An Introduction to Chromatic Homotopy TheoryPart II : Complex Orientations and the Morava K -Theories

Agnes Beaudry

May 16, 2019

Last Time

(1) Spectra Sp and the stable homotopy category SH.

§ SH is a closed symmetric monoidal triangulated categorywhose objects represent (co)homology theories.

§ SH is obtained from Sp by inverting theπ˚ “ S0

˚-isomorphisms.

(2) Bousfield Localization and E -local spectra SHE

§ SHE is obtained from Sp by inverting the E˚-isomorphisms.§ E -Local spectra Y : E˚pZq “ 0 ùñ rZ ,Y s “ 0§ E -Localization X Ñ LEX :

X

��

// Y E -localoo

E˚-iso //

E -local // LEX

;;

Part II – Complex Orientations and the Morava K -Theories

(1) Complex Orientations

(2) Formal Group Laws

(3) Height

(4) Landweber Exact Functor Theorem

(5) Morava K -Theories

(6) Chromatic Fracture Square

Ring Objects

A homotopy ring spectrum E is a ring object in SH. In particular, E has aunit and a multiplication:

S0 Ñ E , E ^E Ñ E

which satisfy the diagrams of a ring object in SH.

It is homotopy commutative if the following diagram commutes in SH:

E ^Eσ //

""

E ^E

||E

For such E , π˚E is a graded ring. It is graded commutative if E is commutative.

Complex Orientations

A complex orientation for a homotopy ring spectrum E is a class

x “ xE P rE2pCP8q

whose restriction under the map

E2pCP8q Ñ E2pCP1q – E2pS2q – π0E

is the unit.

Eilenberg-MacLane Spectra

Let γ1 Ñ CP8 be the tautological line bundle. Recall that

HZ˚pCP8q – Zrxs

wherex “ xHZ P HZ2pCP8q

is the first Chern class of γ1. This is a complex orientation. In fact,

HZ˚pCP8q – HZ˚rrxss.

Complex K -Theory

Let β P π2K be the Bott class and rγ1 ´ 1s P rK0pCP8q. Then

K˚pCP8q – Zrβ˘1srrrγ1 ´ 1sss

The classx “ xK “ β´1rγ1 ´ 1s P rK2pCP8q

is a complex orientation. In fact,

K˚pCP8q – K˚rrxss.

Complex Cobordism

Note that CP8 – BUp1q. The zero section

CP8 Ñ Thompγ1q “: MUp1q

is a homotopy equivalence and gives a class

x “ xMU P rCP8,MU2s “ ĄMU2pCP8q

which is a complex orientation. In fact,

MU˚pCP8q – MU˚rrxss

Exercise : Chern Classes

For a complex oriented theory E with orientation x “ xE ,

E˚ppCP8qmq – E˚rrx1, . . . , xmss xi P rE2ppCP8qmq

andE˚pBUq – E˚rrc1, c2, . . .ss ci P rE2i pBUq

Hint. Use the Atiyah–Hirzebruch spectral sequence.

The ci are the Chern classes of E . The map

E˚pBUq Ñ E˚pBUp1qq – E˚pCP8q

maps c1 to x .

Tensor Product of Line Bundles

CP8 ˆ CP8b // CP8

classifying the tensor product of line bundles gives map

E˚rrxss – E˚pCP8qE˚pbq // E˚pCP8 ˆ CP8q – E˚rrx , yss

E˚pbqpxq “ F px , yq “: x `F y

For

CP8 i // CP8

classifying the C-linear dual of γ1,

E˚piqpxq “: ipxq

In fact,c1p`1 b `2q “ F pc1p`1q, c1p`2qq

Properties of b imply:

(1) px `F yq `F z “ x `F py `F zq

(2) x `F y “ y `F x

(3) x `F 0 “ x , 0`F y “ y .

(4) x `F ipxq “ 0

Formal Group Laws

For R˚ “ R´˚ a graded ring and x , y P pR˚rrx , yssq´2. Define a category

FGLpR˚q

Objects. A formal group law (fgl) over R˚ is a power series

x `F y “ F px , yq P pR˚rrx , yssq´2

satisfying the properties (1)-(4).

