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A Survey of Elliptic Cohomology - towards a proof

Abstract This entry attempts to give an outline of a proof of Lurie’s main theorem.

This is a sub-entry of

see there for background and context.

Here are the entries on the previous sessions:


Towards a proof

Recall the main theorem.

Theorem

(J. Lurie)

For

  • AA any E-∞ ring

  • and E(A)E(A) is the space of oriented derived elliptic curves over AA (the realization of the topological category of elliptic curves over AA).

There exists a derived Deligne-Mumford stack M Der=(M,O Der)M^{Der} = (M, O^{Der}) such that we have an equivalence

Hom(SpecA,M Der)E(A) Hom(Spec A, M^{Der}) \simeq E(A)

natural in AA.

And O DerO^{Der} provides the lift of Goerss-Hopkins-Miller.

Preliminaries

Recall that for AA an E E_\infty-ring a derived elliptic curve FF is a commutative derived group scheme over AA such that F 0F_0 over π 0A\pi_0 A is an elliptic curve.

Denote by E(A)E' (A) the space of preoriented (derived) elliptic curves (so equipped with a map P F(SpecA)\mathbb{C} P^\infty \to F ( \mathrm{Spec} A ). And E(A)E(A) the space of oriented elliptic curves.

Note that a map SpecAM Der\mathrm{Spec} A \to M^{Der} is a map Specπ 0AM 1,1\mathrm{Spec} \pi_0 A \to M_{1,1} and a map of rings O(Specπ 0A)AO (\mathrm{Spec} \pi_0 A) \to A.

Representability

In his thesis, Lurie proves the following.

Proposition. Let FF be a functor from connective E E_\infty-ring spectra to spaces s.t.

  1. The restriction of FF to discrete rings is represented by a (classical) DM-stack XX, i.e. F(R)NervX(R)F(R) \simeq \mathrm{Nerv} X(R);

  2. FF is a sheaf with respect to the etale topology;

  3. FF has a good deformation theory.

Then there exists a derived DM-stack (X,O^)(X , \hat O ) representing FF s.t. O^(U)\hat O (U) is connective for UU affine.

Examples.

  1. The functor EE': observe that every classical elliptic curve over a discrete RR has a unique preorientation. Hence EE' is represented by DM-stack (M,O)(M, O' ).

  2. The functor EE: the theorem doesn’t apply as a discrete ring cannot be weakly periodic.

Claim. (M,O)(M, O') represents EE' for all E E_\infty-rings, so we dropped connectivity.

Proof. Recall the map Aτ 0AA \to \tau_{\ge 0} A to the connected cover.

  1. Let AA be an E E_\infty-ring, then we have an equivalence
Hom(O(Specπ 0τ 0A),τ 0A)Hom(O(Specπ 0A),A).\mathrm{Hom} (O' (\mathrm{Spec} \pi_0 \tau_{\ge 0} A) , \tau_{\ge 0} A ) \to \mathrm{Hom} (O' (\mathrm{Spec} \pi_0 A) , A).
  1. E(τ 0AE(A)E' ( \tau_{\ge 0} A \simeq E' (A) since every elliptic curve is by definition flat.

We need the following to prove the claim.

Proposition. The functor MM τ 0AA,M \mapsto M \otimes_{\tau_{\ge 0} A} A, from flat modules over τ 0A\tau_{\ge 0} A to flat modules over AA is an equivalence.

Proof of proposition (sketch). Let M,NM,N be AA-modules then there is a spectral sequence

Tor p π *A(π *M,π *N) qπ p+q(M AN).\mathrm{Tor}_{p}^{\pi_* A} (\pi_* M, \pi_* N )_q \Rightarrow \pi_{p+q} (M \otimes_A N ) .

Suppose MM is flat, then

Tor p π *A(π *M,π *N)=π *N π 0Aπ 0M\mathrm{Tor}_{p}^{\pi_* A} (\pi_* M , \pi_* N) = \pi_* N \otimes_{\pi_0 A} \pi_0 M

if p=0p=0 and 0 otherwise. Thus,

π *(M AN)π *N π 0Aπ 0M.\pi_* (M \otimes_A N ) \simeq \pi_* N \otimes_{\pi_0 A} \pi_0 M .

