A Survey of Elliptic Cohomology - formal groups and cohomology




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This is a sub-entry of

see there for background and context.

This entry disscusses basics of formal group laws arising from periodic multiplicative cohomology theories



rough notes from a talk

the following are rough unpolished notes taken more or less verbatim from some seminar talk – needs somebody to go through it and polish it

Formal groups and elliptic cohomology.

In all of the following, all cohomology theories are multiplicative and all formal group laws are one-dimensional (and commutative).

Last time. we saw that orienting a periodic even cohomology theory gives a formal group law over the cohomology ring A 0()A^0(\bullet). (Note: A 0A^0 and not A A^\bullet because of the periodicity property.)

Today we discuss a generalization of the above statement: orienting a weakly periodic even cohomology theory AA gives a formal group over A 0()A^0(\bullet). In particular, elliptic cohomology theories give elliptic curves over A 0()A^0(\bullet).

Formal group laws and Landweber’s criterion

Formal group laws of dimension 11 over RR are classified by morphisms from the Lazard ring to RR. We can define A f n(X)=MP n(X) MP()RA_f^n(X)=MP^n(X)\otimes_{MP(\bullet)}R. Here MPMP denotes complex cobordism, in particular MP()MP(\bullet) is isomorphic to Lazard's ring.

Definition. A sequence v 0,,v nv_0,\ldots,v_n of elements of RR is regular if endomorphisms of R/(v 0,,v k1)R/(v_0,\ldots,v_{k-1}) given by multiplication by v kv_k are injective for all 0kn0\le k\le n.

Landweber criterion Let f(x,y)f(x,y) be a formal group law and pp a prime, v iv_i the coefficient of x p ix^{p^i} in [p] f(x)=x+ f+ fx[p]_f(x)=x+_f\cdots+_fx. If v 0,,v iv_0,\ldots,v_i form a regular sequence for all pp and ii then f(x,y)f(x,y) gives a cohomology theory via the formula with tensor product above.

Example. g a(x,y)=x+yg_a(x,y)=x+y, [p] a(x)=px[p]_a(x)=px, v 0=pv_0=p, v i=0v_i=0 for all i1i\ge1; regularity condtions imply that the zero map R/(p)R/(p)R/(p)\to R/(p) must be injective. The last statement implies that RR contains the rational numbers as a subring.

Note that HP *(X,R)= kH n+2k(X,R)HP^*(X,R)=\prod_k H^{n+2k}(X,R) is a cohomology theory over any ring RR.

Example. g m(x,y)=xyg_m(x,y)=xy, [p] m(x)=(x+1) p1[p]_m(x)=(x+1)^p-1, v 0=pv_0=p, v 1=1v_1=1, v i=0v_i=0 for all i>1i \gt 1. The regularity conditions are trivial. Hence we know that K *(X)=MP *(X) MP()K^*(X)=MP^*(X)\otimes_{MP(\bullet)} \mathbb{Z} is a cohomology theory.

Formal groups from formal group laws

Given a commutative topological RR-algebra AA and a formal group law f(x,y)f(x,y) if f(a,b)f(a,b) converges for all a,bAa,b\in A and the formula giving an inverse to aa converges for all aAa\in A, we get an abelian group (A,+ f)(A,+_f), where a+ fb=f(a,b)a+_f b=f(a,b).

Example. For any AA the pair (N(A),+ f)(N(A),+_f) is an abelian group, where N(A)N(A) denotes the set of nilpotent elements of AA.

Example. Let AA be an oriented complex oriented cohomology theory. Then computing Chern classes of line bundles is the same as evaluating the formal group law of AA on some algebra. Recall that line bundles on XX are classified by maps from XX to P \mathbb{C}P^\infty, pairs of line bundles are classified by maps to P ×P \mathbb{C}P^\infty \times \mathbb{C}P^\infty, and tensor product of line bundles gives a map P ×P P \mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty. Now apply cohomology functor to the sequence XP ×P P X\to \mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty. We have a degree 0 element tt in the cohomology of P \mathbb{C}P^\infty. Its image in the cohomology of P ×P \mathbb{C}P^\infty \times \mathbb{C}P^\infty is a formal group law. The image of this formal group law in the cohomology of XX makes sense if XX is a finite cell complex so that A 0(X)A^0(X) is a nilpotent algebra.

Question: When do two formal group laws yield isomorphic groups?

