xx INTRODUCTION
presheaf of homology theories on the flat site of the moduli stack of formal groups
MFG . (2) The map of stacks, Mell MFG defined by sending an elliptic curve
to its associated formal group, is flat.
Chapter 5: Sheaves in homotopy theory. By the above construction,
using the Landweber exact functor theorem, we have a presheaf
Ohom
of homology
theories (previously called Ell) on the moduli stack of elliptic curves. One might try
to define a single ‘universal elliptic homology theory’ as the limit limU∈U
Ohom(U),
where U is an affine cover of the moduli stack. However, the category of homology
theories does not admit limits. If, though, we can rigidify the presheaf
Ohom
of
homology theories to a presheaf
Otop
of spectra, then we can use instead a homotopy
limit construction in the category of spectra. The main theorem is that there does
indeed exist such a presheaf, in fact a sheaf, of spectra.
Theorem (Goerss–Hopkins–Miller). There exists a sheaf
Otop
of E∞-ring
spectra on (Mell
et
, the ´ etale site of the moduli stack of elliptic curves (whose
objects are ´ etale maps to Mell ), such that the associated presheaf of homology the-
ories, when restricted to those maps whose domain is affine, is the presheaf
Ohom
built using the Landweber exact functor theorem.
In this theorem, it is essential that the sheaf
Otop
is a functor to a point-set-
level, not homotopy, category of E∞-ring spectra. Moreover, the functor is defined
on all ´ etale maps N Mell , not just those where N is affine; (in fact, N can be
itself a stack, as long as the map to Mell is ´ etale). Given a cover N = {Ni N} of
an object N , we can assemble the n-fold ‘intersections’ Nij := Ni ×N Nj, Nijk :=
Ni ×N Nj ×N Nk, and so forth, into a simplicial object
N• = Ni


Nij



Nijk




· · · .
The sheaf condition is that the natural map from the totalization (homotopy limit)
of the cosimplicial spectrum
Otop
(
N•
)
=
Otop
(
Ni
)


Otop
(
Nij
)



Otop
(
Nijk
)
· · ·
to
Otop(N
) is an equivalence.
Now consider a cover N = {Ni Mell } of Mell by affine schemes. The afore-
mentioned cosimplicial spectrum for this cover has an associated tower of fibrations
. . .
Tot2 Otop(N•)

Tot1 Otop(N•)

Tot0 Otop(N•)
whose inverse limit is Tot Otop(N•) = Otop(Mell ) = TMF . The spectral sequence
associated to this tower has as E2 page the Cech cohomology
ˇ
H
q
N
(Mell , πpOtop) of
N with coefficients in πpOtop. Since N is a cover by affines, the Cech cohomology
of that cover is the same as the sheaf cohomology of Mell with coefficients in the
sheafification πpOtop of πpOtop; (that sheafification happens to agree with πpOtop
on maps to Mell whose domain is affine). Altogether, we get a spectral sequence,
the so-called descent spectral sequence, that converges to the homotopy groups of
the spectrum of global sections:
Epq
2
=
Hq(Mell
,
πpOtop)
πp−qTMF.
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