INTRODUCTION xxvii

There is also a ‘global’ version of this obstruction theory, where one tries to

lift a whole diagram I of E∗-algebras in E∗E-comodules to the category of E∞-

ring spectra. Here, in general, the obstructions live in the Hochschild–Mitchell

cohomology group of the diagram I with coeﬃcients in Andr´ e–Quillen cohomology.

This diagrammatic enhancement of the obstruction theory is used to build the sheaf

OK(1).top

Obstruction theory for OK(2):

top

The stack Mell

ss

is a 0-dimensional substack

of

Mell . More precisely, it is the disjoint union of classifying stacks BG where G

ranges over the automorphism groups of the various supersingular elliptic curves.

The Serre–Tate theorem identifies the formal completion of these groups G with the

automorphism groups of the associated formal group. Consequently, to construct

the sheaf

Otop

K(2)

on the category of ´ etale aﬃnes mapping to Mell ss , it suﬃces to

construct the stalks of the sheaf at each point of Mell ss , together with the action

of these automorphism groups. The spectrum associated to a stalk is a Morava

E-theory, the uniqueness of which is the Goerss–Hopkins–Miller theorem: that

theorem says that there is an essentially unique (unique up to a contractible space

of choices) way to construct an E∞-ring spectrum E(k, G), from a pair (k, G) of a

formal group G of finite height over a perfect field k, whose underlying homology

theory is the Landweber exact homology theory associated to (k, G). Altogether

then, given a formal aﬃne scheme Spf(R), with maximal ideal I ⊂ R, and an ´etale

map Spf(R) → Mell

ss

classifying an elliptic curve C over R, the value of the sheaf

OK(2)

top

is

Otop

K(2)

(Spf(R)) :=

i

E(ki,

C0i)).(

In this formula, the product is indexed by the set i in the expression of the quotient

R/I =

i

ki as a product of perfect fields, and C0 i is the formal group associated

to the base change to ki of the elliptic curve C0 over R/I.

Obstruction theory for OK(1):

top

We first explain the approach described

in Chapter 11. Over the stack Mell ord , there is a presheaf of homology theories

given by the Landweber exact functor theorem. This presheaf assigns to an el-

liptic curve classified by an ´ etale map Spec(R) → Mell

ord

the homology theory

X → BP∗(X)⊗BP∗ R. Ordinary elliptic curves have height 1, and so the represent-

ing spectrum is K(1)-local. In the setup of the Goerss–Hopkins obstruction theory

for this situation, we take E∗ to be p-adic K-theory, which has the structure of a

θ-algebra. The moduli problem that we are trying to solve is that of determining

the space of all lifts:

E∞-ringsK(1)

Kpˆ

I :=

(

Aff /Mell

ord

)

´ et

❧

❧

❧

❧

❧

❧

❧

Algθ.

In the general obstruction theory, the obstructions live in certain Hochschild–

Mitchell cohomology groups of the diagram I. For this particular diagram, the

obstruction groups simplify, and are equivalent to just diagram cohomology of I

with coeﬃcients in Andr´ e–Quillen cohomology. The diagram cohomology of I is