Definiton. A homomorphism of formal group laws ff and gg over AA is a formal power series ϕA[x]\phi\in A[x] such that ϕ(f(x,y))=g(ϕ(x),ϕ(y))\phi(f(x,y))=g(\phi(x),\phi(y)). (The constant term of~ϕ\phi is zero.) Hence formal group laws form a category.

Example. If RR contains rational numbers as a subring, then we have two canonical homomorphisms. The first one is exp:g ag m\exp\colon g_a\to g_m, where exp(x)= k>0x k/k!\exp(x)=\sum_{k \gt 0}x^k/k!. Its inverse is log:g mg a\log\colon g_m\to g_a, where log(x)= k>0(1) k+1x k/k\log(x)=\sum_{k \gt 0}(-1)^{k+1}x^k/k. This shows up in cohomology as Chern character. (Isomorphism from K n(X) K^n(X)\otimes_{\mathbb{Z}} \mathbb{Q} to kH n+2k(X,)\prod_kH^{n+2k}(X,\mathbb{Q}).

Formal groups. A formal group is a group in the category of formal schemes.

A formal scheme Y^\hat{Y} is defined for any closed immersion of schemes YXY \hookrightarrow X. Intuitively the formal scheme Y^\hat Y is the \infty-jet bundle in the normal direction of YY inside of XX.

Definition. The locally ringed space Y^\hat Y is defined as the topological space YY with structure sheaf \lim O_X/{\mathcal I}^n, where \lim O_X/{\mathcal I\mathcal{I} is the defining sheaf of ideals of the closed immersion YXY\hookrightarrow X. (Where YY is a closed subscheme of XX.)

Examples. X=Y^X=\hat Y when Y=XY=X. Speck[t]=X\mathrm{Spec} k[t]=X, Y=V(t)Y=V(t), X^=k[t,t 1]\hat X=k[t,t^{-1}].

In fact not every locally noetherian formal scheme can be obtained as a completion of a single noetherian scheme in another scheme; such formal schemes are called algebraizable.

Definition. (formal spectrum) The formal spectrum SpfR\mathrm{Spf} R of a commutative noetherian ring RR with a specified ideal IRI \subset R whose powers define a local basis of a topology around 00 which is Hausdorff, is the locally ringed space with the underlying topological space SpecR/I\mathrm{Spec} R/I whose global sections of the structure sheaf are the limit

O SpfR(SpfR)=lim n(R/I n). O_{\mathrm{Spf} R}(\mathrm{Spf} R)=\lim_n (R/I^n) \,.

(This is incomplete description, one needs to talk sheaves of ideals instead)

formal group laws from elliptic curve

Recall from the above that a given a formal group law F(x,y)R[[x,y]]F(x,y) \in R[ [x,y] ] we get te structure of a formal group on the formal spectrum SpfSpf by taking the product to be given by

SpfR[[x,y]]Spf[[x]]×SpfR[[y]] SpfR[[z]] f(x,y) z \array{ Spf R[[x,y]] \simeq Spf[[x]] \times Spf R[[y]] &\to& Spf R[[z]] \\ f(x,y)&\leftarrow |& z }

Isomorphic formal group laws give [isomorphism|isomorphic]] (of formal groups) if GG a formal group has GSpfR[[z]]G \simeq Spf R[ [z] ]; we must choose such an iso to get a formal group law.

Now we get formal groups from elliptic curves over RR

Definition An elliptic curve over a commutative ring RR is a group object in the category of schemes over RR that is a relative 1-dimensional, , smooth curve, proper curve over RR.

This implies that it has genus 1. (by a direct argument of the Chern class of the tangent bundle.)

Given an elliptic curve over RR, ESpecRE \to Spec R, we get a formal group E^\hat E by completing DD along its identity section σ 0\sigma_0

ESpec(R)σ 0E E \to Spec(R) \stackrel{\sigma_0}{\to} E

(the one dual to the map that maps everything to 0R0 \in R), we get a ringed space (E^,O^ E,0)(\hat E, \hat O_{E,0})

example if RR is a field kk, then the structure sheaf O^ E,0k[[z]]\hat O_{E,0} \simeq k[ [z] ]


O^ E×E,(0,0)O^ E,0^ kO^ E,0k[[x,y]] \hat O_{E \times E, (0,0)} \simeq \hat O_{E,0} \hat \otimes_k \hat O_{E,0} \simeq k[[x,y]]

example (Jacobi quartics)

y 2=12δx 2+ϵx 4 y^2 = 1- 2 \delta x^2 + \epsilon x^4

defines EE over R=[Y Z,ϵ,δ]R = \mathbb{Z}[Y_Z,\epsilon, \delta].

