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
Obstruction theory for OK(2):
The stack Mell
is a 0-dimensional substack
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
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
classifying an elliptic curve C over R, the value of the sheaf
(Spf(R)) :=
In this formula, the product is indexed by the set i in the expression of the quotient
R/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):
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
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:
I :=
Aff /Mell
´ et

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
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