MORAVA /^-THEORIES AND LOCALISATION
7
Proposition 2.2. For any spectrum X, there is a natural topology on E°X making
it into a profinite (and thus compact Hausdorff) Abelian group. Moreover, ifA(X)
is the category of pairs (Y,u) where Y is finite and u:Y X, then E°X is
homeomorphic to lim E°Y.
(ytu)€A(X)
Proof If X is a finite spectrum then E*X is a finitely generated module over
JE*, and it follows easily that the 7n-adic topology on E°X is profinite. For an
arbitrary spectrum X. define F°X = lim E°Y. with the inverse limit
*- (Y,u)eA(X)
topology. By [HPS97, Proposition 2.3.16], F is a cohomology theory with values
in the category of profinite groups and continuous homomorphisms. There is an
evident map E°X —» F°X which is an isomorphism when X is finite (because
(X,lx) is a terminal object of A(X) in that case). It follows easily that E°X =
F°X for all X.
Corollary 2.3. For any spectrum X, the module E*X is L-complete in the sense
of Definition A.5.
Proof This is immediate when X is finite. It thus follows for general X because the
category of L-complete modules is closed under inverse limits (by Theorem A.6).
Proposition 2.4. For any spectrum X, the module E*X is finitely generated over
E* if and only if K*X is finitely generated over K*.
Proof The cofibration
E2&
'^E/h ^ E/Ik - E/Ik+1 gives a short exact se-
quence
(E/Ikr(X)/vk- (E/Ik+1)*(X) -»ann(z,fe,
(E/Ik)-&+1{X)).
It follows that if (E/Ik)*X isfinitelygenerated then the same is true of (E/Ik+i)*X.
Conversely, suppose that (E/Ik+i)*X is finitely generated over L?*, and write
M = (E/Ik)*X. The above sequence shows that M/vkM is a submodule of
(E/Ik+i)*(X) and thus is finitely generated. This means that there is a finitely
generated free module F over E*/Ik and a map f:F—M such that the induced
map F/vkF » M/vkM is surjective. If we let N be the cokernel of /, we con-
clude that N is an L-complete module over E* with N = vkN. It follows from
Proposition A.8 that N = 0, so / is surjective and M is finitely generated.
It follows that K*X = (E/In)*X is finitely generated if and only if E*X =
(E/I^yX is finitely generated.
Proposition 2.5. Let X be a spectrum. Suppose E*X is pro-free (in the sense of
Definition A. 10). Then K*X = (E*X)/In. Conversely, if K*X is concentrated in
even degrees, then E*X is pro-free and concentrated in even degrees.
Proof The first statement holds because the sequence (^o,... ,vn_i) is regular on
E*X, by Theorem A.9. Conversely, suppose that (E/Ik+i)*X is concentrated in
even degrees. Consider the short exact sequence
js//0*(*)M^^
It follows that
(E/Ik)odd(X)/vk
= 0. As
(E/Ik)odd(X)
is L-complete, we conclude
from Proposition A.8 that it must be zero. It also follows from the sequence that
annfafc,
(E/Ik)even(X))
= 0, so that vk acts injectively on (E/Iky(X). Finally, we
also see from the sequence that (E/Ik+i)*(X) = (E/Ik)*(X)/vk.
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