## Algebraic theories
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## Algebras and modules
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∞-algebra over an (∞,1)-operad
* associated bundle
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## Higher algebras
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* multiplicative cohomology theory
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## Model category presentations
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/ homotopy T-algebra
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model structure on algebras over an operad
## Geometry on formal duals of algebras
* Isbell duality
* derived geometry
* Deligne conjecture
* delooping hypothesis
* monoidal Dold-Kan correspondence
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.
There exists a derived Deligne-Mumford stack such that we have an equivalence
natural in .
And provides the lift of Goerss-Hopkins-Miller.
Recall that for an -ring a derived elliptic curve is a commutative derived group scheme over such that over is an elliptic curve.
Denote by the space of preoriented (derived) elliptic curves (so equipped with a map . And the space of oriented elliptic curves.
Note that a map is a map and a map of rings .
In his thesis, Lurie proves the following.
Proposition. Let be a functor from connective -ring spectra to spaces s.t.
The restriction of to discrete rings is represented by a (classical) DM-stack , i.e. ;
is a sheaf with respect to the etale topology;
has a good deformation theory.
Then there exists a derived DM-stack representing s.t. is connective for affine.
The functor : observe that every classical elliptic curve over a discrete has a unique preorientation. Hence is represented by DM-stack .
The functor : the theorem doesn’t apply as a discrete ring cannot be weakly periodic.
Claim. represents for all -rings, so we dropped connectivity.
Proof. Recall the map to the connected cover.
- Let be an -ring, then we have an equivalence
- since every elliptic curve is by definition flat.
We need the following to prove the claim.
Proposition. The functor from flat modules over to flat modules over is an equivalence.
Proof of proposition (sketch). Let be -modules then there is a spectral sequence
Suppose is flat, then
if and 0 otherwise. Thus,
So we have
This is an equivalence and 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 are coherent sheaves over , where a coherent sheaf is an assignment which behaves well under finite limits. Let be the line bundle of invariant differentials on so that is
Recall that a preorientation determines a map . So define a sheaf of -rings as which is characterized (maybe) by
- This formula comes from a much simpler situation…Let be (an honest to God) ring, then
- Suppose with affine, restricted to is trivialized then and . Hence, and we get the formula above .
More generally, for we have that is weakly periodic, so
is an equivalence.
- classifies oriented elliptic curve. Let over be a preoriented elliptic curve over , so we have the classifying map and this is an orientation iff
is an isomorphism. This map can be identified with , so the preorientation is an orientation iff there is a unique factorization through .
Claim. To prove the theorem it is enough to show
is an isomorphism;
For odd, the sheaf .
Proof. Suppose (1) and (2) hold. Let be etale for discrete. We must show that is an elliptic cohomology theory associated to . Condition (1) ensures , (2) guarantees evenness and from above we have weakly periodic. We must show that which follows from having an orientation.
Reducing to a Local Calculation
We wish to show that for odd. From above, it suffices to show that
is zero for all . Note that is a quotient of which is coherent. Suppose is an etale cover of then iff . We can find an etale cover by a disjoint union of level 3 and level 4 modular forms denoted .
That is equivalent to for all . It is not difficult to show that all residue fields are finite in this case.
Now it is enough to show condition (2) formally locally as .
In the previous section we had a moduli stack preoriented elliptic curves . The structure sheaf of took values in connected -rings. We had a refinement which was a moduli stack for oriented elliptic curves. From the orientation condition we deduced that the structure sheaf took values in weakly periodic -rings.
Further, we showed how to reduce the main theorem to a (formal) local computation. That is, we only need to consider , 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: -divisible groups.
Let be a complete, local ring (e.g. the -adic integers ) and an elliptic curve over . What do we need in order to lift to an elliptic curve over ?
Let be a prime (say the characteristic of ) and an elliptic curve over . Using the multiplication by map we can define a sheaf of Abelian groups
where is the kernel of the map . That is, corresponds to the -torsion points of .
Definition. A -divisible group over is a sheaf of Abelian groups on the flat site of schemes over such that
is a finite, flat, commutative -group scheme. Note that finite means that is affine and whose global sections is a finite -module.
