Definition 1.4. Let G be a stable homotopy category, and R a ring object in 6.
We say that an object X £ G is R-nilpotent if it lies in the ideal generated by R.
We now recall some different notions of finiteness in a stable homotopy category.
We have replaced "strongly dualisable" by "dualisable" for brevity. Also recall that
DZ = F(Z, S) is the usual duality functor.
Definition 1.5. Let G be a stable homotopy category, and Z an object of G. We
say that Z is
(a) small if for any collection of objects {Xi}, the natural map 0[Z,Xi]
[Z,]jXi] is an isomorphism.
(b) F-small if for any collection of objects {Xi}y the natural map ]\F(Z,Xi)
F(Zy]jXi) is an isomorphism.
(c) A-finite (for any family A of objects of G) if Z lies in the thick subcategory
generated by A.
(d) dualisable if for any X, the natural map DZAX F(Z, X) is an equivalence.
A triangulated category with a compatible closed symmetric monoidal structure
is an algebraic stable homotopy category if there is a set S of small objects such
that the localising subcategory generated by S is the whole category. An algebraic
stable homotopy category is called monogenic if we can take 9 = {S}.
Finally, we recall that limits and colimits generally do not exist in triangulated
categories, but sometimes suitable weak versions do exist. In particular, given a
sequence Xo X\ ... , we can form the sequential colimit as the cofibre of
the usual self-map of V ^ - We denote this by holimXi , as it is the homotopy
colimit of a suitable lift of the sequence to a model category. Similarly, we denote
the sequential limit of a sequence ... » Xi Xo by holimXi.
In case we have a more complicated functor F : 3 G to a stable homotopy
category, we say that a weak colimit X of F is a minimal weak colimit if the induced
map limJTo F HX is an isomorphism for all homology functors H. We write
X = mwlimF. Minimal weak colimits are unique in algebraic stable homotopy
categories when they exist, and are extremely useful. See [HPS97, Section 2] for
In this section we assemble some basic results about E theory. We begin with
the fact that E*X is L-complete in the sense of Appendix A. In Section 2.1 we
show that the ring structure on E is canonical by showing there are no even degree
phantom maps between evenly graded Landweber exact spectra. We recall the
modified Adams spectral sequence briefly in Section 2.2. Finally, we briefly discuss
operations in .E-theory in Section 2.3.
Proposition 2.1. If X is a finite spectrum and R is one of E, E(ri), E/Ik,
E(n)/Ik or K(ri) then R*X is finitely generated over i?*.
Proof. We first recall that in each case R has an associative ring structure, so
that R*X is a module over i?*. The ring structure is not canonical but the module
structure is induced by the MU*-module structure so it is canonical. In each case i?*
is Noetherian and the claim follows easily by induction on the number of cells.
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