The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to...
Transcript of The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to...
![Page 1: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/1.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The cofiber Cτ and Motivic Chromatic stuffMotivic Homotopy Theory
Bogdan GheorghePhD student of Dan Isaksen
Wayne State University
Operations in Highly Structured Homology TheoriesBanff, May 22-27, 2016
![Page 2: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/2.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Motivic Homotopy Theory
![Page 3: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/3.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Unstable Motivic Spaces
I will only work over the base scheme SpecC.
Motivic spaces are
1 start with the category Sm/C of C-schemes (smooth, fin. type)
2 add colimits by embedding it in
3 sPre(Sm/C) has point-wise model structures from sSet∗4 Bousfield localize to
force Nisnevich covers to be homotopy colimitsmake “the interval” A1
C contractible
Theorem (Morel-Voevodsky)
This gives a symmetric monoidal model category SpcC, and there is arealization functor R by taking C-points
SpcCR
GGA⊥GDGG
Sing
Top.
![Page 4: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/4.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Unstable Motivic Spaces
I will only work over the base scheme SpecC. Motivic spaces are
1 start with the category Sm/C of C-schemes (smooth, fin. type)
2 add colimits by embedding it in
3 sPre(Sm/C) has point-wise model structures from sSet∗4 Bousfield localize to
force Nisnevich covers to be homotopy colimitsmake “the interval” A1
C contractible
Theorem (Morel-Voevodsky)
This gives a symmetric monoidal model category SpcC, and there is arealization functor R by taking C-points
SpcCR
GGA⊥GDGG
Sing
Top.
![Page 5: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/5.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Unstable Motivic Spaces
I will only work over the base scheme SpecC. Motivic spaces are
1 start with the category Sm/C of C-schemes (smooth, fin. type)
2 add colimits by embedding it in presheaves
3 sPre(Sm/C) has point-wise model structures from sSet∗4 Bousfield localize to
force Nisnevich covers to be homotopy colimitsmake “the interval” A1
C contractible
Theorem (Morel-Voevodsky)
This gives a symmetric monoidal model category SpcC, and there is arealization functor R by taking C-points
SpcCR
GGA⊥GDGG
Sing
Top.
![Page 6: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/6.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Unstable Motivic Spaces
I will only work over the base scheme SpecC. Motivic spaces are
1 start with the category Sm/C of C-schemes (smooth, fin. type)
2 add htpy colimits by embedding it in simplicial presheaves
3 sPre(Sm/C) has point-wise model structures from sSet∗4 Bousfield localize to
force Nisnevich covers to be homotopy colimitsmake “the interval” A1
C contractible
Theorem (Morel-Voevodsky)
This gives a symmetric monoidal model category SpcC, and there is arealization functor R by taking C-points
SpcCR
GGA⊥GDGG
Sing
Top.
![Page 7: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/7.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Unstable Motivic Spaces
I will only work over the base scheme SpecC. Motivic spaces are
1 start with the category Sm/C of C-schemes (smooth, fin. type)
2 add htpy colimits by embedding it in simplicial presheaves
3 sPre(Sm/C) has point-wise model structures from sSet∗
4 Bousfield localize to
force Nisnevich covers to be homotopy colimitsmake “the interval” A1
C contractible
Theorem (Morel-Voevodsky)
This gives a symmetric monoidal model category SpcC, and there is arealization functor R by taking C-points
SpcCR
GGA⊥GDGG
Sing
Top.
![Page 8: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/8.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Unstable Motivic Spaces
I will only work over the base scheme SpecC. Motivic spaces are
1 start with the category Sm/C of C-schemes (smooth, fin. type)
2 add htpy colimits by embedding it in simplicial presheaves
3 sPre(Sm/C) has point-wise model structures from sSet∗4 Bousfield localize to
force Nisnevich covers to be homotopy colimitsmake “the interval” A1
C contractible
Theorem (Morel-Voevodsky)
This gives a symmetric monoidal model category SpcC, and there is arealization functor R by taking C-points
SpcCR
GGA⊥GDGG
Sing
Top.
![Page 9: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/9.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Unstable Motivic Spaces
I will only work over the base scheme SpecC. Motivic spaces are
1 start with the category Sm/C of C-schemes (smooth, fin. type)
2 add htpy colimits by embedding it in simplicial presheaves
3 sPre(Sm/C) has point-wise model structures from sSet∗4 Bousfield localize to
force Nisnevich covers to be homotopy colimitsmake “the interval” A1
C contractible
Theorem (Morel-Voevodsky)
This gives a symmetric monoidal model category SpcC
, and there is arealization functor R by taking C-points
SpcCR
GGA⊥GDGG
Sing
Top.
![Page 10: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/10.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Unstable Motivic Spaces
I will only work over the base scheme SpecC. Motivic spaces are
1 start with the category Sm/C of C-schemes (smooth, fin. type)
2 add htpy colimits by embedding it in simplicial presheaves
3 sPre(Sm/C) has point-wise model structures from sSet∗4 Bousfield localize to
force Nisnevich covers to be homotopy colimitsmake “the interval” A1
C contractible
Theorem (Morel-Voevodsky)
This gives a symmetric monoidal model category SpcC, and there is arealization functor R by taking C-points
SpcCR
GGA⊥GDGG
Sing
Top.
![Page 11: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/11.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Motivic Spheres
There are two types of spheres in SpcC.
1 The constant U [GGA ∆1/∂∆1 = S1, which realizes to S1 ∈ Top.This is called the simplicial sphere and denoted by S1,0.
2 The scheme Gm =(A1
C)×
, which realizes to S1 ∈ Top.This is called the geometric sphere and denoted by S1,1.
This gives bigraded spheres Sn+k,n =(S1,0
)∧k ∧ (S1,1)∧n
for n, k ≥ 0,and thus bigraded homotopy groups, and bigraded everything. . . .
The first index Sm,n is the topological dimension.The second index Sm,n is called the weight.
![Page 12: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/12.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Motivic Spheres
There are two types of spheres in SpcC.
1 The constant U [GGA ∆1/∂∆1 = S1, which realizes to S1 ∈ Top.This is called the simplicial sphere and denoted by S1,0.
2 The scheme Gm =(A1
C)×
, which realizes to S1 ∈ Top.This is called the geometric sphere and denoted by S1,1.
This gives bigraded spheres Sn+k,n =(S1,0
)∧k ∧ (S1,1)∧n
for n, k ≥ 0,and thus bigraded homotopy groups, and bigraded everything. . . .
The first index Sm,n is the topological dimension.The second index Sm,n is called the weight.
![Page 13: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/13.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Motivic Spheres
There are two types of spheres in SpcC.
1 The constant U [GGA ∆1/∂∆1 = S1, which realizes to S1 ∈ Top.This is called the simplicial sphere and denoted by S1,0.
2 The scheme Gm =(A1
C)×
, which realizes to S1 ∈ Top.This is called the geometric sphere and denoted by S1,1.
This gives bigraded spheres Sn+k,n =(S1,0
)∧k ∧ (S1,1)∧n
for n, k ≥ 0,and thus bigraded homotopy groups, and bigraded everything. . . .
The first index Sm,n is the topological dimension.The second index Sm,n is called the weight.
![Page 14: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/14.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Motivic Spheres
There are two types of spheres in SpcC.
1 The constant U [GGA ∆1/∂∆1 = S1, which realizes to S1 ∈ Top.This is called the simplicial sphere and denoted by S1,0.
2 The scheme Gm =(A1
C)×
, which realizes to S1 ∈ Top.This is called the geometric sphere and denoted by S1,1.
This gives bigraded spheres Sn+k,n =(S1,0
)∧k ∧ (S1,1)∧n
for n, k ≥ 0,and thus bigraded homotopy groups, and bigraded everything. . . .
The first index Sm,n is the topological dimension.The second index Sm,n is called the weight.
![Page 15: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/15.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Motivic Spheres
There are two types of spheres in SpcC.
1 The constant U [GGA ∆1/∂∆1 = S1, which realizes to S1 ∈ Top.This is called the simplicial sphere and denoted by S1,0.
2 The scheme Gm =(A1
C)×
, which realizes to S1 ∈ Top.This is called the geometric sphere and denoted by S1,1.
This gives bigraded spheres Sn+k,n =(S1,0
)∧k ∧ (S1,1)∧n
for n, k ≥ 0,and thus bigraded homotopy groups, and bigraded everything. . . .
The first index Sm,n is the topological dimension.The second index Sm,n is called the weight.
![Page 16: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/16.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Motivic Spectra and Examples
Theorem (Morel-Voevodsky, Jardine, Hu)
There is a symmetric monoidal (with the smash product − ∧−) modelcategory of motivic spectra SptC
, and the realization pair stabilizes toan adjunction
SptCR
GGA⊥GDGG
Sing
Spt.
A lot of classical spectra have their motivic analogues. We have
Spheres Sm,n
Eilenberg-Maclane spectra HFpComplex K-theory KGL and kgl, with |β| = (2, 1)
(Algebraic) Cobordism MGL, with |xi| = (2i, i)
. . . etc
and they all realize to their classical analogues.
![Page 17: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/17.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Motivic Spectra and Examples
Theorem (Morel-Voevodsky, Jardine, Hu)
There is a symmetric monoidal (with the smash product − ∧−) modelcategory of motivic spectra SptC, and the realization pair stabilizes toan adjunction
SptCR
GGA⊥GDGG
Sing
Spt.
A lot of classical spectra have their motivic analogues. We have
Spheres Sm,n
Eilenberg-Maclane spectra HFpComplex K-theory KGL and kgl, with |β| = (2, 1)
(Algebraic) Cobordism MGL, with |xi| = (2i, i)
. . . etc
and they all realize to their classical analogues.
