonzero differential.

§ h26 is still open.

Question

What are the differentials that h2j , j ě 7 support?

Hopf classes and Kervaire classes

Easy to show for all j ,

§ d2ph2j q “ 0,

§ d3ph2j q “ 0.

Theorem (Burklund - X.)

§ d4ph2j q “ 0, for all j .

Work in progress: For j large enough,

§ d5ph2j q “ 0,

§ d6ph2j q “ 0.

Hopf classes, Kervaire classes and ???

Question

What about h3j , h4j , ¨ ¨ ¨ ?

§ h4j “ 0 in Ext for j ě 1.ñ Only need to discuss h3j .

§ h3j survives for 0 ď j ď 4.

Theorem (Isaksen - Wang - X.)

§ d4ph35q “ h30g3.§ h20g3 detects θ4θ5.

Theorem (Burklund - X.)

§ d4ph3j q “ h30gj´2` possible other classes, for j ě 6.

The open case θ6Theorem (Isaksen - Wang - X.)

§ h30h35 survives and detects xθ4, 2, θ5y .

§ For any α P xθ4, 2, θ5y, if ηα “ 0, and 0 P xα, η, 2y, then θ6 exists.

126

2 $$

η

!!126

θ5

125

α2

θ563

2

312

124

α

62

θ5 22

30 θ4((0

S126f // X

g // S0

Cellular Motivic Stable Homotopy Category SHpkqcell

§ S1,0: simplicial sphere S1

§ S1,1: A1 ´ t0u “ Gm

§ Stabilization with respect to S2,1

§ 1: the motivic sphere spectrum

§ HFp: motivic mod p Eilenberg-Mac Lane spectrum

§ p1: HFp-completed sphere

§ MGL: algebraic cobordism spectrum

§ Motivic analogue of classical computational tools exist!

Cellular Motivic Stable Homotopy Category SHpkqcell

§ S1,0: simplicial sphere S1

§ S1,1: A1 ´ t0u “ Gm§ Stabilization with respect to S2,1

§ 1: the motivic sphere spectrum

§ HFp: motivic mod p Eilenberg-Mac Lane spectrum

§ p1: HFp-completed sphere

§ MGL: algebraic cobordism spectrum

§ Motivic analogue of classical computational tools exist!

Over Spec CTheorem (Voevodsky)

π˚,˚HFp “ Fprτ s, |τ | “ p0,´1q.

§ τ : Σ0,´1p1Ñ p1.§ p1{τ : the cofiber of τ .

Theorem (Isaksen)

motAdamsNovikovSSpp1{τq collapses at E2.

Theorem (Gheorghe - Wang - X.)

p1{τ -Modcell » StablepBP˚BP-Comodq

Alternative proofs: Krause, and Pstra̧gowski.

Isomorphism of spectral sequences

Theorem (Gheorghe - Wang - X.)

algNovikovSSpBP˚q – motAdamsSSpp1{τq.

Exts,2wBP˚BPpFp, Ia´s{I a´s`1q

Algebraic Novikov SS

��

– // Exta,2w´s`a,wA pFprτ s,Fpq

Motivic Adams SS

��Exts,2wBP˚BPpBP˚,BP˚q

– // π2w´s,w pp1{τq.

Strategy of Stem-wise Computations

§ Compute Ext over C.§ Compute algNovikovSSpBP˚q, including all differentials.

§ algNovikovSSpBP˚q – motAdamsSSpp1{τq§

p1 ÝÑ p1{τ ÝÑ Σ1,´1p1

pull back Adams differentials from p1{τ to p1.§ Apply ad hoc arguments such as shuffling Toda brackets.

§ Invert τ .


§ Compute Ext over C.§ Compute algNovikovSSpBP˚q, including all differentials.§ algNovikovSSpBP˚q – motAdamsSSpp1{τq

§p1 ÝÑ p1{τ ÝÑ Σ1,´1p1


§ Invert τ .


§ Compute Ext over C.§ Compute algNovikovSSpBP˚q, including all differentials.§ algNovikovSSpBP˚q – motAdamsSSpp1{τq§

p1 ÝÑ p1{τ ÝÑ Σ1,´1p1


§ Invert τ .

General Questions

Questions

§ Can this p1{τ method be applied to other fields?

§ What about the non-cellular part?

General Questions

Questions

§ Can this p1{τ method be applied to other fields?§ What about the non-cellular part?

Chow-Novikov t-structure on SHpkq

§ X : smooth projective scheme over k ,

§ ξ: virtual vector bundle over X ,

§ ThpX , ξq: its Thom spectrum.

