nLab 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

• A Survey of Elliptic Cohomology

see there for background and context.

Here are the entries on the previous sessions:

• A Survey of Elliptic Cohomology - cohomology theories

• A Survey of Elliptic Cohomology - formal groups and cohomology

• A Survey of Elliptic Cohomology - E-infinity rings and derived schemes

• A Survey of Elliptic Cohomology - elliptic curves

• A Survey of Elliptic Cohomology - equivariant cohomology

• A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations

• A Survey of Elliptic Cohomology - A-equivariant cohomology

• A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves

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Recall the main theorem.

Theorem

(J. Lurie)

For

• $A$ any E-∞ ring

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

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

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

natural in $A$.

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

Preliminaries

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

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

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

Representability

In his thesis, Lurie proves the following.

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

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

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

3. $F$ has a good deformation theory.

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

Examples.

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

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

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

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

1. Let $A$ be an $E_\infty$-ring, then we have an equivalence

$$\mathrm{Hom} (O' (\mathrm{Spec} \pi_0 \tau_{\ge 0} A) , \tau_{\ge 0} A ) \to \mathrm{Hom} (O' (\mathrm{Spec} \pi_0 A) , A).$$

1. $E' ( \tau_{\ge 0} A \simeq E' (A)$ since every elliptic curve is by definition flat.

We need the following to prove the claim.

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

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

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

Suppose $M$ is flat, then

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

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

$$\pi_* (M \otimes_A N ) \simeq \pi_* N \otimes_{\pi_0 A} \pi_0 M .$$

So we have

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

This is an equivalence and $F$ 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 $\pi_i O'$ are coherent sheaves over $M_{1,1}$, where a coherent sheaf is an assignment $\mathrm{Spec} R \to M_{1,1}$ which behaves well under finite limits. Let $\omega$ be the line bundle of invariant differentials on $M_{1,1}$ so that is

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

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

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

Remark.

• This formula comes from a much simpler situation…Let $R$ be (an honest to God) ring, $x \in R$ then

$$R[x^{-1}] \simeq \lim (R \stackrel{x}{\to} R \stackrel{x}{\to} \dots )$$

• Suppose $U \to M_{1,1}$ with $U$ affine, $\omega$ restricted to $U$ is trivialized then $O' (U) =A$ and $\beta \in \pi_2 A$. Hence, $O' [\beta^{-1} ] (U) = A [ \beta^{-1} ]$ and we get the formula above $\pi_i (A [ \beta^{-1} ]) = (\pi_i A) [\beta^{-1} ]$.

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

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

is an equivalence.

• $(M, O)$ classifies oriented elliptic curve. Let $F$ over $U = \mathrm{Spec} B$ be a preoriented elliptic curve over $B$, so we have the classifying map $f: O' (U) = A \to B$ and this is an orientation iff

$$\pi_n B \otimes_{\pi_0 B} \omega \to \pi_{n+2} B$$

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

Reduction

Claim. To prove the theorem it is enough to show

1. $O_{1,1} = \pi_0 O' \to \pi_0 O$ is an isomorphism;

2. For $n$ odd, the sheaf $\pi_n O =0$.

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

Reducing to a Local Calculation

We wish to show that $\pi_n O = 0$ for $n$ odd. From above, it suffices to show that

$$f_k : \pi_{n+2k} O' \otimes \omega^{-k} \to \pi_n O$$

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

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

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

Key Ingredients

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

Further, we showed how to reduce the main theorem to a (formal) local computation. That is, we only need to consider $\hat R_m$, 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: $p$-divisible groups.

$p$-divisible Groups

Let $R$ be a complete, local ring (e.g. the $p$-adic integers $\mathbb{Z}_p$) and $E_0$ an elliptic curve over $R_0 := R/M = \mathbf{F}_q$. What do we need in order to lift $E_0$ to an elliptic curve over $R$?

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

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

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

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

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

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

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

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

Serre-Tate Theory

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

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

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 $A$ be an $E_\infty$-ring. A functor $\mathfrak{I}$ from commutative $A$-algebras to topological Abelian groups is a $p$-divisible group if

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

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

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

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

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

Proposition. If $\mathfrak{I} = E [p^\infty]$, for $E$ 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 $A$ be an $E_\infty$-ring such that $\pi_0 A$ is a complete, local, Noetherian ring and $\pi_i A$ are finitely generated $\pi_0 A$-modules. Let $E_0$ be a (derived) elliptic curve over $\pi_0 A / M$, then there is an equivalence of $\infty$-categories: elliptic curves over $A$ that restrict to $E_0$ and $p$-divisible groups over $A$ that restrict to $E_0 [p^\infty]$.

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 $R$ be a ring as in the theorem such that $N \cdot R =0$ for some $N \in \mathbb{N}$ and let $I \subset R$ be a nilpotent ideal, so $I^{r+1} =0$, and set $R_0 = R/I$. Let $\mathfrak{I} \colon R-\mathrm{alg} \to \mathrm{Ab}$ and

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

Further, define

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

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

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

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

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

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

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

Note that if $E$ is an elliptic curve over $R$ then the $E_0$ above is given by

$$E_0 = E \times_{\mathrm{Spec} \; R} \mathrm{Spec} \; R/I .$$

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

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

Adding in Completions

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

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

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

Deformation Theory

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

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

Deformations of FGLs and Lubin-Tate Spectra

Recall that for $F,G$ formal group laws over $R$ a morphism is $f \in R [ [x] ]$ with $f(0) =0$ such that $f(X +_F Y) = f(X) +_G f(Y)$. Now, let $k$ be a field of characteristic $p$ and $F,G$ formal group laws over $k$. Let $f: F \to G$ be a morphism, then

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

where $n$ is the height? of $f$. For $F$ a formal group law the height of $F$, $\mathrm{ht} \; F$, is the height of $[p]F$, that is the multiplication by $p$ map.

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

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

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

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

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

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

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

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

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

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

Theorem (Hopkins-Miller II). $\pi : E_{\infty}^{LT} \to \mathbf{FG}$ 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 $\mathbf{FG} \to E_\infty$ along the inclusion $E_{\infty}^{LT} \hookrightarrow E_\infty$.

The Final Sprint

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

$$0 \to \hat \mathfrak{I} \to \mathfrak{I} \to \mathfrak{I}_{et} \to 0 .$$

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

The Supersingular Case

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

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

so $E_{k, \widehat{E_0}}$ classifies deformations to oriented $p$-divisible groups, hence $\mathbf{O}_k \simeq E_{k, \widehat{E_0}}$. By construction $E_{k,\widehat{E_0}}$ 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 $p$-divisible groups of height 1 are classified by $E_{k, \widehat{E_0}} [ p^\infty]$. By analyzing extensions etale $k$-algebras we see that $\mathbf{O}_k \simeq E_{k,\widehat{E_0}}^{\mathbb{C}P^\infty}$. The homotopy groups are then $E_{k , \widehat{E_0}} [ [t] ]$ and are odd as the degree of $t$ is 0.