For instance, the constant sheaf is a -divisible group.
Now let , in addition to above, be Noetherian with residue field for .
Theorem (Serre-Tate). Let be an elliptic curve over , then there is an equivalence of categories between elliptic curves over that restrict to and the category of -divisible groups over such that the restriction of to is .
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 be an -ring. A functor from commutative -algebras to topological Abelian groups is a -divisible group if
is a sheaf;
, where as above ;
is a derived commutative group scheme over which is finite and flat.
If is a -divisible group over , then is a finite -module of dimension called the rank of .
Proposition. If , for an elliptic curve, then has rank 2.
One can verify the proposition over pretty easily; it is more subtle over a finite field. We have a derived version of the Serre-Tate theorem.
Theorem (Serre-Tate). Let be an -ring such that is a complete, local, Noetherian ring and are finitely generated -modules. Let be a (derived) elliptic curve over , then there is an equivalence of -categories: elliptic curves over that restrict to and -divisible groups over that restrict to .
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 be a ring as in the theorem such that for some and let be a nilpotent ideal, so , and set . Let and
Lemma. If is a formal group over , then is annihilated by .
Proposition. Let , be elliptic curves or -divisible groups over and let . Denote by and the restriction of and to -algebras. Then
and have no -torsion;
For there is a unique homomorphism which lifts ;
lifts to if and only if annihilates .
Note that if is an elliptic curve over then the 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 be a ring with nilpotent and a nilpotent ideal. Let . We have a categorical equivalence: elliptic curves over and the category of triples ; where is an elliptic curve over , is a -divisble group over , and is a natural isomorphism.
Adding in Completions
We really want to consider elliptic curves over completed by an ideal , this is the from far above. We can reduce this problem to that of elliptic curves over and a system of -divisible groups over by combining the Serre-Tate theorem and the following theorem of Grothendieck.
Theorem (Formal GAGA). Let be exceedingly nice schemes over and , be their formal completions, then there is a bijection
Fix a morphism , that is an elliptic curve . Let be the sheaf over that classifies deformations of to oriented elliptic curves over where is an -ring with a complete, local ring.
Let us assume for the moment that (more generally we pass to an affine cover). One can show that moreover classifies oriented -divisible groups which deform .
Deformations of FGLs and Lubin-Tate Spectra
Recall that for formal group laws over a morphism is with such that . Now, let be a field of characteristic and formal group laws over . Let be a morphism, then
where is the height? of . For a formal group law the height of , , is the height of , that is the multiplication by map.
Let be a formal group law over , then a deformation of consists of a complete local ring () and a formal group law over such that , where is the canonical surjection. To such deformations are isomorphic if there is an isomorphism which induces the identity on .
Theorem (Lubin-Tate). Let as above with then there exists a complete local ring with residue field and a formal group law over , which reduces to such that there is a bijection of sets
If , then where . More generally, if , then , that is, the Witt ring.
Let as in the theorem, then we define over , degree , by We then define a homology theory as
Define a category (really a stack and let and vary) of pairs where and . By associating to any such pair its Lubin-Tate theory we get a functor from to multiplicative cohomology theories.
Theorem (Hopkins-Miller I). This functor lifts to -rings.
The philosophy is that there should be a sheaf of -rings on the stack of formal groups with global sections the sphere spectrum. Then tmf and taf are low height approximations.
Let be the subcategory of -rings such that the associated cohomology theory is isomorphic to some .
Theorem (Hopkins-Miller II). 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 along the inclusion .
The Final Sprint
Let us sketch how to proceed…Let be a -divisible group over with complete and local. Then fits in an exact sequence
There are two cases: either the underlying elliptic curve is super singular (), else is ordinary.
The Supersingular Case
In this case , so one can show for every deformation. Now an orientation means
so classifies deformations to oriented -divisible groups, hence . By construction 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 -divisible groups of height 1 are classified by . By analyzing extensions etale -algebras we see that . The homotopy groups are then and are odd as the degree of is 0.