Definition (Chow-Novikov t-structure)

§ SHpkqtě0: full subcategory generated by ThpX , ξqunder colimits and extensions.

§ SHpkqtă0: objects Y such that for any object X P SHpkqtě0,

rX ,Y sSHpkq “ 0.

This defines a t-structure.

Chow-Novikov t-structure on SHpkq§ Hopkins-Morel, Levine:

MGL˚,˚ is in ě 0 Chow-Novikov degrees.Chow-Novikov degree 0 part “ MU˚.

§ X P SHpkqtě0 ñMGL˚,˚X is in ě 0 Chow-Novikov degrees.

§ Y P SHpkqtă0 ñ

MGL˚,˚Y “ colim π˚,˚Σ´2n,´nThpGr ,V q ^ Y“ colim rΣ˚,˚1^ Σ2n,nDpThpGr ,V qq,Y sSHpkq“ 0 on ě 0 Chow-Novikov degrees.

MGL˚,˚Y is in ă 0 Chow-Novikov degrees.§ X P SHpkq♥ ñ MGL˚,˚X is in Chow-Novikov degree 0.

Question

What is this heart?




§ Y P SHpkqtă0 ñ


MGL˚,˚Y is in ă 0 Chow-Novikov degrees.

§ X P SHpkq♥ ñ MGL˚,˚X is in Chow-Novikov degree 0.

Question

What is this heart?




§ Y P SHpkqtă0 ñ


MGL˚,˚Y is in ă 0 Chow-Novikov degrees.§ X P SHpkq♥ ñ MGL˚,˚X is in Chow-Novikov degree 0.

Question

What is this heart?


MGL-ModU // SHpkqF

oo // τtď01-Mod

Theorem (Bachmann - Kong - Wang - X.)

§ MGL-Mod♥ » FunˆphPMopMGL,Setq,

§ SHpkq♥ » τtď01-Mod♥ » C -Comod, C “ FU.

Restricting to cellular subcategories,


§ MGL-Mod♥cell » MU˚-Mod,

§ SHpkq♥cell » MU˚MU-Comod,§ τtď01cell-Modcell » StablepMU˚MU-Comodq


MGL-ModU // SHpkqF

oo // τtď01-Mod







§ SHpkq♥cell » MU˚MU-Comod,

§ τtď01cell-Modcell » StablepMU˚MU-Comodq


MGL-ModU // SHpkqF

oo // τtď01-Mod







§ SHpkq♥cell » MU˚MU-Comod,§ τtď01cell-Modcell » StablepMU˚MU-Comodq

Motivic Adams and Adams-Novikov spectral sequences

§ p1: HFp-completed sphere,

§ p1t“0: cellularization of τtď0p1,

§ p1t“0-Modcell » StablepBP˚BP-Comodq,§ X P p1t“0-Mod♥cell ñ BPGL˚,˚X is a BP˚BP-comodule.

Work in progress: (Bachmann - Kong - Wang - X.)For X P p1t“0-Mod♥,

§ motAdamsNovikovSSpX q collapses,§ motAdamsSSpX q – algNovikovSSpBPGL˚,˚X q.




§ p1t“0-Modcell » StablepBP˚BP-Comodq,

§ X P p1t“0-Mod♥cell ñ BPGL˚,˚X is a BP˚BP-comodule.








§ motAdamsNovikovSSpX q collapses,

§ motAdamsSSpX q – algNovikovSSpBPGL˚,˚X q.

Postnikov Tower

Postnikov tower for p1 w.r.t. the Chow-Novikov t-structure:

��p1tě2

��

//p1t“2

p1tě1

��

//p1t“1

p1 p1 // p1t“0

§ p1t“n: cellular spectrum such that

BPGL˚,˚p1t“n “ Chow-Novikov degree n part of BPGL˚,˚.

Computing Stable Stems over k

Apply the motivic Adams spectral sequences:

��motASSpp1tě2q

��

// motASSpp1t“2q algNSSpBPGL˚,˚p1t“2q

motASSpp1tě1q

��

// motASSpp1t“1q algNSSpBPGL˚,˚p1t“1q

motASSpp1q motASSpp1q // motASSpp1t“0q algNSSpBPGL˚,˚p1t“0q

Stable Stems over C

§ π˚,˚BPGL “ Zprτ, v1, v2, ¨ ¨ ¨ s,

§ Chow-Novikov degree: |τ | “ 2, |vi | “ 0.§ Chow-Novikov degree 2k: Σ0,´kBP˚,

§ Chow-Novikov degree 2k ` 1: 0.§ p1“2k “ Σ0,´kp1{τ , k ě 0§ p1“2k`1 “ ˚.

motASSpΣ0,´2p1q

��

// motASSpΣ0,´2p1{τq algNSSpΣ0,´2BP˚q

motASSpΣ0,´1p1q

��


motASSpp1q motASSpp1q // motASSpp1{τq algNSSpBP˚q

Stable Stems over C

§ π˚,˚BPGL “ Zprτ, v1, v2, ¨ ¨ ¨ s,§ Chow-Novikov degree: |τ | “ 2, |vi | “ 0.

