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
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