![Page 18: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/18.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Motivic Spectra and Examples
Theorem (Morel-Voevodsky, Jardine, Hu)
There is a symmetric monoidal (with the smash product − ∧−) modelcategory of motivic spectra SptC, and the realization pair stabilizes toan adjunction
SptCR
GGA⊥GDGG
Sing
Spt.
A lot of classical spectra have their motivic analogues.
We have
Spheres Sm,n
Eilenberg-Maclane spectra HFpComplex K-theory KGL and kgl, with |β| = (2, 1)
(Algebraic) Cobordism MGL, with |xi| = (2i, i)
. . . etc
and they all realize to their classical analogues.
![Page 19: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/19.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Motivic Spectra and Examples
Theorem (Morel-Voevodsky, Jardine, Hu)
There is a symmetric monoidal (with the smash product − ∧−) modelcategory of motivic spectra SptC, and the realization pair stabilizes toan adjunction
SptCR
GGA⊥GDGG
Sing
Spt.
A lot of classical spectra have their motivic analogues. We have
Spheres Sm,n
Eilenberg-Maclane spectra HFpComplex K-theory KGL and kgl, with |β| = (2, 1)
(Algebraic) Cobordism MGL, with |xi| = (2i, i)
. . . etc
and they all realize to their classical analogues.
![Page 20: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/20.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Motivic Spectra and Examples
Theorem (Morel-Voevodsky, Jardine, Hu)
There is a symmetric monoidal (with the smash product − ∧−) modelcategory of motivic spectra SptC, and the realization pair stabilizes toan adjunction
SptCR
GGA⊥GDGG
Sing
Spt.
A lot of classical spectra have their motivic analogues. We have
Spheres Sm,n
Eilenberg-Maclane spectra HFp
Complex K-theory KGL and kgl, with |β| = (2, 1)
(Algebraic) Cobordism MGL, with |xi| = (2i, i)
. . . etc
and they all realize to their classical analogues.
![Page 21: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/21.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Motivic Spectra and Examples
Theorem (Morel-Voevodsky, Jardine, Hu)
There is a symmetric monoidal (with the smash product − ∧−) modelcategory of motivic spectra SptC, and the realization pair stabilizes toan adjunction
SptCR
GGA⊥GDGG
Sing
Spt.
A lot of classical spectra have their motivic analogues. We have
Spheres Sm,n
Eilenberg-Maclane spectra HFpComplex K-theory KGL and kgl
, with |β| = (2, 1)
(Algebraic) Cobordism MGL, with |xi| = (2i, i)
. . . etc
and they all realize to their classical analogues.
![Page 22: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/22.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Motivic Spectra and Examples
Theorem (Morel-Voevodsky, Jardine, Hu)
There is a symmetric monoidal (with the smash product − ∧−) modelcategory of motivic spectra SptC, and the realization pair stabilizes toan adjunction
SptCR
GGA⊥GDGG
Sing
Spt.
A lot of classical spectra have their motivic analogues. We have
Spheres Sm,n
Eilenberg-Maclane spectra HFpComplex K-theory KGL and kgl, with |β| = (2, 1)
(Algebraic) Cobordism MGL, with |xi| = (2i, i)
. . . etc
and they all realize to their classical analogues.
![Page 23: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/23.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Motivic Spectra and Examples
Theorem (Morel-Voevodsky, Jardine, Hu)
There is a symmetric monoidal (with the smash product − ∧−) modelcategory of motivic spectra SptC, and the realization pair stabilizes toan adjunction
SptCR
GGA⊥GDGG
Sing
Spt.
A lot of classical spectra have their motivic analogues. We have
Spheres Sm,n
Eilenberg-Maclane spectra HFpComplex K-theory KGL and kgl, with |β| = (2, 1)
(Algebraic) Cobordism MGL
, with |xi| = (2i, i)
. . . etc
and they all realize to their classical analogues.
![Page 24: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/24.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Motivic Spectra and Examples
Theorem (Morel-Voevodsky, Jardine, Hu)
There is a symmetric monoidal (with the smash product − ∧−) modelcategory of motivic spectra SptC, and the realization pair stabilizes toan adjunction
SptCR
GGA⊥GDGG
Sing
Spt.
A lot of classical spectra have their motivic analogues. We have
Spheres Sm,n
Eilenberg-Maclane spectra HFpComplex K-theory KGL and kgl, with |β| = (2, 1)
(Algebraic) Cobordism MGL, with |xi| = (2i, i)
. . . etc
and they all realize to their classical analogues.
![Page 25: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/25.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Motivic Spectra and Examples
Theorem (Morel-Voevodsky, Jardine, Hu)
There is a symmetric monoidal (with the smash product − ∧−) modelcategory of motivic spectra SptC, and the realization pair stabilizes toan adjunction
SptCR
GGA⊥GDGG
Sing
Spt.
A lot of classical spectra have their motivic analogues. We have
Spheres Sm,n
Eilenberg-Maclane spectra HFpComplex K-theory KGL and kgl, with |β| = (2, 1)
(Algebraic) Cobordism MGL, with |xi| = (2i, i)
. . . etc
and they all realize to their classical analogues.
![Page 26: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/26.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The cofiber Cτ
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Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The Motivic Adams Spectral sequence
We now fix p = 2 for the remaining of the talk.
Theorem (Voevodsky)
The coefficients are HF2∗,∗(S0,0) = M2
∼= F2[τ ] for |τ | = (0, 1).
The HF2-Steenrod Algebra is AC ∼= M2
⟨Sq1, Sq2, . . .
⟩/Adem.
The HF2 motivic Adams spectral sequence for S0,0 takes the form
ExtAC(M2,M2) =⇒ π∗,∗(S0,02),
and the element τ ∈ Ext0 survives to a map S0,−1 τGGGA S0,0
2, butdoes not exist before 2-completion.
Therefore, we work in the 2-completed category, and S0,0 meansthe 2-completed sphere.
![Page 28: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/28.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The Motivic Adams Spectral sequence
We now fix p = 2 for the remaining of the talk.
Theorem (Voevodsky)
The coefficients are HF2∗,∗(S0,0) = M2
∼= F2[τ ] for |τ | = (0, 1).
The HF2-Steenrod Algebra is AC ∼= M2
⟨Sq1, Sq2, . . .
⟩/Adem.
The HF2 motivic Adams spectral sequence for S0,0 takes the form
ExtAC(M2,M2) =⇒ π∗,∗(S0,02),
and the element τ ∈ Ext0 survives to a map S0,−1 τGGGA S0,0
2, butdoes not exist before 2-completion.
Therefore, we work in the 2-completed category, and S0,0 meansthe 2-completed sphere.
![Page 29: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/29.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The Motivic Adams Spectral sequence
We now fix p = 2 for the remaining of the talk.
Theorem (Voevodsky)
The coefficients are HF2∗,∗(S0,0) = M2
∼= F2[τ ] for |τ | = (0, 1).
The HF2-Steenrod Algebra is AC ∼= M2
⟨Sq1, Sq2, . . .
⟩/Adem.
The HF2 motivic Adams spectral sequence for S0,0 takes the form
ExtAC(M2,M2) =⇒ π∗,∗(S0,02),
and the element τ ∈ Ext0 survives to a map S0,−1 τGGGA S0,0
2, butdoes not exist before 2-completion.
Therefore, we work in the 2-completed category, and S0,0 meansthe 2-completed sphere.
![Page 30: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/30.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The Motivic Adams Spectral sequence
We now fix p = 2 for the remaining of the talk.
Theorem (Voevodsky)
The coefficients are HF2∗,∗(S0,0) = M2
∼= F2[τ ] for |τ | = (0, 1).
The HF2-Steenrod Algebra is AC ∼= M2
⟨Sq1, Sq2, . . .
⟩/Adem.
The HF2 motivic Adams spectral sequence for S0,0 takes the form
ExtAC(M2,M2) =⇒ π∗,∗(S0,02),
and the element τ ∈ Ext0 survives to a map S0,−1 τGGGA S0,0
2
, butdoes not exist before 2-completion.
Therefore, we work in the 2-completed category, and S0,0 meansthe 2-completed sphere.
![Page 31: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/31.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The Motivic Adams Spectral sequence
We now fix p = 2 for the remaining of the talk.
Theorem (Voevodsky)
The coefficients are HF2∗,∗(S0,0) = M2
∼= F2[τ ] for |τ | = (0, 1).
The HF2-Steenrod Algebra is AC ∼= M2
⟨Sq1, Sq2, . . .
⟩/Adem.
The HF2 motivic Adams spectral sequence for S0,0 takes the form
ExtAC(M2,M2) =⇒ π∗,∗(S0,02),
and the element τ ∈ Ext0 survives to a map S0,−1 τGGGA S0,0
2, butdoes not exist before 2-completion.
Therefore, we work in the 2-completed category, and S0,0 meansthe 2-completed sphere.
![Page 32: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/32.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The realization functor and τ
The map S0,−1 τGGA S0,0 realizes to
S0 idGGA S0
and realization has the computational effect of setting τ = 1.
From the motivic A.s.s. to the classical A.s.s.
copies of M2 become copies of F2
copies of M2/τn disappear, i.e., τ -torsion disappears.
For example η4 ∈ π4,4 is not zero, but is τ -torsion as τη4 = 0, and soη4 realizes to 0 which is consistent with the classical η4 = 0 ∈ π4.
Question
What happens when we let τ = 0 ?
![Page 33: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/33.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The realization functor and τ
The map S0,−1 τGGA S0,0 realizes to
S0 idGGA S0
and realization has the computational effect of setting τ = 1.
From the motivic A.s.s. to the classical A.s.s.
copies of M2 become copies of F2
copies of M2/τn disappear, i.e., τ -torsion disappears.
For example η4 ∈ π4,4 is not zero, but is τ -torsion as τη4 = 0, and soη4 realizes to 0 which is consistent with the classical η4 = 0 ∈ π4.