So we have

F:Mod flat(τ 0A)Mod flat(A):G. F: \mathrm{Mod}^{flat} (\tau_{\ge 0} A ) \leftrightarrow \mathrm{Mod}^{flat} (A) : G .

This is an equivalence and FF respects the monoidal structure, hence the equivalence extends to the categories of algebras and the proposition (and hence claim) is proved.

We need to know that π iO\pi_i O' are coherent sheaves over M 1,1M_{1,1}, where a coherent sheaf is an assignment SpecRM 1,1\mathrm{Spec} R \to M_{1,1} which behaves well under finite limits. Let ω\omega be the line bundle of invariant differentials on M 1,1M_{1,1} so that is

ω=e *Ω F| SpecR. \omega = e^* \Omega_{F |_{\mathrm{Spec} R}}.

Recall that a preorientation determines a map β:ωπ 2(O)\beta : \omega \to \pi_2 (O' ). So define a sheaf of E E_\infty-rings OO as O[β 1]O' [ \beta^{-1} ] which is characterized (maybe) by

π nO=lim(π n+2kO O 1,1ω k). \pi_n O = \lim ( \pi_{n+2k} O' \otimes_{O_{1,1}} \omega^{-k} ) .

Remark.

  • This formula comes from a much simpler situation…Let RR be (an honest to God) ring, xRx \in R then
R[x 1]lim(RxRx) R[x^{-1}] \simeq \lim (R \stackrel{x}{\to} R \stackrel{x}{\to} \dots )
  • Suppose UM 1,1U \to M_{1,1} with UU affine, ω\omega restricted to UU is trivialized then O(U)=AO' (U) =A and βπ 2A\beta \in \pi_2 A. Hence, O[β 1](U)=A[β 1]O' [\beta^{-1} ] (U) = A [ \beta^{-1} ] and we get the formula above π i(A[β 1])=(π iA)[β 1]\pi_i (A [ \beta^{-1} ]) = (\pi_i A) [\beta^{-1} ].

More generally, for UM 1,1U \to M_{1,1} we have that O(U)O(U) is weakly periodic, so

π 2O(U) π 0O(U)π nO(U)π n+2O(U) \pi_2 O (U) \otimes_{\pi_0 O(U)} \pi_n O(U) \to \pi_{n+2} O (U)

is an equivalence.

  • (M,O)(M, O) classifies oriented elliptic curve. Let FF over U=SpecBU = \mathrm{Spec} B be a preoriented elliptic curve over BB, so we have the classifying map f:O(U)=ABf: O' (U) = A \to B and this is an orientation iff
π nB π 0Bωπ n+2B \pi_n B \otimes_{\pi_0 B} \omega \to \pi_{n+2} B

is an isomorphism. This map can be identified with ×β\times \beta, so the preorientation is an orientation iff there is a unique factorization through O(U)=A[β 1]O(U) = A [ \beta^{-1} ].

Reduction

Claim. To prove the theorem it is enough to show

  1. O 1,1=π 0Oπ 0OO_{1,1} = \pi_0 O' \to \pi_0 O is an isomorphism;

  2. For nn odd, the sheaf π nO=0\pi_n O =0.

Proof. Suppose (1) and (2) hold. Let f:SpecRM 1,1f: \mathrm{Spec} R \to M_{1,1} be etale for RR discrete. We must show that O(SpecR)O (\mathrm{Spec} R) is an elliptic cohomology theory associated to ff. Condition (1) ensures π 0AR\pi_0 A \simeq R, (2) guarantees evenness and from above we have weakly periodic. We must show that SpfA 0(P )E^ f\mathrm{Spf} A^{0} (\mathbb{C}P^\infty ) \simeq \hat E_f which follows from having an orientation.