§ Chow-Novikov degree 2k: Σ0,´kBP˚,


motASSpΣ0,´2p1q

��


motASSpΣ0,´1p1q

��



Stable Stems over C

§ π˚,˚BPGL “ Zprτ, v1, v2, ¨ ¨ ¨ s,§ Chow-Novikov degree: |τ | “ 2, |vi | “ 0.§ Chow-Novikov degree 2k : Σ0,´kBP˚,

§ Chow-Novikov degree 2k ` 1: 0.

§ p1“2k “ Σ0,´kp1{τ , k ě 0§ p1“2k`1 “ ˚.

motASSpΣ0,´2p1q

��


motASSpΣ0,´1p1q

��



Stable Stems over C

§ π˚,˚BPGL “ Zprτ, v1, v2, ¨ ¨ ¨ s,§ Chow-Novikov degree: |τ | “ 2, |vi | “ 0.§ Chow-Novikov degree 2k : Σ0,´kBP˚,


motASSpΣ0,´2p1q

��


motASSpΣ0,´1p1q

��



Stable Stems over R

§ π˚,˚HFp “ Fprρ, τ s|τ | “ p0,´1q, |ρ| “ p´1,´1q.

§ ρ : Σ´1,´11Ñ 1, ρ : Σ´1,´1p1Ñ p1§ p1{ρ: cofiber of ρ

§ τ : Σ0,´1p1{ρ ÝÑ p1{ρ§ p1{pρ, τq: its cofiber - a 4 cell complex.

§ p1t“0 » p1{pρ, τq.

Stable Stems over R



§ τ : Σ0,´1p1{ρ ÝÑ p1{ρ

§ p1{pρ, τq: its cofiber - a 4 cell complex.

§ p1t“0 » p1{pρ, τq.

Stable Stems over R



§ τ : Σ0,´1p1{ρ ÝÑ p1{ρ§ p1{pρ, τq: its cofiber - a 4 cell complex.

§ p1t“0 » p1{pρ, τq.

Stable Stems over R at p “ 2

Theorem (Hu-Kriz, Hill)

BPGL˚,˚ “ Z2

»

—

—

—

—

–

ρ,v0, τ

2v0, τ4v0, τ

6v0, τ8v0, ¨ ¨ ¨

v1, τ4v1, τ

8v1, ¨ ¨ ¨v2, τ

8v2, ¨ ¨ ¨¨ ¨ ¨

fi

ffi

ffi

ffi

ffi

fl

{

»

—

—

—

—

–

v0 “ 2ρv0 “ 0ρ3v1 “ 0ρ7v2 “ 0¨ ¨ ¨

fi

ffi

ffi

ffi

ffi

fl

and the generators satisfy the further relations

τ 2i`1¨jvi ¨ τ 2

k`1¨lvk “ τ 2i`1pj`2k´i lqvivk

when i ď k, as if the class τ were an element in this ring.

Low Chow-Novikov degrees over R§ Chow-Novikov = 0: Z2rv0, v1, ¨ ¨ ¨ s “ BP˚.

§ Chow-Novikov = 1: ρ ¨ Z2rv0, v1, ¨ ¨ ¨ s{ρv0 “ Σ´1,´1BP˚{2.§ Chow-Novikov = 2: ρ2 ¨ Z2rv0, v1, ¨ ¨ ¨ s{ρv0 “ Σ´2,´2BP˚{2.§ Chow-Novikov = 3:ρ3 ¨ Z2rv0, v1, ¨ ¨ ¨ s{pρv0, ρ3v1q “ Σ´3,´3BP˚{p2, v1q.

§ Chow-Novikov = 4:ρ4BP˚ ‘ τ 2v0BP˚ “ Σ´4,´4BP˚{p2, v1q ‘ Σ0,´2BP˚.

§ Chow-Novikov = 5:ρ5BP˚ ‘ ppρ ¨ τ 2v0 “ 0qBP˚ “ Σ´5,´5BP˚{p2, v1q.

§ Chow-Novikov = 6: ρ6BP˚ ‘ ρ2 ¨ τ 2v0BP˚ “ Σ´6,´6BP˚{p2, v1q.§ Chow-Novikov = 7: ρ7BP˚ “ Σ´7,´7BP˚{p2, v1, v2q.