Question
What happens when we let τ = 0 ?
![Page 34: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/34.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The realization functor and τ
The map S0,−1 τGGA S0,0 realizes to
S0 idGGA S0
and realization has the computational effect of setting τ = 1.
From the motivic A.s.s. to the classical A.s.s.
copies of M2 become copies of F2
copies of M2/τn disappear, i.e., τ -torsion disappears.
For example η4 ∈ π4,4 is not zero
, but is τ -torsion as τη4 = 0, and soη4 realizes to 0 which is consistent with the classical η4 = 0 ∈ π4.
Question
What happens when we let τ = 0 ?
![Page 35: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/35.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The realization functor and τ
The map S0,−1 τGGA S0,0 realizes to
S0 idGGA S0
and realization has the computational effect of setting τ = 1.
From the motivic A.s.s. to the classical A.s.s.
copies of M2 become copies of F2
copies of M2/τn disappear, i.e., τ -torsion disappears.
For example η4 ∈ π4,4 is not zero, but is τ -torsion as τη4 = 0
, and soη4 realizes to 0 which is consistent with the classical η4 = 0 ∈ π4.
Question
What happens when we let τ = 0 ?
![Page 36: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/36.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The realization functor and τ
The map S0,−1 τGGA S0,0 realizes to
S0 idGGA S0
and realization has the computational effect of setting τ = 1.
From the motivic A.s.s. to the classical A.s.s.
copies of M2 become copies of F2
copies of M2/τn disappear, i.e., τ -torsion disappears.
For example η4 ∈ π4,4 is not zero, but is τ -torsion as τη4 = 0, and soη4 realizes to 0 which is consistent with the classical η4 = 0 ∈ π4.
Question
What happens when we let τ = 0 ?
![Page 37: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/37.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The realization functor and τ
The map S0,−1 τGGA S0,0 realizes to
S0 idGGA S0
and realization has the computational effect of setting τ = 1.
From the motivic A.s.s. to the classical A.s.s.
copies of M2 become copies of F2
copies of M2/τn disappear, i.e., τ -torsion disappears.
For example η4 ∈ π4,4 is not zero, but is τ -torsion as τη4 = 0, and soη4 realizes to 0 which is consistent with the classical η4 = 0 ∈ π4.
Question
What happens when we let τ = 0 ?
![Page 38: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/38.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The cofiber Cτ and its homotopy
Let’s look at the 2-cell complex Cτ that fits in the cofiber sequence
S0,−1 τGGA S0,0 i
GGA Cτp
GGA S1,−1.
Although it realizes to a tiny ∗ ∈ Top, its homotopy is a miracle:
Theorem (Hu-Kriz-Ormsby, Isaksen)
There is an isomorphism of bigraded abelian groups
π∗,∗(Cτ)∼=
GGA E2(S0;BP ),
where E2(S0;BP ) is a (harmless) regrading of the Adams-NovikovE2-page for the sphere S0, i.e., ExtBP∗BP (BP∗, BP∗).
![Page 39: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/39.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The cofiber Cτ and its homotopy
Let’s look at the 2-cell complex Cτ that fits in the cofiber sequence
S0,−1 τGGA S0,0 i
GGA Cτp
GGA S1,−1.
Although it realizes to a tiny ∗ ∈ Top
, its homotopy is a miracle:
Theorem (Hu-Kriz-Ormsby, Isaksen)
There is an isomorphism of bigraded abelian groups
π∗,∗(Cτ)∼=
GGA E2(S0;BP ),
where E2(S0;BP ) is a (harmless) regrading of the Adams-NovikovE2-page for the sphere S0, i.e., ExtBP∗BP (BP∗, BP∗).
![Page 40: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/40.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The cofiber Cτ and its homotopy
Let’s look at the 2-cell complex Cτ that fits in the cofiber sequence
S0,−1 τGGA S0,0 i
GGA Cτp
GGA S1,−1.
Although it realizes to a tiny ∗ ∈ Top, its homotopy is a miracle:
Theorem (Hu-Kriz-Ormsby, Isaksen)
There is an isomorphism of bigraded abelian groups
π∗,∗(Cτ)∼=
GGA E2(S0;BP ),
where E2(S0;BP ) is a (harmless) regrading of the Adams-NovikovE2-page for the sphere S0, i.e., ExtBP∗BP (BP∗, BP∗).
![Page 41: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/41.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The cofiber Cτ and its homotopy
Let’s look at the 2-cell complex Cτ that fits in the cofiber sequence
S0,−1 τGGA S0,0 i
GGA Cτp
GGA S1,−1.
Although it realizes to a tiny ∗ ∈ Top, its homotopy is a miracle:
Theorem (Hu-Kriz-Ormsby, Isaksen)
There is an isomorphism of bigraded abelian groups
π∗,∗(Cτ)∼=
GGA E2(S0;BP ),
where E2(S0;BP ) is a (harmless) regrading of the Adams-NovikovE2-page for the sphere S0, i.e., ExtBP∗BP (BP∗, BP∗).
![Page 42: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/42.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The cofiber Cτ and its homotopy
Let’s look at the 2-cell complex Cτ that fits in the cofiber sequence
S0,−1 τGGA S0,0 i
GGA Cτp
GGA S1,−1.
Although it realizes to a tiny ∗ ∈ Top, its homotopy is a miracle:
Theorem (Hu-Kriz-Ormsby, Isaksen)
There is an isomorphism of bigraded abelian groups
π∗,∗(Cτ)∼=
GGA E2(S0;BP ),
where E2(S0;BP ) is a (harmless) regrading of the Adams-NovikovE2-page for the sphere S0, i.e., ExtBP∗BP (BP∗, BP∗).
![Page 43: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/43.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Very cool question
Question
Is there a ring structure on Cτ inducing the product on E2-AN(S0) ?
![Page 44: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/44.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Notice the big vanishing regions for Cτ
The classical E2-AN(S0) has big vanishing areas:
f
s
zero
zerozero
zero
Adams-Novikov filtration > stem
negative Adams-Novikov filtration
negative stem.
These vanishing areas give via the isomorhism π∗,∗(Cτ) ∼= E2-AN(S0)
w
s
non-vanishing homotopy
zero
zero zero
zero
zero
lots of vanishing in πs,w(Cτ),
w
s
non-vanishing region
zero
zero zero
zero
zero
![Page 45: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/45.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Notice the big vanishing regions for Cτ
The classical E2-AN(S0) has big vanishing areas:
f
s
zero
zerozero
zero
Adams-Novikov filtration > stem
negative Adams-Novikov filtration
negative stem.
These vanishing areas give via the isomorhism π∗,∗(Cτ) ∼= E2-AN(S0)
w
s
non-vanishing homotopy
zero
zero zero
zero
zero
lots of vanishing in πs,w(Cτ),
w
s
non-vanishing region
zero
zero zero
zero
zero
![Page 46: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/46.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Notice the big vanishing regions for Cτ
The classical E2-AN(S0) has big vanishing areas:
f
s
zero
zero
zero
zero
Adams-Novikov filtration > stem
negative Adams-Novikov filtration
negative stem.
These vanishing areas give via the isomorhism π∗,∗(Cτ) ∼= E2-AN(S0)
w
s
non-vanishing homotopy
zero
zero zero
zero
zero
lots of vanishing in πs,w(Cτ),
w
s
non-vanishing region
zero
zero zero
zero
zero
![Page 47: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/47.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Notice the big vanishing regions for Cτ
The classical E2-AN(S0) has big vanishing areas:
f
s
zero
zerozero
zero
Adams-Novikov filtration > stem
negative Adams-Novikov filtration
negative stem.
These vanishing areas give via the isomorhism π∗,∗(Cτ) ∼= E2-AN(S0)
w
s
non-vanishing homotopy
zero
zero zero
zero
zero
lots of vanishing in πs,w(Cτ),
w
s
non-vanishing region
zero
zero zero
zero
zero
![Page 48: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/48.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Notice the big vanishing regions for Cτ
The classical E2-AN(S0) has big vanishing areas:
f
s
zero
zerozero
zero
Adams-Novikov filtration > stem
negative Adams-Novikov filtration
negative stem.
These vanishing areas give via the isomorhism π∗,∗(Cτ) ∼= E2-AN(S0)
w
s
non-vanishing homotopy
zero
zero zero
zero
zero
lots of vanishing in πs,w(Cτ),
w
s
non-vanishing region
zero
zero zero
zero
zero
![Page 49: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/49.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Notice the big vanishing regions for Cτ
The classical E2-AN(S0) has big vanishing areas:
f
s
zero
zerozero
zero
Adams-Novikov filtration > stem
negative Adams-Novikov filtration
negative stem.
These vanishing areas give via the isomorhism π∗,∗(Cτ) ∼= E2-AN(S0)
w
s
non-vanishing homotopy
zero
zero zero
zero
zero
lots of vanishing in πs,w(Cτ),
w
s
non-vanishing region
zero
zero zero
zero
zero
![Page 50: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/50.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Notice the big vanishing regions for Cτ
The classical E2-AN(S0) has big vanishing areas:
f
s
zero
zerozero
zero
Adams-Novikov filtration > stem
negative Adams-Novikov filtration
negative stem.
These vanishing areas give via the isomorhism π∗,∗(Cτ) ∼= E2-AN(S0)
w
s
non-vanishing homotopy
zero
zero zero
zero
zero
lots of vanishing in πs,w(Cτ),
w
s
non-vanishing region
zero
zero zero
zero
zero
and lots of vanishing in [Σs,wCτ,Cτ ].
![Page 51: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/51.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Notice the big vanishing regions for Cτ
The classical E2-AN(S0) has big vanishing areas:
f
s
zero
zerozero
zero
Adams-Novikov filtration > stem
negative Adams-Novikov filtration
negative stem.