Reducing to a Local Calculation

We wish to show that π nO=0\pi_n O = 0 for nn odd. From above, it suffices to show that

f k:π n+2kOω kπ nO f_k : \pi_{n+2k} O' \otimes \omega^{-k} \to \pi_n O

is zero for all kk. Note that im(f k)\mathrm{im} (f_k ) is a quotient of π n+2kOim(f k)\pi_{n+2k} O' \to \mathrm{im} (f_k ) which is coherent. Suppose pp is an etale cover of M 1,1M_{1,1} then im(f k)=0\mathrm{im} (f_k ) = 0 iff p iim(f k)=0p^{i} \mathrm{im} (f_k )=0. We can find an etale cover by a disjoint union of level 3 and level 4 modular forms denoted SpecR\mathrm{Spec} R.

That M:=p iim(f k)=0M:= p^{i} \mathrm{im} (f_k ) =0 is equivalent to M RR/mM \otimes_R R/m for all mRm \in R. It is not difficult to show that all residue fields R/mR/m are finite in this case.

Now it is enough to show condition (2) formally locally as R^ m/mR m/m\hat R_m /m \simeq R_m / m.

Key Ingredients

In the previous section we had a moduli stack preoriented elliptic curves (M,O)(M,O'). The structure sheaf of OO' took values in connected E E_\infty-rings. We had a refinement (M,O)(M, O) which was a moduli stack for oriented elliptic curves. From the orientation condition we deduced that the structure sheaf took values in weakly periodic E E_\infty-rings.

Further, we showed how to reduce the main theorem to a (formal) local computation. That is, we only need to consider R^ m\hat R_m, the completion of a ring localized at a maximal ideal.

We delay the completion of the proof until later, but now we introduce the key technical tool: pp-divisible groups.

pp-divisible Groups

Let RR be a complete, local ring (e.g. the pp-adic integers p\mathbb{Z}_p) and E 0E_0 an elliptic curve over R 0:=R/M=F qR_0 := R/M = \mathbf{F}_q. What do we need in order to lift E 0E_0 to an elliptic curve over RR?

Let pp be a prime (say the characteristic of F q\mathbf{F}_q) and EE an elliptic curve over RR. Using the multiplication by pp map p n:EEp^n \colon E \to E we can define a sheaf of Abelian groups

E[p ]:=colimE[p n], E[p^\infty] := \mathrm{colim} \; E[p^n],

where E[p n]E[p^n] is the kernel of the map p np^n. That is, E[p n]E[p^n] corresponds to the pp-torsion points of EE.

Definition. A pp-divisible group \mathfrak{I} over RR is a sheaf of Abelian groups on the flat site of schemes over RR such that

  1. p n:p^n : \mathfrak{I} \to \mathfrak{I} is surjective;

  2. =colim[p n]\mathfrak{I} = \mathrm{colim} \; \mathfrak{I} [p^n] where =ker(p n:)\mathfrak{I} = \mathrm{ker} \; (p^n \colon \mathfrak{I} \to \mathfrak{I}).

  3. [p n]\mathfrak{I}[p^n] is a finite, flat, commutative RR-group scheme. Note that finite means that \mathfrak{I} is affine and whose global sections is a finite RR-module.

For instance, the constant sheaf p̲\underline{\mathbb{Z}_p} is a pp-divisible group.

Serre-Tate Theory

Now let RR, in addition to above, be Noetherian with residue field F q\mathbf{F}_q for q=p nq=p^n.

Theorem (Serre-Tate). Let E¯\overline{E} be an elliptic curve over F q\mathbf{F}_q, then there is an equivalence of categories between elliptic curves over RR that restrict to E¯\overline{E} and the category of pp-divisible groups \mathfrak{I} over RR such that the restriction of \mathfrak{I} to F q\mathbf{F}_q is E¯[p ]\overline{E} [p^\infty ].

The theorem is somewhat surprising as, a priori, the latter category sees only torsion phenomena of the elliptic curves.

Derived Serre-Tate Theory

Definition. Let AA be an E E_\infty-ring. A functor \mathfrak{I} from commutative AA-algebras to topological Abelian groups is a pp-divisible group if

  1. B(B)B \mapsto \mathfrak{I} (B) is a sheaf;

  2. p n:p^n : \mathfrak{I} \to \mathfrak{I} is surjective;

  3. (ho)colim[p n](\mathrm{ho})\mathrm{colim} \; \mathfrak{I} [p^n] \simeq \mathfrak{I}, where as above [p n]:=(ho)kerp n\mathfrak{I} [p^n] := (\mathrm{ho}) \mathrm{ker} \; p^n;

  4. [p n]\mathfrak{I}[p^n] is a derived commutative group scheme over AA which is finite and flat.