Morphisms. A morphism f : F Ñ G of fgls is a power series

f pxq P pxR˚rrxssq´2 such that f px `F yq “ f pxq `G f pyq.

The identity is x .

Base Change. If φ : R˚ Ñ S˚ is a ring homomorphism, applying φ to coeffi-cients gives a functor

φ : FGLpR˚q Ñ FGLpS˚q

§ FHQpx , yq “ x ` y is the additive fgl.

§ FK px , yq “ β´1ppβx ` 1qpβy ` 1q ´ 1q is the multiplicative fgl.

Lazard and Quillen’s Theorem – Algebra

MU carries the universal formal group law:

FMUpx , yq “ÿ

i,j

ai,jxiy j

over the Lazard ring

MU˚ – L – Zraij : i , j ě 0s{I

where I is ideal of relations required for FMU to be a fgl.

MU˚ – Zrx1, x2, x3, . . .s, FMUpx , yq “ x ` y ` x1xy ` . . .

Universality. For a F P FGLpR˚q, there is a homomorphism φ : MU˚ Ñ R˚such that

φFMU “ F .

Quillen’s Theorem – Topology

For E complex oriented with fgl FE , there is a map of homotopy ring spectraf : MU Ñ E such that

pπ˚f qFMU “ FE px , yq

n-Series

For any n P N and formal group law F ,

rnsF pxq “ x `F . . .`F xlooooooomooooooon

n-times

is a morphism rnsF : F Ñ F . This is called the n-series.

For example:

§ FHQpx , yq “ x ` y sornsFHQ pxq “ nx

§ FK px , yq “ β´1ppβx ` 1qpβy ` 1q ´ 1q so

rnsFK pxq “ β´1ppβx ` 1qn ´ 1q

Fix p a prime. Let vn P MU2ppn´1q be the coefficient of xpn

in

rpsFMUpxq “ x `F . . .`F x

looooooomooooooon

p-times

“ px ` . . .` v1xp ` . . .` vnx

pn ` . . . .

vn “

#

p n “ 0

xpn´1 mod pp, x1, . . . , xpn´2q

Height

Let F P FGLpR˚q be classified by φ : MU˚ Ñ R˚. If φpvnq is a unit in R˚,then F has height ď n at p. If in addition φpvi q “ 0 for 0 ď i ă n, then F hasheight “ n.

§ FHQ has height 0:rpsFHQ pxq “ px “ v0x .

§ FK has height ď 1:

rpsFK pxq “ px ` pp. . .q ` βp´1xp .

§ KZ{p “ K ^SZ{p has height “ 1:

rpsFKZ{p pxq “ βp´1xp .

For a ring homomorphism φ : MU˚ Ñ R˚ consider the functor CW` Ñ Ab:

X ÞÑ R˚ bMU˚ĄMU˚pX q.

This is a homology theory provided that Exactness holds.

Landweber Exact Functor Theorem

(1) If for every p and n ě 0,

R˚{pp, v1, . . . , vn´1qvn // R˚{pp, v1, . . . , vn´1q

is injective, thenX ÞÑ R˚ bMU˚ MU˚pX q

is a homology theory.

(2) If E is complex oriented and MU˚ Ñ E˚ satisfies (1), then

E˚pX q – E˚ bMU˚ MU˚pX q.

Landweber Exact Functor Theorem

(1) If for every p and n ě 0,

R˚{pp, v1, . . . , vn´1qvn // R˚{pp, v1, . . . , vn´1q

is injective, thenX ÞÑ R˚ bMU˚ MU˚pX q

is a homology theory.

(2) If E is complex oriented and MU˚ Ñ E˚ is as in (1), then

E˚pX q – E˚ bMU˚ MU˚pX q.

BP and Johnson-Wilson Spectra

§ The Brown-Peterson BP is the spectrum coming from

MU˚ Ñ BP˚ “ π˚MUppq{pxi , i ‰ pk ´ 1q – Zppqrv1, v2, v3, . . .s

In fact, MUppq splits as a wedge of suspensions of BP.