These vanishing areas give via the isomorhism π∗,∗(Cτ) ∼= E2-AN(S0)
w
s
non-vanishing homotopy
zero
zero zero
zero
zero
lots of vanishing in πs,w(Cτ),
w
s
non-vanishing region
zero
zero zero
zero
zero
we will use [Σ0,−1Cτ,Cτ ] = 0.
![Page 52: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/52.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Notice the big vanishing regions for Cτ
The classical E2-AN(S0) has big vanishing areas:
f
s
zero
zerozero
zero
Adams-Novikov filtration > stem
negative Adams-Novikov filtration
negative stem.
These vanishing areas give via the isomorhism π∗,∗(Cτ) ∼= E2-AN(S0)
w
s
non-vanishing homotopy
zero
zero zero
zero
zero
lots of vanishing in πs,w(Cτ),
w
s
non-vanishing region
zero
zero zero
zero
zero
and also use [Σ1,−1Cτ,Cτ ] = 0.
![Page 53: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/53.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The ring structure of Cτ
Smash with − ∧ Cτ the defining cofiber sequence of Cτ
S0,−1 τGGA S0,0 i
GGA Cτp
GGA S1,−1.
S0,−1 ∧ Cτ S0,0 ∧ Cτ Cτ ∧ Cτ S1,−1 ∧ Cτ
Cτ.
τ i p
∼= ∃ µ ?
τ ∈ [Σ0,−1Cτ,Cτ ] = 0
there is a left unital multipication µand a splitting Cτ ∧ Cτ ' Cτ ∨ Σ1,−1Cτ
[Σ1,−1Cτ,Cτ ] = 0
µ is uniqueµ is the projection on the first factor Cτ
![Page 54: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/54.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The ring structure of Cτ
Smash with − ∧ Cτ the defining cofiber sequence of Cτ
S0,−1 ∧ Cττ
GGA S0,0 ∧ Cτi
GGA Cτ ∧ Cτp
GGA S1,−1 ∧ Cτ.
S0,−1 ∧ Cτ S0,0 ∧ Cτ Cτ ∧ Cτ S1,−1 ∧ Cτ
Cτ.
τ i p
∼= ∃ µ ?
τ ∈ [Σ0,−1Cτ,Cτ ] = 0
there is a left unital multipication µand a splitting Cτ ∧ Cτ ' Cτ ∨ Σ1,−1Cτ
[Σ1,−1Cτ,Cτ ] = 0
µ is uniqueµ is the projection on the first factor Cτ
![Page 55: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/55.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The ring structure of Cτ
Smash with − ∧ Cτ the defining cofiber sequence of Cτ
S0,−1 ∧ Cτ S0,0 ∧ Cτ Cτ ∧ Cτ S1,−1 ∧ Cτ
Cτ.
τ i p
∼= ∃ µ ?
τ ∈ [Σ0,−1Cτ,Cτ ] = 0
there is a left unital multipication µand a splitting Cτ ∧ Cτ ' Cτ ∨ Σ1,−1Cτ
[Σ1,−1Cτ,Cτ ] = 0
µ is uniqueµ is the projection on the first factor Cτ
![Page 56: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/56.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The ring structure of Cτ
Smash with − ∧ Cτ the defining cofiber sequence of Cτ
S0,−1 ∧ Cτ S0,0 ∧ Cτ Cτ ∧ Cτ S1,−1 ∧ Cτ
Cτ.
τ i p
∼= ∃ µ ?
τ ∈ [Σ0,−1Cτ,Cτ ] = 0
there is a left unital multipication µand a splitting Cτ ∧ Cτ ' Cτ ∨ Σ1,−1Cτ
[Σ1,−1Cτ,Cτ ] = 0
µ is uniqueµ is the projection on the first factor Cτ
![Page 57: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/57.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The ring structure of Cτ
Smash with − ∧ Cτ the defining cofiber sequence of Cτ
S0,−1 ∧ Cτ S0,0 ∧ Cτ Cτ ∧ Cτ S1,−1 ∧ Cτ
Cτ.
τ i p
∼= ∃ µ ?
τ ∈ [Σ0,−1Cτ,Cτ ] = 0
there is a left unital multipication µand a splitting Cτ ∧ Cτ ' Cτ ∨ Σ1,−1Cτ
[Σ1,−1Cτ,Cτ ] = 0
µ is uniqueµ is the projection on the first factor Cτ
![Page 58: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/58.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The good ring structure on Cτ
Theorem (G.)
The multiplication on Cτ extends (uniquely) to an E∞-ring structure.
Corollary
The isomorphism π∗,∗(Cτ) ∼= E2-AN(S0) is an isomorphism of higherrings, i.e., preserves all higher products.
Theorem (G.)
In fact every Cτn admits a unique E∞-ring structure.
![Page 59: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/59.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The good ring structure on Cτ
Theorem (G.)
The multiplication on Cτ extends (uniquely) to an E∞-ring structure.
Corollary
The isomorphism π∗,∗(Cτ) ∼= E2-AN(S0) is an isomorphism of higherrings, i.e., preserves all higher products.
Theorem (G.)
In fact every Cτn admits a unique E∞-ring structure.
![Page 60: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/60.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The good ring structure on Cτ
Theorem (G.)
The multiplication on Cτ extends (uniquely) to an E∞-ring structure.
Corollary
The isomorphism π∗,∗(Cτ) ∼= E2-AN(S0) is an isomorphism of higherrings, i.e., preserves all higher products.
Theorem (G.)
In fact every Cτn admits a unique E∞-ring structure.
![Page 61: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/61.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Operations and Co-operations on Cτ
Recall the maps i and p in the defining cofiber sequence of Cτ
S0,−1 τGGA S0,0 i
GGA Cτp
GGA S1,−1.
Proposition (G.)
The E∞-ring spectrum Cτ ∧ Cτ has homotopy ring
π∗,∗ (Cτ ∧ Cτ) ∼= E2-AN(S0)[x] /x2
The A∞-endomorphism spectrum End(Cτ) has homotopy ring
π∗,∗ (End(Cτ)) ∼= E2-AN(S0) 〈x〉/
ax− (−1)|a|xa = i ◦ p(a)x2 = 0
![Page 62: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/62.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Operations and Co-operations on Cτ
Recall the maps i and p in the defining cofiber sequence of Cτ
S0,−1 τGGA S0,0 i
GGA Cτp
GGA S1,−1.
Proposition (G.)
The E∞-ring spectrum Cτ ∧ Cτ has homotopy ring
π∗,∗ (Cτ ∧ Cτ) ∼= E2-AN(S0)[x] /x2
The A∞-endomorphism spectrum End(Cτ) has homotopy ring
π∗,∗ (End(Cτ)) ∼= E2-AN(S0) 〈x〉/
ax− (−1)|a|xa = i ◦ p(a)x2 = 0
![Page 63: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/63.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Operations and Co-operations on Cτ
Recall the maps i and p in the defining cofiber sequence of Cτ
S0,−1 τGGA S0,0 i
GGA Cτp
GGA S1,−1.
Proposition (G.)
The E∞-ring spectrum Cτ ∧ Cτ has homotopy ring
π∗,∗ (Cτ ∧ Cτ) ∼= E2-AN(S0)[x] /x2
The A∞-endomorphism spectrum End(Cτ) has homotopy ring
π∗,∗ (End(Cτ)) ∼= E2-AN(S0) 〈x〉/
ax− (−1)|a|xa = i ◦ p(a)x2 = 0
![Page 64: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/64.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Applications to Motivic Chromatic Homotopytheory
![Page 65: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/65.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Parts of the classical Chromatic story
1 Start with complex cobordism MU .
2 Quillen showed MU(p) ' ∨BP .
3 From BP , we construct the fields K(n).
Here are some cool properties of these guys:
1 The Morava K-theories K(n) are essentially the only gradedfields, and K(n)∗-acyclic spectra the only thick subcategoriesof FinSpt.
2 MU detects nilpotence, and p-locally BP does too.
3 Every X ∈ FinSpt(p) has a well-defined type, and any spectrum
of type n admits a periodic self-map inducing ·vkn in K(n).
![Page 66: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/66.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Parts of the classical Chromatic story
1 Start with complex cobordism MU .
2 Quillen showed MU(p) ' ∨BP .
3 From BP , we construct the fields K(n).
Here are some cool properties of these guys:
1 The Morava K-theories K(n) are essentially the only gradedfields, and K(n)∗-acyclic spectra the only thick subcategoriesof FinSpt.
2 MU detects nilpotence, and p-locally BP does too.
3 Every X ∈ FinSpt(p) has a well-defined type, and any spectrum
of type n admits a periodic self-map inducing ·vkn in K(n).
![Page 67: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/67.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Parts of the classical Chromatic story
1 Start with complex cobordism MU .
2 Quillen showed MU(p) ' ∨BP .
3 From BP , we construct the fields K(n).
Here are some cool properties of these guys:
1 The Morava K-theories K(n) are essentially the only gradedfields, and K(n)∗-acyclic spectra the only thick subcategoriesof FinSpt.
2 MU detects nilpotence, and p-locally BP does too.
3 Every X ∈ FinSpt(p) has a well-defined type, and any spectrum
of type n admits a periodic self-map inducing ·vkn in K(n).
![Page 68: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/68.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Parts of the classical Chromatic story
1 Start with complex cobordism MU .
2 Quillen showed MU(p) ' ∨BP .
3 From BP , we construct the fields K(n).
Here are some cool properties of these guys:
1 The Morava K-theories K(n) are essentially the only gradedfields, and K(n)∗-acyclic spectra the only thick subcategoriesof FinSpt.
2 MU detects nilpotence, and p-locally BP does too.
3 Every X ∈ FinSpt(p) has a well-defined type, and any spectrum
of type n admits a periodic self-map inducing ·vkn in K(n).