If \mathfrak{I} is a pp-divisible group over F q\mathbf{F}_q, then [p]\mathfrak{I} [p] is a finite F q\mathbf{F}_q-module of dimension rr called the rank of \mathfrak{I}.

Proposition. If =E[p ]\mathfrak{I} = E [p^\infty], for EE an elliptic curve, then \mathfrak{I} has rank 2.

One can verify the proposition over \mathbb{C} pretty easily; it is more subtle over a finite field. We have a derived version of the Serre-Tate theorem.

Theorem (Serre-Tate). Let AA be an E E_\infty-ring such that π 0A\pi_0 A is a complete, local, Noetherian ring and π iA\pi_i A are finitely generated π 0A\pi_0 A-modules. Let E 0E_0 be a (derived) elliptic curve over π 0A/M\pi_0 A / M, then there is an equivalence of \infty-categories: elliptic curves over AA that restrict to E 0E_0 and pp-divisible groups over AA that restrict to E 0[p ]E_0 [p^\infty].

Proof of the Classical Theorem

We give a proof of the classical result, however the proof is quite formal and should carry over to the derived setting.

Let RR be a ring as in the theorem such that NR=0N \cdot R =0 for some NN \in \mathbb{N} and let IRI \subset R be a nilpotent ideal, so I r+1=0I^{r+1} =0, and set R 0=R/IR_0 = R/I. Let :RalgAb\mathfrak{I} \colon R-\mathrm{alg} \to \mathrm{Ab} and

I(A):=ker((A)(A/IA). \mathfrak{I}_I (A) := \mathrm{ker} \; ( \mathfrak{I} (A) \to \mathfrak{I} (A / I \cdot A ) .

Further, define

¯(A):=ker((A)(A/nilRA)) I(A). \overline{\mathfrak{I}} (A) := \mathrm{ker} \; (\mathfrak{I}(A) \to \mathfrak{I} (A / \mathrm{nil}R \cdot A)) \subset \mathfrak{I}_I (A) .

Lemma. If \mathfrak{I} is a formal group over RR, then I\mathfrak{I}_I is annihilated by N rN^r.

Proposition. Let \mathfrak{I}, \mathfrak{H} be elliptic curves or pp-divisible groups over RR and let N=p nN=p^n. Denote by 0\mathfrak{I}_0 and 0\mathfrak{H}_0 the restriction of \mathfrak{I} and \mathfrak{H} to R 0R_0-algebras. Then

  1. Hom Rgrp(,)\mathrm{Hom}_{R-grp} (\mathfrak{I}, \mathfrak{H}) and Hom R 0grp( 0, 0)\mathrm{Hom}_{R_0-grp} (\mathfrak{I}_0 , \mathfrak{H}_0 ) have no NN-torsion;

  2. Hom(,)Hom( 0, 0)\mathrm{Hom} (\mathfrak{I} , \mathfrak{H}) \to \mathrm{Hom} (\mathfrak{I}_0 , \mathfrak{H}_0 ) is injective;

  3. For f 0: 0 0f_0 \colon \mathfrak{I}_0 \to \mathfrak{H}_0 there is a unique homomorphism N rf:N^r f \colon \mathfrak{I} \to \mathfrak{H} which lifts N rf 0N^r \cdot f_0;

  4. f 0: 0 0f_0 \colon \mathfrak{I}_0 \to \mathfrak{H}_0 lifts to f:f \colon \mathfrak{I} \to \mathfrak{H} if and only if N rfN^r f annihilates [N r]\mathfrak{I}[N^r] \subset \mathfrak{I}.

Note that if EE is an elliptic curve over RR then the E 0E_0 above is given by

E 0=E× SpecRSpecR/I.E_0 = E \times_{\mathrm{Spec} \; R} \mathrm{Spec} \; R/I .