The corresponding formal group law is Euler’s formal group law

f(x,y)=x12δy 2+ϵy 4+y12δx 2+ϵx 41ϵx 2y 2 f(x,y) = \frac{x\sqrt{1- 2 \delta y^2 + \epsilon y^4} + y \sqrt{1- 2 \delta x^2 + \epsilon x^4}} {1- \epsilon x^2 y^2}

if Δ:=ϵ(δ 2ϵ) 20\Delta := \epsilon(\delta^2 - \epsilon)^2 \neq 0 then this is a non-trivial elliptic curve.

If Δ=0\Delta = 0 then f(x,y)G m,G af(x,y) \simeq G_m, G_a (additive or multiplicative formal group law corresponding to integral cohomology and K-theory, respectively).

weakly periodic cohomology theories and formal groups

A multiplicative cohomology theory AA is weakly periodic if the natural map

A 2(*) A 0(*)A n(*)A n+2(*) A^2({*}) \otimes_{A^0({*})} A^n({*}) \stackrel{\simeq}{\to} A^{n+2}({*})

is an isomorphism for all nn \in \mathbb{Z}.

Compare with the notion of a periodic cohomology theory.

Relation to formal groups

One reason why weakly periodic cohomology theories are of interest is that their cohomology ring over the space P \mathbb{C}P^\infty defines a formal group.

To get a formal group from a weakly periodic, even multiplicative cohomology theory A A^\bullet, we look at the induced map on A A^\bullet from a morphism

i 0:*P i_0 : {*} \to \mathbb{C}P^\infty

and take the kernel

J:=ker(i 0 *:A 0(P )A 0(*)) J := ker(i_0^* : A^0(\mathbb{C}P^\infty) \to A^0({*}))

to be the ideal that we complete along to define the formal scheme SpfA 0(P )Spf A^0(\mathbb{C}P^\infty) (see there for details).

Notice that the map from the point is unique only up to homotopy, so accordingly there are lots of chocies here, which however all lead to the same result.

The fact that AA is weakly periodic allows to reconstruct the cohomology theory essentially from this formal scheme.

To get a formal group law from this we proceed as follows: if the Lie algebra Lie(SpfA 0(P ))Lie(Spf A^0(\mathbb{C}P^\infty)) of the formal group

Lie(SpfA 0(P ))ker(i 0 *)/ker(i 0 *) 2 Lie(Spf A^0(\mathbb{C}P^\infty)) \simeq ker(i_0^*)/ker(i_0^*)^2

is a free A 0(*)A^0({*})-module, we can pick a generator tt and this gives an isomorphism

Spf(A 0(P ))Spf(A 0(*)[[t]]) Spf(A^0(\mathbb{C}P^\infty)) \simeq Spf(A^0({*})[[t]])

if A 0(P )A 0(*)[[t]]A^0(\mathbb{C}P^\infty) A^0({*})[ [t] ] then i 0 *i_0^* “forgets the tt-coordinate”.

Definition An elliptic cohomology theory over RR is

So we have on one side

E^ SpfA 0(P ) SpecR SpecA 0(*) σ 0 E^ SpfA 0(P ) \array{ \hat E &\stackrel{\simeq}{\to}& Spf A^0(\mathbb{C}P^\infty) \\ \downarrow && \downarrow \\ Spec R &\stackrel{\simeq}{\to}& Spec A^0({*}) \\ \downarrow^{\sigma_0} && \downarrow \\ \hat E &\stackrel{\simeq}{\to}& Spf A^0(\mathbb{C}P^\infty) }

We can check that the Landweber exactness criterion is satisfied for the formal group law of the Jacobi quartic?, i.e. for Euler's formal group law? over [Δ 1,ϵ,δ,1/2]\mathbb{Z}[\Delta^{-1}, \epsilon, \delta, 1/2], so this provides an example of an elliptic cohomology theory.

A G n(X)=MP(X) MP(*)[Δ 1,ϵ,δ,1/2] A^n_G(X) = M P (X) \otimes_{M P({*})} \mathbb{Z}[\Delta^{-1}, \epsilon, \delta, 1/2]
Revised on October 7, 2010 11:38:08 by Urs Schreiber (