§ Chow-Novikov = 8: ρ8BP˚ ‘ τ4v0BP˚‘τ 4v1BP˚τ 4v0¨v1´τ 4v1¨v0

“ Σ´8,´8BP˚{p2, v1, v2q ‘ pΣ0,´4BP˚ ` Σ2,´3BP˚{2q.

Low Chow-Novikov degrees over R§ Chow-Novikov = 0: Z2rv0, v1, ¨ ¨ ¨ s “ BP˚.§ Chow-Novikov = 1: ρ ¨ Z2rv0, v1, ¨ ¨ ¨ s{ρv0 “ Σ´1,´1BP˚{2.

§ Chow-Novikov = 2: ρ2 ¨ Z2rv0, v1, ¨ ¨ ¨ s{ρv0 “ Σ´2,´2BP˚{2.§ Chow-Novikov = 3:ρ3 ¨ Z2rv0, v1, ¨ ¨ ¨ s{pρv0, ρ3v1q “ Σ´3,´3BP˚{p2, v1q.






Low Chow-Novikov degrees over R§ Chow-Novikov = 0: Z2rv0, v1, ¨ ¨ ¨ s “ BP˚.§ Chow-Novikov = 1: ρ ¨ Z2rv0, v1, ¨ ¨ ¨ s{ρv0 “ Σ´1,´1BP˚{2.§ Chow-Novikov = 2: ρ2 ¨ Z2rv0, v1, ¨ ¨ ¨ s{ρv0 “ Σ´2,´2BP˚{2.

§ Chow-Novikov = 3:ρ3 ¨ Z2rv0, v1, ¨ ¨ ¨ s{pρv0, ρ3v1q “ Σ´3,´3BP˚{p2, v1q.






Low Chow-Novikov degrees over R§ Chow-Novikov = 0: Z2rv0, v1, ¨ ¨ ¨ s “ BP˚.§ Chow-Novikov = 1: ρ ¨ Z2rv0, v1, ¨ ¨ ¨ s{ρv0 “ Σ´1,´1BP˚{2.§ Chow-Novikov = 2: ρ2 ¨ Z2rv0, v1, ¨ ¨ ¨ s{ρv0 “ Σ´2,´2BP˚{2.§ Chow-Novikov = 3:ρ3 ¨ Z2rv0, v1, ¨ ¨ ¨ s{pρv0, ρ3v1q “ Σ´3,´3BP˚{p2, v1q.






Low Chow-Novikov degrees over R

0 // Σ|vn`1|BP˚{Invn`1 // BP˚{In // BP˚{In`1 // 0.

BP˚{In can be inductively realized by p1{pρ, τ, 2, v1, ¨ ¨ ¨ , vnq.§ p1t“0 » p1{pρ, τq.§ p1t“1 » Σ´1,´1p1{pρ, τ, 2q.§ p1t“2 » Σ´2,´2p1{pρ, τ, 2q.§ p1t“3 » Σ´3,´3p1{pρ, τ, 2, v1q.§ p1t“4 » Σ´4,´4p1{pρ, τ, 2, v1q _ Σ0,´2p1{pρ, τq.§ p1t“5 » Σ´5,´5p1{pρ, τ, 2, v1q.§ p1t“6 » Σ´6,´6p1{pρ, τ, 2, v1q.§ p1t“7 » Σ´7,´7p1{pρ, τ, 2, v1, v2q.§ p1t“8 » Σ´8,´8p1{pρ, τ, 2, v1, v2q _ X , X “ cofiber of:

Σ´2,´3p1{pρ, τqpv0,v1q // Σ´2,´3p1{pρ, τq _ Σ0,´4p1{pρ, τq



BP˚{In can be inductively realized by p1{pρ, τ, 2, v1, ¨ ¨ ¨ , vnq.

§ p1t“0 » p1{pρ, τq.§ p1t“1 » Σ´1,´1p1{pρ, τ, 2q.§ p1t“2 » Σ´2,´2p1{pρ, τ, 2q.§ p1t“3 » Σ´3,´3p1{pρ, τ, 2, v1q.§ p1t“4 » Σ´4,´4p1{pρ, τ, 2, v1q _ Σ0,´2p1{pρ, τq.§ p1t“5 » Σ´5,´5p1{pρ, τ, 2, v1q.§ p1t“6 » Σ´6,´6p1{pρ, τ, 2, v1q.§ p1t“7 » Σ´7,´7p1{pρ, τ, 2, v1, v2q.§ p1t“8 » Σ´8,´8p1{pρ, τ, 2, v1, v2q _ X , X “ cofiber of:




BP˚{In can be inductively realized by p1{pρ, τ, 2, v1, ¨ ¨ ¨ , vnq.§ p1t“0 » p1{pρ, τq.§ p1t“1 » Σ´1,´1p1{pρ, τ, 2q.§ p1t“2 » Σ´2,´2p1{pρ, τ, 2q.§ p1t“3 » Σ´3,´3p1{pρ, τ, 2, v1q.§ p1t“4 » Σ´4,´4p1{pρ, τ, 2, v1q _ Σ0,´2p1{pρ, τq.§ p1t“5 » Σ´5,´5p1{pρ, τ, 2, v1q.§ p1t“6 » Σ´6,´6p1{pρ, τ, 2, v1q.§ p1t“7 » Σ´7,´7p1{pρ, τ, 2, v1, v2q.§ p1t“8 » Σ´8,´8p1{pρ, τ, 2, v1, v2q _ X , X “ cofiber of:


Stable Stems over R at p “ 2

motASSpp1tě3q

��

// motASSpΣ´3,´3p1{pρ, τ, 2, v1qq

algNSSpΣ´3,´3BP˚{p2, v1qq

motASSpp1tě2q

��

// motASSpΣ´2,´2p1{pρ, τ, 2qq

algNSSpΣ´2,´2BP˚{2q

motASSpp1tě1q

��

// motASSpΣ´1,´1p1{pρ, τ, 2qq

algNSSpΣ´1,´1BP˚{2q

motASSpp1q motASSpp1q // motASSpp1{pρ, τqq algNSSpBP˚q

More purely algebraic parts!

Stable Stems over Fq at p “ 2

§ q: power of an odd prime,

§ Fq: finite field with q elements,

§ π˚,˚HF2 “#

F2rτ, us{u2, Sq1τ “ 0 if q ” 1 mod 4,F2rτ, ρs{ρ2, Sq1τ “ ρ if q ” 3 mod 4.

|τ | “ p0,´1q, |u| “ |ρ| “ p´1,´1q.

§ π˚,˚HZ2 “#

Z2 if p˚, ˚q “ p0, 0q,pZ{pqw ´ 1qq^2 if p˚, ˚q “ p´1,´wq and w ě 0.

§ π˚,˚BPGL “ π˚,˚HZ2 b BP˚.


§ π˚,˚HZ2 “#


§ π˚,˚BPGL “ π˚,˚HZ2 b BP˚.

§ Chow-Novikov degree 1 part of π˚,˚BPGL “$

’

&

’

%

Σ´1,´1BP˚{2 if q ” 3 mod 4,Σ´1,´1BP˚{4 if q ” 5 mod 8,Σ´1,´1BP˚{8 ¨ 2m if q ” 1 mod 8.


§ π˚,˚HZ2 “#


§ π˚,˚BPGL “ π˚,˚HZ2 b BP˚.§ Chow-Novikov degree 1 part of π˚,˚BPGL “

$

’

&

’

%

Σ´1,´1BP˚{2 if q ” 3 mod 4,Σ´1,´1BP˚{4 if q ” 5 mod 8,Σ´1,´1BP˚{8 ¨ 2m if q ” 1 mod 8.

Stable Stems over Fq at p “ 2§ Hoyois-Kelly-Østvær computed the mod 2 Steenrod algebra over Fq,

§ Wilson-Østvær computed πn,0 for#

0 ď n ď 20 if q ” 1 mod 4,0 ď n ď 18 if q ” 3 mod 4.

Proposition (Wilson-Østvær)

§ d2pτgq “ uh20g , if q ” 5 mod 8,§ d2pτgq “ 0, if q ” 1 mod 8,§ d2pτ 2gq “ ρτh20g or 0, if q ” 3 mod 4.

Using Chow-Novikov degree ď 3 part of π˚,˚BPGL,

Proposition (Bachmann-Kong-Wang-X.)

When q ” 3 mod 4, d2pτ 2gq “ 0.

Stable Stems over Fq at p “ 2§ Hoyois-Kelly-Østvær computed the mod 2 Steenrod algebra over Fq,§ Wilson-Østvær computed πn,0 for

#








#



§ d2pτgq “ uh20g , if q ” 5 mod 8,§ d2pτgq “ 0, if q ” 1 mod 8,

§ d2pτ 2gq “ ρτh20g or 0, if q ” 3 mod 4.





#







Thank you!

stable stems and the chow-novikov t-structure in motivic stable … · 2020. 5. 5. · d2e3 m p3il...

Documents