Using the previous results one can prove the following alternate version of the Serre-Tate theorem.

Theorem (alternative Serre-Tate). Let RR be a ring with pp nilpotent and IRI \subset R a nilpotent ideal. Let R 0=R/IR_0 = R/I. We have a categorical equivalence: elliptic curves over RR and the category of triples {(E,E 0[p ],ϵ)}\{ (E, E_0 [p^\infty] , \epsilon )\}; where EE is an elliptic curve over R 0R_0, E 0[p ]E_0 [p^\infty] is a pp-divisble group over RR, and ϵ:E 0[p ]E[p ] 0\epsilon \colon E_0 [p^\infty] \to E[p^\infty]_0 is a natural isomorphism.

Adding in Completions

We really want to consider elliptic curves over RR completed by an ideal mm, this is the R^ m\hat{R}_m from far above. We can reduce this problem to that of elliptic curves over RR and a system of pp-divisible groups over R/m nR/m^n by combining the Serre-Tate theorem and the following theorem of Grothendieck.

Theorem (Formal GAGA). Let X,YX, Y be exceedingly nice schemes over RR and X^\hat X, Y^\hat Y be their formal completions, then there is a bijection

Hom R(X,Y)Hom R^(X^,Y^).\mathrm{Hom}_{R} (X,Y) \leftrightarrow \mathrm{Hom}_{\hat R} (\hat X, \hat Y).

Deformation Theory

Fix a morphism SpecF qM 1,1\mathrm{Spec} \; \mathbf{F}_q \to M_{1,1}, that is an elliptic curve E 0/kE_0 /k. Let O k\mathbf{O}_k be the sheaf over M 1,1M_{1,1} that classifies deformations of E 0E_0 to oriented elliptic curves over AA where AA is an E E_\infty-ring with π 0A\pi_0 A a complete, local ring.

Let us assume for the moment that M 1,1=SpecRM_{1,1} = \mathrm{Spec} \; R (more generally we pass to an affine cover). One can show that O k\mathbf{O}_k moreover classifies oriented pp-divisible groups which deform E 0[p ]E_0 [p^\infty].

Deformations of FGLs and Lubin-Tate Spectra

Recall that for F,GF,G formal group laws over RR a morphism is fR[[x]]f \in R [ [x] ] with f(0)=0f(0) =0 such that f(X+ FY)=f(X)+ Gf(Y)f(X +_F Y) = f(X) +_G f(Y). Now, let kk be a field of characteristic pp and F,GF,G formal group laws over kk. Let f:FGf: F \to G be a morphism, then

f=x p n+f = x^{p^n} + \dots

where nn is the height? of ff. For FF a formal group law the height of FF, htF\mathrm{ht} \; F, is the height of [p]F[p]F, that is the multiplication by pp map.

Let Γ\Gamma be a formal group law over kk, then a deformation of FF consists of a complete local ring BB (B/M=kB/M = k) and FF a formal group law over BB such that p *F=Γp_* F = \Gamma, where p:BB/Mp: B \to B/M is the canonical surjection. To such deformations F 1,F 2F_1 , F_2 are isomorphic if there is an isomorphism f:F 1F 2f: F_1 \to F_2 which induces the identity on p *F 1=p *F 2=Γp_* F_1 = p_* F_2 = \Gamma.

Theorem (Lubin-Tate). Let k,Γk, \Gamma as above with htΓ<\mathrm{ht} \; \Gamma \lt \infty then there exists a complete local ring E(k,Γ)E(k, \Gamma ) with residue field kk and a formal group law over E(k,Γ)E(k, \Gamma), F univF^{univ} which reduces to Γ\Gamma such that there is a bijection of sets

Hom(E(k,Γ),R)Deform(Γ,R).\mathrm{Hom} (E(k, \Gamma) , R) \to \mathrm{Deform} (\Gamma, R) .

