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.

•