MORAVA K-THEORIES AND LOCALISATION
5
1.2. Categories of localised spectra. We use the following notation.
Notation 1.1.
1. S is the (homotopy) category of p-local spectra. We write S for 5°.
2. £ = Ln is the category of 2£(n)-local spectra, and L = Ln : § £ is the
localisation functor. The corresponding acyclisation functor is written C, so
there is a natural cofibre sequence CX •— X » LX.
3. JV C = M
n
is the monochromatic category. This is defined to be the image of
the functor Mn = C
n
_iL
n
: S » §. Note that there is a natural fibration
MnX LnX Lin—\X = ±jn—\LnX.
4. X = X
n
is the category of if-local spectra, and L = Ln = £#(
n
) : S 3C is
the localisation functor. The corresponding acyclisation functor is written C.
5. F(m) denotes a finite spectrum of type m, and T(m) =
v^t1F(m)
is its
telescope. Recall from [Rav92a, Chapter V] or [HS] that any two F(ra)'s
generate the same thick subcategory and have the same Bousfield class, so
it usually does not matter which one we use. Note also that the Spanier-
Whitehead dual of an F(m) is again an F(m).
There are topological closed model categories whose homotopy categories are §,
L and X [EKMM96, Chapter VIII].
1.3. Stable homotopy categories. In this section we collect some basic defini-
tions from the theory of stable homotopy categories developed in [HPS97]. The
reader will be familiar with most of these: we assemble them here as a convenient
reference. We will not recall the definition of a stable homotopy category, except to
say that a stable homotopy category is a triangulated category with a closed sym-
metric monoidal structure which is compatible with the triangulation and which
has a set of generators in an appropriate sense. The symmetric monoidal structure
is written X AY and the closed structure is written F{X,Y). The unit for the
smash product is written 5.
Definition 1.2. A full subcategory T) of any triangulated category is thick if it is
closed under retracts, cofibres, and suspensions. That is, T is thick if both of the
following conditions hold.
(a) If X V Y e D, then both X and Y are in D; and
(b)If
X - Y - Z - EX
is a cofibre sequence and two of X, y , and Z are in D, then so is the third.
Certain kinds of thick subcategories come up frequently.
Definition 1.3. Let C be a thick subcategory of a stable homotopy category T.
(a) C is a localising subcategory if it is closed under arbitrary coproducts.
(b) 6 is a colocalising subcategory if it is closed under arbitrary products.
(c) C is an ideal if, whenever X eT and Y G C we have X A Y 6.
(d) 6 is a coideal if, whenever X e*D and Y G 6 we have F(X, Y) 6.
If the localising subcategory generated by S is all of D, then every (co)localising
subcategory is a (co)ideal [HPS97, Lemma 1.4.6]. This is true in 8, £, and X (and
any other localisation of §).
The ideal generated by a ring object is particularly important.
Previous Page Next Page