If k=F pk = \mathbf{F}_p, then E(k,Γ) p[[u 1,dotsc,u n1]]E(k, \Gamma) \simeq \mathbb{Z}_p [ [ u_1 , \dotsc , u_{n-1} ] ] where htΓ=n\mathrm{ht} \; \Gamma =n. More generally, if k=F qk= \mathbf{F}_q, then E(k, \Gamma) \simeq W\mathbf{F}_q [u_1 , \dotsc , u_{n-1}]$, that is, the Witt ring.

Let E(k,Γ),F univE(k, \Gamma), F^{univ} as in the theorem, then we define F¯ univ\overline{F}^{univ} over E(k,Γ)[u ±]E(k, \Gamma) [u^\pm], degree u=2u=2, by F¯ univ(X,Y)=u 1F univ(uX,uY). \overline{F}^{univ} (X,Y) = u^{-1} F^{univ} (uX , uY). We then define a homology theory as

(E k,Γ) *(X)=E(K,Γ)[[u ±]] MU *MU *(X).( E_{k,\Gamma})_* (X) = E(K,\Gamma)[ [u^\pm] ] \otimes_{MU_*} MU_* (X).

Define a category (really a stack and let pp and nn vary) FG\mathbf{FG} of pairs (k,Γ)(k,\Gamma) where chark=p\mathrm{char} \; k =p and htΓ=n\mathrm{ht} \; \Gamma =n. By associating to any such pair its Lubin-Tate theory we get a functor from FG\mathbf{FG} to multiplicative cohomology theories.

Theorem (Hopkins-Miller I). This functor lifts to E E_\infty-rings.

The philosophy is that there should be a sheaf of E E_\infty-rings on the stack of formal groups FG\mathbf{FG} with global sections the sphere spectrum. Then tmf and taf are low height approximations.

Let E LTE_{\infty}^{LT} be the subcategory of E E_\infty-rings such that the associated cohomology theory is isomorphic to some E k,ΓE_{k,\Gamma}.

Theorem (Hopkins-Miller II). π:E LTFG\pi : E_{\infty}^{LT} \to \mathbf{FG} is a weak equivalence of topological categories. That is, the lift above is pretty unique.

This implies Hopkins-Miller I by taking a Kan extension FGE \mathbf{FG} \to E_\infty along the inclusion E LTE E_{\infty}^{LT} \hookrightarrow E_\infty.

The Final Sprint

Let us sketch how to proceed…Let \mathfrak{I} be a pp-divisible group over AA with π 0A\pi_0 A complete and local. Then \mathfrak{I} fits in an exact sequence

0^ et0. 0 \to \hat \mathfrak{I} \to \mathfrak{I} \to \mathfrak{I}_{et} \to 0 .

There are two cases: either the underlying elliptic curve E 0E_0 is super singular (htE 0=2\mathrm{ht} \; E_0 =2), else E 0E_0 is ordinary.

The Supersingular Case

In this case E 0^[p ]=E 0[p ]\widehat{E_0} [p^\infty] = E_0 [p^\infty], so one can show ^=\widehat{\mathfrak{I}} = \mathfrak{I} for every deformation. Now an orientation means

^SpfA P [p ],\widehat{\mathfrak{I}} \simeq \mathrm{Spf} \; A^{\mathbb{C}P^\infty}[p^\infty],

so E k,E 0^E_{k, \widehat{E_0}} classifies deformations to oriented pp-divisible groups, hence O kE k,E 0^\mathbf{O}_k \simeq E_{k, \widehat{E_0}}. By construction E k,E 0^E_{k,\widehat{E_0}} is even.

The Ordinary Case

This is more subtle (see DAG IV 3.4.1 for some hints).

First we can analyze the formal part of the exact sequence and see that deformations to formal pp-divisible groups of height 1 are classified by E k,E 0^[p ]E_{k, \widehat{E_0}} [ p^\infty]. By analyzing extensions etale kk-algebras we see that O kE k,E 0^ P \mathbf{O}_k \simeq E_{k,\widehat{E_0}}^{\mathbb{C}P^\infty}. The homotopy groups are then E k,E 0^[[t]]E_{k , \widehat{E_0}} [ [t] ] and are odd as the degree of tt is 0.

Revised on December 14, 2009 14:33:29 by Ryan Grady (195.37.209.182)