![Page 69: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/69.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Parts of the classical Chromatic story
1 Start with complex cobordism MU .
2 Quillen showed MU(p) ' ∨BP .
3 From BP , we construct the fields K(n).
Here are some cool properties of these guys:
1 The Morava K-theories K(n) are essentially the only gradedfields, and K(n)∗-acyclic spectra the only thick subcategoriesof FinSpt.
2 MU detects nilpotence, and p-locally BP does too.
3 Every X ∈ FinSpt(p) has a well-defined type, and any spectrum
of type n admits a periodic self-map inducing ·vkn in K(n).
![Page 70: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/70.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Parts of the classical Chromatic story
1 Start with complex cobordism MU .
2 Quillen showed MU(p) ' ∨BP .
3 From BP , we construct the fields K(n).
Here are some cool properties of these guys:
1 The Morava K-theories K(n) are essentially the only gradedfields
, and K(n)∗-acyclic spectra the only thick subcategoriesof FinSpt.
2 MU detects nilpotence, and p-locally BP does too.
3 Every X ∈ FinSpt(p) has a well-defined type, and any spectrum
of type n admits a periodic self-map inducing ·vkn in K(n).
![Page 71: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/71.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Parts of the classical Chromatic story
1 Start with complex cobordism MU .
2 Quillen showed MU(p) ' ∨BP .
3 From BP , we construct the fields K(n).
Here are some cool properties of these guys:
1 The Morava K-theories K(n) are essentially the only gradedfields, and K(n)∗-acyclic spectra the only thick subcategoriesof FinSpt.
2 MU detects nilpotence, and p-locally BP does too.
3 Every X ∈ FinSpt(p) has a well-defined type, and any spectrum
of type n admits a periodic self-map inducing ·vkn in K(n).
![Page 72: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/72.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Parts of the classical Chromatic story
1 Start with complex cobordism MU .
2 Quillen showed MU(p) ' ∨BP .
3 From BP , we construct the fields K(n).
Here are some cool properties of these guys:
1 The Morava K-theories K(n) are essentially the only gradedfields, and K(n)∗-acyclic spectra the only thick subcategoriesof FinSpt.
2 MU detects nilpotence
, and p-locally BP does too.
3 Every X ∈ FinSpt(p) has a well-defined type, and any spectrum
of type n admits a periodic self-map inducing ·vkn in K(n).
![Page 73: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/73.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Parts of the classical Chromatic story
1 Start with complex cobordism MU .
2 Quillen showed MU(p) ' ∨BP .
3 From BP , we construct the fields K(n).
Here are some cool properties of these guys:
1 The Morava K-theories K(n) are essentially the only gradedfields, and K(n)∗-acyclic spectra the only thick subcategoriesof FinSpt.
2 MU detects nilpotence, and p-locally BP does too.
3 Every X ∈ FinSpt(p) has a well-defined type, and any spectrum
of type n admits a periodic self-map inducing ·vkn in K(n).
![Page 74: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/74.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Parts of the classical Chromatic story
1 Start with complex cobordism MU .
2 Quillen showed MU(p) ' ∨BP .
3 From BP , we construct the fields K(n).
Here are some cool properties of these guys:
1 The Morava K-theories K(n) are essentially the only gradedfields, and K(n)∗-acyclic spectra the only thick subcategoriesof FinSpt.
2 MU detects nilpotence, and p-locally BP does too.
3 Every X ∈ FinSpt(p) has a well-defined type, and any spectrum
of type n admits a periodic self-map inducing ·vkn in K(n).
![Page 75: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/75.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
What is the Motivic Chromatic story ? Let p = 2.
1 There is an algebraic cobordism MGL, with MGL∗,∗ = Z2[τ ][xi].
2 Similarly MGL ' ∨BPGL with BPGL∗,∗ ∼= Z2[τ ][vi].
3 We also get Morava K-theories K(n) with K(n)∗,∗ ∼= F2[τ ][v±1n ].
However the story is more complicated, for example:
1 The K(n)’s are not fields (even though K(n) ∧ Cτ are).
2 There are more thick subcategories [Joachimi].
3 MGL does not detect nilpotence, as η : S1,1GGA S0,0 is not
nilpotent and all |xi| = (2i, 2i−1) are in even degrees.
4 No idea what to say about type.
![Page 76: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/76.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
What is the Motivic Chromatic story ? Let p = 2.
1 There is an algebraic cobordism MGL, with MGL∗,∗ = Z2[τ ][xi].
2 Similarly MGL ' ∨BPGL with BPGL∗,∗ ∼= Z2[τ ][vi].
3 We also get Morava K-theories K(n) with K(n)∗,∗ ∼= F2[τ ][v±1n ].
However the story is more complicated, for example:
1 The K(n)’s are not fields (even though K(n) ∧ Cτ are).
2 There are more thick subcategories [Joachimi].
3 MGL does not detect nilpotence, as η : S1,1GGA S0,0 is not
nilpotent and all |xi| = (2i, 2i−1) are in even degrees.
4 No idea what to say about type.
![Page 77: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/77.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
What is the Motivic Chromatic story ? Let p = 2.
1 There is an algebraic cobordism MGL, with MGL∗,∗ = Z2[τ ][xi].
2 Similarly MGL ' ∨BPGL with BPGL∗,∗ ∼= Z2[τ ][vi].
3 We also get Morava K-theories K(n) with K(n)∗,∗ ∼= F2[τ ][v±1n ].
However the story is more complicated, for example:
1 The K(n)’s are not fields (even though K(n) ∧ Cτ are).
2 There are more thick subcategories [Joachimi].
3 MGL does not detect nilpotence, as η : S1,1GGA S0,0 is not
nilpotent and all |xi| = (2i, 2i−1) are in even degrees.
4 No idea what to say about type.
![Page 78: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/78.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
What is the Motivic Chromatic story ? Let p = 2.
1 There is an algebraic cobordism MGL, with MGL∗,∗ = Z2[τ ][xi].
2 Similarly MGL ' ∨BPGL with BPGL∗,∗ ∼= Z2[τ ][vi].
3 We also get Morava K-theories K(n) with K(n)∗,∗ ∼= F2[τ ][v±1n ].
However the story is more complicated, for example:
1 The K(n)’s are not fields (even though K(n) ∧ Cτ are).
2 There are more thick subcategories [Joachimi].
3 MGL does not detect nilpotence, as η : S1,1GGA S0,0 is not
nilpotent and all |xi| = (2i, 2i−1) are in even degrees.
4 No idea what to say about type.
![Page 79: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/79.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
What is the Motivic Chromatic story ? Let p = 2.
1 There is an algebraic cobordism MGL, with MGL∗,∗ = Z2[τ ][xi].
2 Similarly MGL ' ∨BPGL with BPGL∗,∗ ∼= Z2[τ ][vi].
3 We also get Morava K-theories K(n) with K(n)∗,∗ ∼= F2[τ ][v±1n ].
However the story is more complicated, for example:
1 The K(n)’s are not fields (even though K(n) ∧ Cτ are).
2 There are more thick subcategories [Joachimi].
3 MGL does not detect nilpotence, as η : S1,1GGA S0,0 is not
nilpotent and all |xi| = (2i, 2i−1) are in even degrees.
4 No idea what to say about type.
![Page 80: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/80.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
What is the Motivic Chromatic story ? Let p = 2.
1 There is an algebraic cobordism MGL, with MGL∗,∗ = Z2[τ ][xi].
2 Similarly MGL ' ∨BPGL with BPGL∗,∗ ∼= Z2[τ ][vi].
3 We also get Morava K-theories K(n) with K(n)∗,∗ ∼= F2[τ ][v±1n ].
However the story is more complicated, for example:
1 The K(n)’s are not fields (even though K(n) ∧ Cτ are).
2 There are more thick subcategories [Joachimi].
3 MGL does not detect nilpotence, as η : S1,1GGA S0,0 is not
nilpotent and all |xi| = (2i, 2i−1) are in even degrees.
4 No idea what to say about type.
![Page 81: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/81.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
What is the Motivic Chromatic story ? Let p = 2.
1 There is an algebraic cobordism MGL, with MGL∗,∗ = Z2[τ ][xi].
2 Similarly MGL ' ∨BPGL with BPGL∗,∗ ∼= Z2[τ ][vi].
3 We also get Morava K-theories K(n) with K(n)∗,∗ ∼= F2[τ ][v±1n ].
However the story is more complicated, for example:
1 The K(n)’s are not fields (even though K(n) ∧ Cτ are).
2 There are more thick subcategories [Joachimi].
3 MGL does not detect nilpotence
, as η : S1,1GGA S0,0 is not
nilpotent and all |xi| = (2i, 2i−1) are in even degrees.
4 No idea what to say about type.
![Page 82: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/82.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
What is the Motivic Chromatic story ? Let p = 2.
1 There is an algebraic cobordism MGL, with MGL∗,∗ = Z2[τ ][xi].
2 Similarly MGL ' ∨BPGL with BPGL∗,∗ ∼= Z2[τ ][vi].
3 We also get Morava K-theories K(n) with K(n)∗,∗ ∼= F2[τ ][v±1n ].
However the story is more complicated, for example:
1 The K(n)’s are not fields (even though K(n) ∧ Cτ are).
2 There are more thick subcategories [Joachimi].
3 MGL does not detect nilpotence, as η : S1,1GGA S0,0 is not
nilpotent and all |xi| = (2i, 2i−1) are in even degrees.
4 No idea what to say about type.
![Page 83: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/83.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
What is the Motivic Chromatic story ? Let p = 2.
