Abstract This entry attempts to give an outline of a proof of Lurie’s main theorem.
Here are the entries on the previous sessions:
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 - derived group schemes and (pre-)orientations
A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves
Recall the main 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
natural in $A$.
And $O^{Der}$ provides the lift of Goerss-Hopkins-Miller.
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 $O (\mathrm{Spec} \pi_0 A) \to A$.
In his thesis, Lurie proves the following.
Proposition. Let $F$ be a functor from connective $E_\infty$-ring spectra to spaces s.t.
The restriction of $F$ to discrete rings is represented by a (classical) DM-stack $X$, i.e. $F(R) \simeq \mathrm{Nerv} X(R)$;
$F$ is a sheaf with respect to the etale topology;
$F$ 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.
The functor $E'$: observe that every classical elliptic curve over a discrete $R$ has a unique preorientation. Hence $E'$ is represented by DM-stack $(M, O' )$.
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.
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
Suppose $M$ is flat, then
if $p=0$ and 0 otherwise. Thus,
So we have
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
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
Remark.
More generally, for $U \to M_{1,1}$ we have that $O(U)$ is weakly periodic, so
is an equivalence.
is an isomorphism. This map can be identified with $\times \beta$, so the preorientation is an orientation iff there is a unique factorization through $O(U) = A [ \beta^{-1} ]$.
Claim. To prove the theorem it is enough to show
$O_{1,1} = \pi_0 O' \to \pi_0 O$ is an isomorphism;
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.
We wish to show that $\pi_n O = 0$ for $n$ odd. From above, it suffices to show that
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 $\hat R_m /m \simeq R_m / m$.
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.
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
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
$p^n : \mathfrak{I} \to \mathfrak{I}$ is surjective;
$\mathfrak{I} = \mathrm{colim} \; \mathfrak{I} [p^n]$ where $\mathfrak{I} = \mathrm{ker} \; (p^n \colon \mathfrak{I} \to \mathfrak{I})$.
$\mathfrak{I}[p^n]$ 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.
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.
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
$B \mapsto \mathfrak{I} (B)$ is a sheaf;
$p^n : \mathfrak{I} \to \mathfrak{I}$ is surjective;
$(\mathrm{ho})\mathrm{colim} \; \mathfrak{I} [p^n] \simeq \mathfrak{I}$, where as above $\mathfrak{I} [p^n] := (\mathrm{ho}) \mathrm{ker} \; p^n$;
$\mathfrak{I}[p^n]$ 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 $E_0 [p^\infty]$.
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
Further, define
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
$\mathrm{Hom}_{R-grp} (\mathfrak{I}, \mathfrak{H})$ and $\mathrm{Hom}_{R_0-grp} (\mathfrak{I}_0 , \mathfrak{H}_0 )$ have no $N$-torsion;
$\mathrm{Hom} (\mathfrak{I} , \mathfrak{H}) \to \mathrm{Hom} (\mathfrak{I}_0 , \mathfrak{H}_0 )$ is injective;
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 $N^r \cdot f_0$;
$f_0 \colon \mathfrak{I}_0 \to \mathfrak{H}_0$ 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
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.
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
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 $E_0 [p^\infty]$.
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
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
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
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 $E_{\infty}^{LT} \hookrightarrow E_\infty$.
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
There are two cases: either the underlying elliptic curve $E_0$ is super singular ($\mathrm{ht} \; E_0 =2$), else $E_0$ is ordinary.
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
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.
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.