§ The Johnson-Wilson Spectrum Epnq is the spectrum coming from

MU˚ Ñ Epnq˚ “ v´1n BP˚{pvn`1, . . .q – Zppqrv1, . . . , vn´1, v

˘1n s

The theory Epnq has height ď n.

Morava K -Theory

Fix p and n ě 1. The n’th Morava K -theory spectrum is constructed fromMUppq by killing

pp, x1, x2, . . . , xpn´2, xpn , xpn`1 . . .q

and inverting vn “ xpn´1. It satisfies

π˚Kpnq “ Kpnq˚ – Fprv˘1n s

and is a homotopy associative ring spectrum. It is complex oriented and

rpsFKpnq pxq “ vnxpn ` . . .

so FKpnq has height “ n at p.

§ Kp0q » HQ§ Kp1q » KZ{pCp´1 is the Adams Summand

§ Kp8q » HZ{p§ Kp2q is Kp2q

Note. These are not Landweber Exact Theories.

§ A spectrum is a (p-local) finite spectrum if it is isomorphic up to a shiftin SH to (the p-localization) Σ8X for X a finite CW complex.

§ E is a field if E˚X is free over E˚ for any X .

Some Properties of Kpnq

§ Kpnq is a field and any field E is a wedge of suspensions of Kpnq’s.

§ Kpnq˚ has a Kunneth Isomorphism:

Kpnq˚pX ^Y q – Kpnq˚pX q bKpnq˚ Kpnq˚pY q.

§ Kpnq^Kpmq » ˚ if m ‰ n

§ Let 0 ă n ă 8. If X is finite, then

Kpnq˚X “ 0 ùñ Kpn ´ 1q˚X “ 0.

Let C0 “ Cp,0 Ď Sp be the full subcategory of p-local finite spectra and

Cn “ Cp,n “ tX P C0 : Kpn ´ 1q˚X “ 0u

Then, Cn`1 Ď Cn. A spectrum has type n if it is in CnzCn`1.

An isomorphism f : F Ñ G , given by f pxq P xR˚rrxss, is strict if f pxq “ x` . . ..

Let ĄFGLpR˚q have objects formal group laws over R˚ and morphisms strict isos:

ĄFGL : Graded Rings ÝÑ Groupoids

Quillen’s Theorem – Continued

MU˚MU – MU˚rb1, b2, b3, . . .s bi P MU2iMU

wherefMUpxq “ x `

ÿ

iě1

bixi`1

represents the universal strict isomorphism. That is,

φ : MU˚MU Ñ R˚

is a the data:

F ,G P FGLpR˚q f : Fs. iso.

–// G

In particular, ĄFGL is representable:

objpĄFGLpR˚qq – HomRingspMU˚,R˚q

morpĄFGLpR˚qq – HomRingspMU˚MU,R˚q

The Moduli Stack Picture MFGL at p

0 ≤ 1 ≤ 2 ≤ n-1≤ n

…E(n)

K(n)

⟵ En

MďnFGL

{M“nFGL

oo

Mďn´1FGL

OO

Mďn´1FGL X {M“n

FGLoo

OO

Chromatic Fracture Square (Hopkins–Ravenel)

LetLn :“ LKp0q_..._Kpnq

ThenLn » LEpnq » L

v´1n MU

.

There is a natural transformations

LnX Ñ Ln´1X LnX Ñ LKpnqX

In fact, for any X , there is a homotopy pull-back:

LnX //

��

LKpnqX

��Ln´1X // Ln´1LKpnqX

MďnFGL

{M“nFGL

oo

Mďn´1FGL

OO

Mďn´1FGL X {M“n

FGLoo

OO

Thank you!

π˚Sp2q (Illustration by Isaksen)

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The E∞-page of the classical Adams spectral sequence

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h1W1

π˚Sp2q (Illustration by Isaksen)

π˚Sp2q (Illustration by Isaksen)

Telescope Conjecture (Ravenel)

The first n-rays are detected by LnS0.

Chromatic Splitting Conjecture (Hopkins)

The gluing data for the chromatic layers is simple.