1 There is an algebraic cobordism MGL, with MGL∗,∗ = Z2[τ ][xi].
2 Similarly MGL ' ∨BPGL with BPGL∗,∗ ∼= Z2[τ ][vi].
3 We also get Morava K-theories K(n) with K(n)∗,∗ ∼= F2[τ ][v±1n ].
However the story is more complicated, for example:
1 The K(n)’s are not fields (even though K(n) ∧ Cτ are).
2 There are more thick subcategories [Joachimi].
3 MGL does not detect nilpotence, as η : S1,1GGA S0,0 is not
nilpotent and all |xi| = (2i, 2i−1) are in even degrees.
4 No idea what to say about type.
![Page 84: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/84.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
There is more (non-)Nilpotence and Periodicity
We need something bigger than MGL to detect nilpotence and tocapture all the periodicity.
1 There are more non-nilpotent elements than η ∈ π1,1. Forexample the classes detected by Ph1 ∈ π9,5, or d1 ∈ π32,18.
2 There are more periodicity operators than the vi’s. Forexample, the class that Ph1 detects is η-periodic, and DanIsaksen observed g-periodic classes in π∗,∗(S
0,0).
Michael Andrews et al. suggested that η = w0, and that there shouldbe an infinite family of wi’s behaving like the vi’s. He started theprocess and constructed a w4
1-map on Cη, at the prime p = 2. Hisintuition for these maps comes from using the algebraic Novikov s.s.
Using Cτ , the wi’s fit in the following setting:
![Page 85: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/85.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
There is more (non-)Nilpotence and Periodicity
We need something bigger than MGL to detect nilpotence and tocapture all the periodicity.
1 There are more non-nilpotent elements than η ∈ π1,1. Forexample the classes detected by Ph1 ∈ π9,5, or d1 ∈ π32,18.
2 There are more periodicity operators than the vi’s. Forexample, the class that Ph1 detects is η-periodic, and DanIsaksen observed g-periodic classes in π∗,∗(S
0,0).
Michael Andrews et al. suggested that η = w0, and that there shouldbe an infinite family of wi’s behaving like the vi’s. He started theprocess and constructed a w4
1-map on Cη, at the prime p = 2. Hisintuition for these maps comes from using the algebraic Novikov s.s.
Using Cτ , the wi’s fit in the following setting:
![Page 86: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/86.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
There is more (non-)Nilpotence and Periodicity
We need something bigger than MGL to detect nilpotence and tocapture all the periodicity.
1 There are more non-nilpotent elements than η ∈ π1,1. Forexample the classes detected by Ph1 ∈ π9,5, or d1 ∈ π32,18.
2 There are more periodicity operators than the vi’s.
Forexample, the class that Ph1 detects is η-periodic, and DanIsaksen observed g-periodic classes in π∗,∗(S
0,0).
Michael Andrews et al. suggested that η = w0, and that there shouldbe an infinite family of wi’s behaving like the vi’s. He started theprocess and constructed a w4
1-map on Cη, at the prime p = 2. Hisintuition for these maps comes from using the algebraic Novikov s.s.
Using Cτ , the wi’s fit in the following setting:
![Page 87: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/87.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
There is more (non-)Nilpotence and Periodicity
We need something bigger than MGL to detect nilpotence and tocapture all the periodicity.
1 There are more non-nilpotent elements than η ∈ π1,1. Forexample the classes detected by Ph1 ∈ π9,5, or d1 ∈ π32,18.
2 There are more periodicity operators than the vi’s. Forexample, the class that Ph1 detects is η-periodic
, and DanIsaksen observed g-periodic classes in π∗,∗(S
0,0).
Michael Andrews et al. suggested that η = w0, and that there shouldbe an infinite family of wi’s behaving like the vi’s. He started theprocess and constructed a w4
1-map on Cη, at the prime p = 2. Hisintuition for these maps comes from using the algebraic Novikov s.s.
Using Cτ , the wi’s fit in the following setting:
![Page 88: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/88.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
There is more (non-)Nilpotence and Periodicity
We need something bigger than MGL to detect nilpotence and tocapture all the periodicity.
1 There are more non-nilpotent elements than η ∈ π1,1. Forexample the classes detected by Ph1 ∈ π9,5, or d1 ∈ π32,18.
2 There are more periodicity operators than the vi’s. Forexample, the class that Ph1 detects is η-periodic, and DanIsaksen observed g-periodic classes in π∗,∗(S
0,0).
Michael Andrews et al. suggested that η = w0, and that there shouldbe an infinite family of wi’s behaving like the vi’s. He started theprocess and constructed a w4
1-map on Cη, at the prime p = 2. Hisintuition for these maps comes from using the algebraic Novikov s.s.
Using Cτ , the wi’s fit in the following setting:
![Page 89: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/89.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
There is more (non-)Nilpotence and Periodicity
We need something bigger than MGL to detect nilpotence and tocapture all the periodicity.
1 There are more non-nilpotent elements than η ∈ π1,1. Forexample the classes detected by Ph1 ∈ π9,5, or d1 ∈ π32,18.
2 There are more periodicity operators than the vi’s. Forexample, the class that Ph1 detects is η-periodic, and DanIsaksen observed g-periodic classes in π∗,∗(S
0,0).
Michael Andrews et al. suggested that η = w0, and that there shouldbe an infinite family of wi’s behaving like the vi’s.
He started theprocess and constructed a w4
1-map on Cη, at the prime p = 2. Hisintuition for these maps comes from using the algebraic Novikov s.s.
Using Cτ , the wi’s fit in the following setting:
![Page 90: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/90.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
There is more (non-)Nilpotence and Periodicity
We need something bigger than MGL to detect nilpotence and tocapture all the periodicity.
1 There are more non-nilpotent elements than η ∈ π1,1. Forexample the classes detected by Ph1 ∈ π9,5, or d1 ∈ π32,18.
2 There are more periodicity operators than the vi’s. Forexample, the class that Ph1 detects is η-periodic, and DanIsaksen observed g-periodic classes in π∗,∗(S
0,0).
Michael Andrews et al. suggested that η = w0, and that there shouldbe an infinite family of wi’s behaving like the vi’s. He started theprocess and constructed a w4
1-map on Cη, at the prime p = 2.
Hisintuition for these maps comes from using the algebraic Novikov s.s.
Using Cτ , the wi’s fit in the following setting:
![Page 91: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/91.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
There is more (non-)Nilpotence and Periodicity
We need something bigger than MGL to detect nilpotence and tocapture all the periodicity.
1 There are more non-nilpotent elements than η ∈ π1,1. Forexample the classes detected by Ph1 ∈ π9,5, or d1 ∈ π32,18.
2 There are more periodicity operators than the vi’s. Forexample, the class that Ph1 detects is η-periodic, and DanIsaksen observed g-periodic classes in π∗,∗(S
0,0).
Michael Andrews et al. suggested that η = w0, and that there shouldbe an infinite family of wi’s behaving like the vi’s. He started theprocess and constructed a w4
1-map on Cη, at the prime p = 2. Hisintuition for these maps comes from using the algebraic Novikov s.s.
Using Cτ , the wi’s fit in the following setting:
![Page 92: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/92.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
There is more (non-)Nilpotence and Periodicity
We need something bigger than MGL to detect nilpotence and tocapture all the periodicity.
1 There are more non-nilpotent elements than η ∈ π1,1. Forexample the classes detected by Ph1 ∈ π9,5, or d1 ∈ π32,18.
2 There are more periodicity operators than the vi’s. Forexample, the class that Ph1 detects is η-periodic, and DanIsaksen observed g-periodic classes in π∗,∗(S
0,0).
Michael Andrews et al. suggested that η = w0, and that there shouldbe an infinite family of wi’s behaving like the vi’s. He started theprocess and constructed a w4
1-map on Cη, at the prime p = 2. Hisintuition for these maps comes from using the algebraic Novikov s.s.
Using Cτ , the wi’s fit in the following setting:
![Page 93: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/93.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
wBP and Morava K-theories K(wi)
Theorem (G.)
1 For every n, there is an E∞-ring spectrum K(wn) with homotopy
π∗,∗(K(wn)) ∼= F2[w±1n ]
,
which is a graded field and with the correct cohomology.
2 There is a (almost certainly E∞) ring spectrum wBP withhomotopy
π∗,∗(wBP ) ∼= F2[w0, w1, . . .],
and with the correct cohomology.
Question
Where do the wi’s come from ?
![Page 94: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/94.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
wBP and Morava K-theories K(wi)
Theorem (G.)
1 For every n, there is an E∞-ring spectrum K(wn) with homotopy
π∗,∗(K(wn)) ∼= F2[w±1n ],
which is a graded field and with the correct cohomology.
2 There is a (almost certainly E∞) ring spectrum wBP withhomotopy
π∗,∗(wBP ) ∼= F2[w0, w1, . . .],
and with the correct cohomology.
Question
Where do the wi’s come from ?
![Page 95: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/95.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
wBP and Morava K-theories K(wi)
Theorem (G.)
1 For every n, there is an E∞-ring spectrum K(wn) with homotopy
π∗,∗(K(wn)) ∼= F2[w±1n ],
which is a graded field and with the correct cohomology.
2 There is a (almost certainly E∞) ring spectrum wBP withhomotopy
π∗,∗(wBP ) ∼= F2[w0, w1, . . .]
,
and with the correct cohomology.
Question
Where do the wi’s come from ?
![Page 96: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/96.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
wBP and Morava K-theories K(wi)
Theorem (G.)
1 For every n, there is an E∞-ring spectrum K(wn) with homotopy
π∗,∗(K(wn)) ∼= F2[w±1n ],
which is a graded field and with the correct cohomology.
2 There is a (almost certainly E∞) ring spectrum wBP withhomotopy
π∗,∗(wBP ) ∼= F2[w0, w1, . . .],
and with the correct cohomology.
Question
Where do the wi’s come from ?
![Page 97: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/97.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
wBP and Morava K-theories K(wi)
Theorem (G.)
