INTRODUCTION xxi

Chapter 6: Bousfield localization and the Hasse square. We would like

a sheaf of spectra

Otop

on the moduli stack of elliptic curves Mell . As we will see,

this moduli stack is built out of its p-completions Op

top

and its rationalization. The

p-completion Op top is in turn built from certain localizations of Otop with respect

to the first and second Morava K-theories.

Localizing a spectrum X at a spectrum E is a means of systematically ignoring

the part of X that is invisible to E. A spectrum A is called E-acyclic if A ∧ X is

contractible. A spectrum B is called E-local if there are no nontrivial maps from

an E-acyclic spectrum into B. Finally, a spectrum Y is an E-localization of X if

it is E-local and there is a map X → Y that is an equivalence after smashing with

E. This localization is denoted LEX or XE.

The localization LpX := LM(Z/p)X of a spectrum X at the mod p Moore spec-

trum is the p-completion of X (when X is connective); we denote this localization

map ηp : X → LpX. The localization LQX := LHQX at the rational Eilenberg–

MacLane spectrum is the rationalization of X; we denote this localization map

ηQ : X → LQX.

Any spectrum X can be reconstructed from its p-completion and rationaliza-

tion by means of the ‘Sullivan arithmetic square’, which is the following homotopy

pullback square:

X

p

LpX

LQX LQ

p

LpX .

ηp

LQ( ηp)

ηQ ηQ

The above pullback square is a special case of the localization square

LE∨F X LEX

LF X LF LEX,

ηE

LF (ηE )

ηF ηF

which is a homotopy pullback square if one assumes that E∗(LF X) = 0.

An application of this localization square gives the so-called ‘chromatic fracture

square’:

LK(1)∨K(2)X LK(2)X

LK(1)X LK(1)LK(2)X.

ηK(2)

LK(1)(ηK(2))

ηK(1) ηK(1)

Here K(1) and K(2) are the first and second Morava K-theory spectra.

When the spectrum in question is an elliptic spectrum, the above square sim-

plifies into the ‘Hasse square’: for any elliptic spectrum E, there is a pullback