MORAVA K-THEORIES AND LOCALISATION 9
Every even finite spectrum is evenly generated. The collection of evenly gener-
ated spectra is closed under even suspensions, coproducts, and retracts, but does
not form a thick subcategory. We will see in Proposition 2.19 that evenly generated
spectra are closed under the smash product.
Lemm a 2.11. MU is evenly generated.
Proof Any map from a finite spectrum to MU factors though a skeleton of MU.
Any skeleton of MU is an even finite.
The following result is essentially due to Hopkins.
Proposition 2.12. Suppose M G £ ' . Then M is evenly generated.
Proof Suppose / : Z M is a map from a finite spectrum to M. Then / is a
class in M°Z. Spanier-Whitehead duality implies that M*Z = M*
8MU*
MU*Z.
We can thus write / = Y^Li k 8 c% say. As M* is concentrated in even degrees we
see that |c$| = \bi\ is even. Each map Ci thus has a factorisation
a = (z -% Wi -2*
zlbilMU),
where Wi is a skeleton of E'^'MC/, and so is an even finite. Write W = W\ V.. . V
Wmi letg: Z —» W be the map with components gi and let h: W —» M be the map
with components bi 0 e* 6 M*
8MU*
MU*Wi = [Wi,M]*. This gives the desired
factorisation / = hg.
There are several different characterisations of evenly generated spectra.
Proposition 2.13. The spectrum X is evenly generated if and only if X can be
written as the minimal weak colimit [HPS97, Section 2.2] of a filtered system {Ma}
of even finite spectra.
Proof. First suppose that X can be written as such a minimal weak colimit. Then,
by smallness, any map from a finite to X will factor through one of the terms in the
minimal weak colimit, and so through an even finite. Thus X is evenly generated.
Conversely, suppose X is evenly generated. We replace £5F by a small skeleton of
£ ? without change of notation. Let Agg-(X) be the category of pairs (J7, w), where
U G S5F and u: U X. Let A(X) be the category of pairs (W, w), where W is
any finite spectrum and w: W X. We know from [HPS97, Theorem 4.2.4] that
X is the minimal weak colimit of A(X). It will therefore be enough to show that
the obvious inclusion Ag^X ) —• A(X) is cofinal.
We first show that As3?(X), like A(X), is filtered. Consider two objects (U,u)
and (V,v) of Agg-(X). We need to show that there is an object (W,w) and maps
(U,u) - (W,w) «- ( V » in A(X). Clearly we can take W = U V V, and let
w: W X be the map with components u and v. We also need to show that
when fyg: (U,u) (V,v) are two maps in Ag^pQ, there is an object (W,w) and
a map h: (V9v) (W,w) with /i / = hg. To see this, let W' be the cofibre of
/ g and h,:V—Wl the evident map. We have vf = u = vg so v(f g) = 0
so v = w'h! for some t(/: W' X. Because X is evenly generated, the map
wf
factors as
Wf
W -^ X for some even finite W. We can evidently take h^kh'.
It is now easy to check that the inclusion Ag^(X) » A(X) is cofinal, as required.
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