1 For every n, there is an E∞-ring spectrum K(wn) with homotopy
π∗,∗(K(wn)) ∼= F2[w±1n ],
which is a graded field and with the correct cohomology.
2 There is a (almost certainly E∞) ring spectrum wBP withhomotopy
π∗,∗(wBP ) ∼= F2[w0, w1, . . .],
and with the correct cohomology.
Question
Where do the wi’s come from ?
![Page 98: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/98.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The vi’s and the Steenrod Algebra
Voevodsky computed the motivic HF2-Steenrod Algebra, its dual is
A∗,∗ ∼= M2[ξ1, ξ2, . . . , τ0, τ1, . . .]/τ2i = τξi+1
,
and denote by Qi ∈ A the dual of τi in the monomial basis.
1 The Qi’s are primitive and exterior.
2 HF∗,∗2 (BPGL) ∼= A//E(Q0, Q1, . . .).
3 By a change of rings, its Adams s.s. collapses giving
π∗,∗(BPGL)2 ∼= Z2[τ ][v1, v2, . . .].
![Page 99: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/99.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The vi’s and the Steenrod Algebra
Voevodsky computed the motivic HF2-Steenrod Algebra, its dual is
A∗,∗ ∼= M2[ξ1, ξ2, . . . , τ0, τ1, . . .]/τ2i = τξi+1
,
and denote by Qi ∈ A the dual of τi in the monomial basis.
1 The Qi’s are primitive and exterior.
2 HF∗,∗2 (BPGL) ∼= A//E(Q0, Q1, . . .).
3 By a change of rings, its Adams s.s. collapses giving
π∗,∗(BPGL)2 ∼= Z2[τ ][v1, v2, . . .].
![Page 100: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/100.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The vi’s and the Steenrod Algebra
Voevodsky computed the motivic HF2-Steenrod Algebra, its dual is
A∗,∗ ∼= M2[ξ1, ξ2, . . . , τ0, τ1, . . .]/τ2i = τξi+1
,
and denote by Qi ∈ A the dual of τi in the monomial basis.
1 The Qi’s are primitive
and exterior.
2 HF∗,∗2 (BPGL) ∼= A//E(Q0, Q1, . . .).
3 By a change of rings, its Adams s.s. collapses giving
π∗,∗(BPGL)2 ∼= Z2[τ ][v1, v2, . . .].
![Page 101: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/101.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The vi’s and the Steenrod Algebra
Voevodsky computed the motivic HF2-Steenrod Algebra, its dual is
A∗,∗ ∼= M2[ξ1, ξ2, . . . , τ0, τ1, . . .]/τ2i = τξi+1
,
and denote by Qi ∈ A the dual of τi in the monomial basis.
1 The Qi’s are primitive and exterior.
2 HF∗,∗2 (BPGL) ∼= A//E(Q0, Q1, . . .).
3 By a change of rings, its Adams s.s. collapses giving
π∗,∗(BPGL)2 ∼= Z2[τ ][v1, v2, . . .].
![Page 102: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/102.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The vi’s and the Steenrod Algebra
Voevodsky computed the motivic HF2-Steenrod Algebra, its dual is
A∗,∗ ∼= M2[ξ1, ξ2, . . . , τ0, τ1, . . .]/τ2i = τξi+1
,
and denote by Qi ∈ A the dual of τi in the monomial basis.
1 The Qi’s are primitive and exterior.
2 HF∗,∗2 (BPGL) ∼= A//E(Q0, Q1, . . .).
3 By a change of rings, its Adams s.s. collapses giving
π∗,∗(BPGL)2 ∼= Z2[τ ][v1, v2, . . .].
![Page 103: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/103.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The vi’s and the Steenrod Algebra
Voevodsky computed the motivic HF2-Steenrod Algebra, its dual is
A∗,∗ ∼= M2[ξ1, ξ2, . . . , τ0, τ1, . . .]/τ2i = τξi+1
,
and denote by Qi ∈ A the dual of τi in the monomial basis.
1 The Qi’s are primitive and exterior.
2 HF∗,∗2 (BPGL) ∼= A//E(Q0, Q1, . . .).
3 By a change of rings, its Adams s.s. collapses giving
π∗,∗(BPGL)2 ∼= Z2[τ ][v1, v2, . . .].
![Page 104: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/104.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The wi’s and the Steenrod Algebra
Voevodsky computed the motivic HF2-Steenrod Algebra, its dual is
A∗,∗ ∼= M2[ξ1, ξ2, . . . , τ0, τ1, . . .]/τ2i = τξi+1
,
and denote by Ri ∈ A the dual of ξi in the monomial basis.
The wi’s would like to arise from the Ri’s, but they are not exterior.
Remark
The Ri’s are exterior modulo τ .
Since τη4 = 0 ∈ π∗,∗, we need to mod out by τ if we wantpolynomial homotopy in the wi’s.
![Page 105: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/105.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The wi’s and the Steenrod Algebra
Voevodsky computed the motivic HF2-Steenrod Algebra, its dual is
A∗,∗ ∼= M2[ξ1, ξ2, . . . , τ0, τ1, . . .]/τ2i = τξi+1
,
and denote by Ri ∈ A the dual of ξi in the monomial basis.
The wi’s would like to arise from the Ri’s, but they are not exterior.
Remark
The Ri’s are exterior modulo τ .
Since τη4 = 0 ∈ π∗,∗, we need to mod out by τ if we wantpolynomial homotopy in the wi’s.
![Page 106: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/106.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The wi’s and the Steenrod Algebra
Voevodsky computed the motivic HF2-Steenrod Algebra, its dual is
A∗,∗ ∼= M2[ξ1, ξ2, . . . , τ0, τ1, . . .]/τ2i = τξi+1
,
and denote by Ri ∈ A the dual of ξi in the monomial basis.
The wi’s would like to arise from the Ri’s, but they are not exterior.
Remark
The Ri’s are exterior modulo τ .
Since τη4 = 0 ∈ π∗,∗, we need to mod out by τ if we wantpolynomial homotopy in the wi’s.
![Page 107: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/107.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The wi’s and the Steenrod Algebra
Voevodsky computed the motivic HF2-Steenrod Algebra, its dual is
A∗,∗ ∼= M2[ξ1, ξ2, . . . , τ0, τ1, . . .]/τ2i = τξi+1
,
and denote by Ri ∈ A the dual of ξi in the monomial basis.
The wi’s would like to arise from the Ri’s, but they are not exterior.
Remark
The Ri’s are exterior modulo τ .
Since τη4 = 0 ∈ π∗,∗, we need to mod out by τ if we wantpolynomial homotopy in the wi’s.
![Page 108: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/108.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The wi’s and the Steenrod Algebra
Voevodsky computed the motivic HF2-Steenrod Algebra, its dual is
A∗,∗ ∼= M2[ξ1, ξ2, . . . , τ0, τ1, . . .]/τ2i = τξi+1
,
and denote by Ri ∈ A the dual of ξi in the monomial basis.
The wi’s would like to arise from the Ri’s, but they are not exterior.
Remark
The Ri’s are exterior modulo τ .
Since τη4 = 0 ∈ π∗,∗
, we need to mod out by τ if we wantpolynomial homotopy in the wi’s.
![Page 109: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/109.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The wi’s and the Steenrod Algebra
Voevodsky computed the motivic HF2-Steenrod Algebra, its dual is
A∗,∗ ∼= M2[ξ1, ξ2, . . . , τ0, τ1, . . .]/τ2i = τξi+1
,
and denote by Ri ∈ A the dual of ξi in the monomial basis.
The wi’s would like to arise from the Ri’s, but they are not exterior.
Remark
The Ri’s are exterior modulo τ .
Since τη4 = 0 ∈ π∗,∗, we need to mod out by τ if we wantpolynomial homotopy in the wi’s.
![Page 110: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/110.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The wi’s from HF2 ∧ Cτ
Therefore, let H = HF2 ∧ Cτ and it has coefficients H∗,∗ ∼= F2.
Wecan compute its Steenrod algebra since we understand End(Cτ), andits dual is
A∗,∗ ∼= F2[ξ1, ξ2, . . .]⊗ E(τ0, τ1, . . .)⊗ E(x),
where x is a τ -Bockstein and the Ri’s are now primitive and exterior.
We are looking for a spectrum with the property
H∗,∗(wBP ) ∼= A//E(R1, R2, . . .),
its Adams s.s. would collapse and give π∗,∗(wBP )2 ∼= F2[w0, w1, . . .].
![Page 111: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/111.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The wi’s from HF2 ∧ Cτ
Therefore, let H = HF2 ∧ Cτ and it has coefficients H∗,∗ ∼= F2. Wecan compute its Steenrod algebra since we understand End(Cτ)
, andits dual is
A∗,∗ ∼= F2[ξ1, ξ2, . . .]⊗ E(τ0, τ1, . . .)⊗ E(x),
where x is a τ -Bockstein and the Ri’s are now primitive and exterior.
We are looking for a spectrum with the property
H∗,∗(wBP ) ∼= A//E(R1, R2, . . .),
its Adams s.s. would collapse and give π∗,∗(wBP )2 ∼= F2[w0, w1, . . .].
![Page 112: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/112.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The wi’s from HF2 ∧ Cτ
Therefore, let H = HF2 ∧ Cτ and it has coefficients H∗,∗ ∼= F2. Wecan compute its Steenrod algebra since we understand End(Cτ), andits dual is
A∗,∗ ∼= F2[ξ1, ξ2, . . .]⊗ E(τ0, τ1, . . .)⊗ E(x),
where x is a τ -Bockstein
and the Ri’s are now primitive and exterior.
We are looking for a spectrum with the property
H∗,∗(wBP ) ∼= A//E(R1, R2, . . .),
its Adams s.s. would collapse and give π∗,∗(wBP )2 ∼= F2[w0, w1, . . .].
