xxvi INTRODUCTION
Once
Otop
has been constructed, it will turn out that Op
top
is the p-completion of
Otop,
and OQ
top
is its rationalization, so that the above diagram is the arithmetic
square for
Otop.
This thus leaves one to construct each Op
top
and OQ
top
and the gluing
map αarith. The sheaf OQ
top
is not difficult to construct. Its value on an ´ etale map
Spec(R) (Mell )Q is given by OQ
top
(Spec R) = H(R∗), the rational Eilenberg–
MacLane spectrum associated to a certain evenly graded ring R∗. This ring is
specified by R2t := Γ(ω⊗t|Spec
R
), where ω is the sheaf of invariant differentials.
The construction of Op
top
is more subtle. The first step in its construction is
to employ a natural stratification of (Mell )p. Each elliptic curve has an associated
formal group which either has height equal to 1 if the curve is ordinary, or equal to
2 if the curve is supersingular. This gives a stratification of the moduli space with
exactly two strata:
Mell
ord
ιord
−→ (Mell )p
ιss
←− Mell
ss
.
The sheaf Op top is presented by a Hasse square, gluing together a sheaf
Otop
K(1)
on
Mell ord and a sheaf
Otop
K(2)
on Mell ss . (This notation is used because the sheaves
Otop
K(i)
are also the K(i)-localizations of
Otop,
where K(i) is the ith Morava K-theory at
the prime p.)
Op
top
is the limit
Op
top
ιss,∗OK(2)top
ιord,∗OK(1)
top
(
ιss,∗OK(2)
top
)
K(1)
for a certain ‘chromatic’ attaching map
αchrom : ιord,∗OK(1)
top
−→
(
ιss,∗OK(2)
top
)
K(1)
.
The sheaf Op
top
is thus equivalent to the following triple of data: a sheaf
OK(1)top
on Mell
ord
, a sheaf OK(2)
top
on Mell
ss
, and a gluing map αchrom as above. We have
now arrived at the core of the construction of tmf : the construction of these three
objects. This construction proceeds via Goerss–Hopkins obstruction theory.
That obstruction theory is an approach to solving the following problem: one
wants to determine the space of all E∞-ring spectra subject to some conditions,
such as having prescribed homology. More specifically, for any generalized homol-
ogy theory E∗, and any choice of E∗-algebra A in E∗E-comodules, one can calculate
the homotopy type of the moduli space of E∞-ring spectra with E∗-homology iso-
morphic to A. Goerss and Hopkins describe that moduli space as the homotopy
limit of a sequence of spaces, where the homotopy fibers are certain Andr´e–Quillen
cohomology spaces of A. As a consequence, there is a sequence of obstructions
to specifying a point of the moduli space, i.e., an
E∞-ring spectrum whose E∗-
homology is A. The obstructions lie in Andr´ e–Quillen cohomology groups of E∗-
algebras in E∗E-comodules. That obstruction theory is used to build the sheaf
Otop
K(2)
.
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