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.