![Page 113: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/113.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The wi’s from HF2 ∧ Cτ
Therefore, let H = HF2 ∧ Cτ and it has coefficients H∗,∗ ∼= F2. Wecan compute its Steenrod algebra since we understand End(Cτ), andits dual is
A∗,∗ ∼= F2[ξ1, ξ2, . . .]⊗ E(τ0, τ1, . . .)⊗ E(x),
where x is a τ -Bockstein and the Ri’s are now primitive and exterior.
We are looking for a spectrum with the property
H∗,∗(wBP ) ∼= A//E(R1, R2, . . .),
its Adams s.s. would collapse and give π∗,∗(wBP )2 ∼= F2[w0, w1, . . .].
![Page 114: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/114.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The wi’s from HF2 ∧ Cτ
Therefore, let H = HF2 ∧ Cτ and it has coefficients H∗,∗ ∼= F2. Wecan compute its Steenrod algebra since we understand End(Cτ), andits dual is
A∗,∗ ∼= F2[ξ1, ξ2, . . .]⊗ E(τ0, τ1, . . .)⊗ E(x),
where x is a τ -Bockstein and the Ri’s are now primitive and exterior.
We are looking for a spectrum with the property
H∗,∗(wBP ) ∼= A//E(R1, R2, . . .)
,
its Adams s.s. would collapse and give π∗,∗(wBP )2 ∼= F2[w0, w1, . . .].
![Page 115: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/115.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
The wi’s from HF2 ∧ Cτ
Therefore, let H = HF2 ∧ Cτ and it has coefficients H∗,∗ ∼= F2. Wecan compute its Steenrod algebra since we understand End(Cτ), andits dual is
A∗,∗ ∼= F2[ξ1, ξ2, . . .]⊗ E(τ0, τ1, . . .)⊗ E(x),
where x is a τ -Bockstein and the Ri’s are now primitive and exterior.
We are looking for a spectrum with the property
H∗,∗(wBP ) ∼= A//E(R1, R2, . . .),
its Adams s.s. would collapse and give π∗,∗(wBP )2 ∼= F2[w0, w1, . . .].
![Page 116: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/116.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
How about wMU ?
The degree of the wi’s on π∗,∗(wBP ) are |wi| = (2i+1 − 3, 2i − 1)
, so
|w0| = (1, 1)
|w1| = (5, 3)
nothing in (9, 5)
|w2| = (13, 7)
etc,
which is the same pattern as the vi’s of BP∗ between the xi’s of MU∗.
Corollary
There is a (almost certainly E∞) ring spectrum wMU with homotopy
π∗,∗(wMU) ∼= F2[y1, y2, . . .],
where |yi| = (4i+ 1, 2i+ 1), and which splits as a wedge of wBP ’s.
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Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
How about wMU ?
The degree of the wi’s on π∗,∗(wBP ) are |wi| = (2i+1 − 3, 2i − 1), so
|w0| = (1, 1)
|w1| = (5, 3)
nothing in (9, 5)
|w2| = (13, 7)
etc,
which is the same pattern as the vi’s of BP∗ between the xi’s of MU∗.
Corollary
There is a (almost certainly E∞) ring spectrum wMU with homotopy
π∗,∗(wMU) ∼= F2[y1, y2, . . .],
where |yi| = (4i+ 1, 2i+ 1), and which splits as a wedge of wBP ’s.
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Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
How about wMU ?
The degree of the wi’s on π∗,∗(wBP ) are |wi| = (2i+1 − 3, 2i − 1), so
|w0| = (1, 1)
|w1| = (5, 3)
nothing in (9, 5)
|w2| = (13, 7)
etc,
which is the same pattern as the vi’s of BP∗ between the xi’s of MU∗.
Corollary
There is a (almost certainly E∞) ring spectrum wMU with homotopy
π∗,∗(wMU) ∼= F2[y1, y2, . . .],
where |yi| = (4i+ 1, 2i+ 1), and which splits as a wedge of wBP ’s.
![Page 119: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/119.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
How about wMU ?
The degree of the wi’s on π∗,∗(wBP ) are |wi| = (2i+1 − 3, 2i − 1), so
|w0| = (1, 1)
|w1| = (5, 3)
nothing in (9, 5)
|w2| = (13, 7)
etc,
which is the same pattern as the vi’s of BP∗ between the xi’s of MU∗.
Corollary
There is a (almost certainly E∞) ring spectrum wMU with homotopy
π∗,∗(wMU) ∼= F2[y1, y2, . . .],
where |yi| = (4i+ 1, 2i+ 1), and which splits as a wedge of wBP ’s.
![Page 120: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/120.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
What’s next ?
Question
Is there an interpretation of wMU ?
Do motivic BP and wBP capture all the chromatic phenomena ?
The K(w0)-local sphere was computed by Andrews-Miller withGuillou-Isaksen
π∗,∗(LK(w0)S
0,0) ∼= F2[η±1][σ, µ9]
/(ησ)2 .
What is the LK(w1)S0,0 ?
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Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
What’s next ?
Question
Is there an interpretation of wMU ?
Do motivic BP and wBP capture all the chromatic phenomena ?
The K(w0)-local sphere was computed by Andrews-Miller withGuillou-Isaksen
π∗,∗(LK(w0)S
0,0) ∼= F2[η±1][σ, µ9]
/(ησ)2 .
What is the LK(w1)S0,0 ?
![Page 122: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/122.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
What’s next ?
Question
Is there an interpretation of wMU ?
Do motivic BP and wBP capture all the chromatic phenomena ?
The K(w0)-local sphere was computed by Andrews-Miller withGuillou-Isaksen
π∗,∗(LK(w0)S
0,0) ∼= F2[η±1][σ, µ9]
/(ησ)2 .
What is the LK(w1)S0,0 ?
![Page 123: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/123.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
What’s next ?
Question
Is there an interpretation of wMU ?
Do motivic BP and wBP capture all the chromatic phenomena ?
The K(w0)-local sphere was computed by Andrews-Miller withGuillou-Isaksen
π∗,∗(LK(w0)S
0,0) ∼= F2[η±1][σ, µ9]
/(ησ)2 .
What is the LK(w1)S0,0 ?
![Page 124: The co ber C˝ and Motivic Chromatic stuMotivic Homotopy Theory The cofiber C˝ Applications to Motivic Chromatic Homotopy theory Bonus Unstable Motivic Spaces I will only work over](https://reader033.fdocuments.net/reader033/viewer/2022053001/5f054be57e708231d41242e9/html5/thumbnails/124.jpg)
Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Bonus
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Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Bonus
1 Bonus 1: S/2 ∧ Cτ admits a v11-self map (instead of v41 on S/2)
2 Bonus 2: kO ∧ Cτ admits a v21-self map (instead of v41 on kO)
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Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Bonus 1: S/2 ∧ Cτ admits a v11-map
There is no map Σ2,1S/2v1
GGA S/2. Indeed
Σ2,1S/2 S2,1 S2,1
S/2 S1,0 S1,0
2
2
η
∃ η@ v1
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Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Bonus 1: S/2 ∧ Cτ admits a v11-map
There is no map Σ2,1S/2v1
GGA S/2. Indeed
Σ2,1S/2 S2,1 S2,1
S/2 S1,0 S1,0
2
2
η∃ η
@ v1
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Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Bonus 1: S/2 ∧ Cτ admits a v11-map
There is no map Σ2,1S/2v1
GGA S/2. Indeed
Σ2,1S/2 S2,1 S2,1
S/2 S1,0 S1,0
2
2
η∃ η
@ v1
since 2 · η is not zero in π2,1S/2 ∼= Z/4.
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Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Bonus 1: S/2 ∧ Cτ admits a v11-map
After smashing with Cτ , there is a map Σ2,1Cτ/2v1
GGA Cτ/2. Indeed
Σ2,1Cτ/2 Σ2,1Cτ Σ2,1Cτ
Cτ/2 Σ1,0Cτ Σ1,0Cτ
2
2
η
∃ η∃ v1
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Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Bonus 1: S/2 ∧ Cτ admits a v11-map
After smashing with Cτ , there is a map Σ2,1Cτ/2v1
GGA Cτ/2. Indeed
Σ2,1Cτ/2 Σ2,1Cτ Σ2,1Cτ
Cτ/2 Σ1,0Cτ Σ1,0Cτ
2
2
η∃ η
∃ v1
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Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Bonus 1: S/2 ∧ Cτ admits a v11-map
After smashing with Cτ , there is a map Σ2,1Cτ/2v1
GGA Cτ/2. Indeed
Σ2,1Cτ/2 Σ2,1Cτ Σ2,1Cτ
Cτ/2 Σ1,0Cτ Σ1,0Cτ
2
2
η∃ η
∃ v1
since 2 · η is zero in[Σ2,1Cτ,Cτ/2
] ∼= Z/2.
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Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Bonus 1: S/2 ∧ Cτ admits a v11-map
After smashing with Cτ , there is a map Σ2,1Cτ/2v1
GGA Cτ/2. Indeed
Σ2,1Cτ/2 Σ2,1Cτ Σ2,1Cτ
Cτ/2 Σ1,0Cτ Σ1,0Cτ
2
2
η∃ η
∃ v1
since 2 · η is zero in[Σ2,1Cτ,Cτ/2
] ∼= Z/2. More concisely, theobstruction to having a v11-map is the bracket 〈2, η, 2〉 = τη2, and thusCτ/2 enjoys it.
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Motivic Homotopy Theory The cofiber Cτ Applications to Motivic Chromatic Homotopy theory Bonus
Thank you for your attention !
w
s5 10
5
α1
α3
α5
α7
α4/4
α6/3
α2/2
zero
zero
Figure: The homotopy groups πs,w(Cτ), with lots of non-nilpotent elements2, α1, α3, α5, α7, . . ..