xxii INTRODUCTION

square

LpE LK(2)E

LK(1)E LK(1)LK(2)E.

By means of the arithmetic square, the construction of the sheaf Otop is reduced

to the construction of its p-completions, of its rationalization, and of the comparison

map between the rationalization and the rationalization of the product of the p-

completions. In turn, via the Hasse square, the construction of the p-completion

Op top of the sheaf Otop is reduced to the construction of the corresponding K(1)-

and K(2)-localizations and of a comparison map between the K(1)-localization and

the K(1)-localization of the K(2)-localization.

Chapter 7: The local structure of the moduli stack of formal groups.

By Landweber’s theorem, flat maps Spec(R) → MFG to the moduli stack of one-

dimensional formal groups give rise to even-periodic homology theories:

X → MP∗(X) ⊗MP0 R.

Here, MP is periodic complex bordism, MP0 = MU∗

∼

=

Z[u1,u2,...] is the Lazard

ring, and the choice of a formal group endows R with the structure of an algebra

over that ring. We wish to understand the geometry of MFG with an eye towards

constructing such flat maps.

The geometric points of MFG can be described as follows. If k is a separably

closed field of characteristic p 0, then formal groups over k are classified by their

height, where again a formal group has height n if the first non-trivial term of its

p-series (the multiplication-by-p map) is the one involving

xpn

. Given a formal

group of height n, classified by Spec(k) → MFG , one may consider ‘infinitesimal

thickenings’ Spec(k) → B, where B is the spectrum of a local (pro-)Artinian algebra

with residue field k, along with an extension

Spec(k) MFG .

B

This is called a deformation of the formal group. The Lubin–Tate theorem says

that a height n formal group admits a universal deformation (a deformation with a

unique map from any other deformation), carried by the ring W(k)[[v1,...vn−1]].

Here, W(k) denotes the ring of Witt vectors of k. Moreover, the map from B :=

Spf(W(k)[[v1,...vn−1]]) to MFG is flat.

The formal groups of interest in elliptic cohomology come from elliptic curves.

The Serre–Tate theorem further connects the geometry of Mell with that of MFG ,

in the neighborhood of supersingular elliptic curves. According to this theorem, the

deformations of a supersingular elliptic curve are equivalent to the deformations of

its associated formal group. The formal neighborhood of a point Spec(k) → Mell

classifying a supersingular elliptic curve is therefore isomorphic to Spf(W(k)[[v1]]),

the formal spectrum of the universal deformation ring.