AXIOMATIC STABLE HOMOTOPY _ A SURVEY
N. P. STRICKLAND
Abstract.We survey various approaches to axiomatic stable homotopy theor*
*y, with examples
including derived categories, categories of (possibly equivariant or loc*
*alized) spectra, and stable
categories of modular representations of finite groups. We focus mainly*
* on representability
theorems, localisation, Bousfield classes, and nilpotence.
1. Introduction
Axiomatic stable homotopy theory is the study of triangulated categories form*
*ally similar to
the homotopy category of spectra in the sense of Boardman [1, 49]. While variou*
*s authors have
used different systems of axioms, there is broad agreement about the main examp*
*les, of which the
following are a sample:
(1) Boardman's category itself, which we denote by B.
(2) The subcategories of E(n)-local and K(n)-local spectra [66, 36].
(3) The homotopy category BG of G-spectra (indexed on a complete universe), *
*where G is a
compact Lie group [48, 50].
(4) The homotopy category of A-modules, where A is a commutative ring spectr*
*um. (Here
and throughout this paper, the phrase "ring spectrum" refers to a strict*
*ly associative
monoid in a suitable geometric category of spectra, such as that defined*
* in [23].)
(5) The derived category DR of R-modules, for a commutative ring R. If we le*
*t HR denote the
associated Eilenberg-MacLane ring spectrum, then DR ' DHR (by [23, Theor*
*em IV.2.4]),
so this is a special case of the previous example.
(6) The stable category StabkGof kG-modules and projective equivalence class*
*es of homo-
morphisms, where G is a finite group and k is a field [70]. This occurs *
*as a quotient of the
category DkG, in which the morphism sets are given by group cohomology; *
*the morphism
sets in StabkGitself are more closely related to Tate cohomology.
(7) The derived category (in a suitable sense) of MU*MU-comodules [33].
There are some further examples that satisfy some authors' axioms but not oth*
*ers, or where
the axioms have not yet been checked (to the best of my knowledge):
(1) The derived category of modules over a noncommutative ring.
(2) The derived category of quasicoherent sheaves over a nonaffine scheme.
(3) The homotopy category of G-spectra indexed by an incomplete G-universe.
(4) Various versions of the category of motivic spectra. (Motivic spaces are*
* discussed in [54];
at the time of writing, there is no published account of the correspondi*
*ng category of
spectra.)
It seems an important problem to decide which of the usual axioms apply to the *
*motivic stable
category, and to see what the axiomatic literature teaches us about this exampl*
*e.
The main topics that have been discussed from an axiomatic point of view are *
*as follows.
(a) Theorems saying that certain (covariant or contravariant) functors are r*
*epresentable, gen-
eralising the Brown representability theorems for (co)homology theories *
*on (finite or infi-
nite) spectra.
(b) Phantom maps.
___________
Date: July 10, 2003.
1991 Mathematics Subject Classification. 55U35.
Key words and phrases. axiomatic stable homotopy,triangulated category,Bousf*
*ield class .
1
2 N. P. STRICKLAND
(c)Various kinds of localisation, generalising Bousfield's theory of localis*
*ation with respect
to a homology theory. Special cases such as finite, cofinite, algebraic o*
*r smashing locali-
sations.
(d) The lattice of Bousfield classes, and various related lattices (some of t*
*hem conjecturally
identical to the Bousfield lattice).
(e)Nilpotence theorems in the spirit of Devinatz, Hopkins and Smith.
(f)Picard groups, Grothendieck rings, and Euler characteristics [28, 51, 52,*
* 46].
(g)Projective classes, and generalisations of the Adams spectral sequence [1*
*5].
(h) Duality theorems generalising those of Verdier and Gross-Hopkins [57, 24].
For (f) to (h) we refer the reader to the cited papers and their bibliographies*
*. This survey will
concentrate on (a) to (e).
The relevant literature consists partly of papers that are explicitly axiomat*
*ic, and partly of pa-
pers that are nominally restricted to some particular category, but whose metho*
*ds allow straight-
forward generalisation to other examples. Some authors are as follows:
(1) Margolis's book [49] treats B from an axiomatic point of view; this was *
*an important
inspiration for much of the later work. Earlier still, there were releva*
*nt papers by Freyd,
Heller and Joel Cohen.
(2) Neeman has written extensively, particularly on questions related to rep*
*resentability and
localisation [13, 17, 59, 56, 62, 55, 61, 58, 63]. Some papers are restr*
*icted to the case of
DR; often R need not be commutative or noetherian. When working axiomat*
*ically, he
has generally assumed that his triangulated category C is öc mpactly gen*
*erated", but not
that C has a symmetric monoidal structure. The class of categories cons*
*idered is thus
rather large, but unfortunately it is not closed under Bousfield localis*
*ation. Recently he
has introduced the more complex notion of a "well-generated" triangulate*
*d category to
repair this problem.
(3) Krause has also written extensively on representability, localisation, a*
*nd versions of the
Bousfield lattice [11, 3, 39, 40, 43, 42, 41, 44, 45, 46, 47].
(4) Beligiannis has written a long paper [2] covering many themes in axiomat*
*ic stable homo-
topy, considered as an analog of relative homological algebra in the con*
*text of triangulated
categories.
(5) Benson, Carlson, Rickard and Gnacadja (working in various combinations) *
*have proved
many results about the categories StabkGand DkG, often using methods tha*
*t transfer
easily to an axiomatic setting [4, 6, 7, 8, 9, 10, 5, 11, 3, 72, 70, 68,*
* 71, 69]. Benson and
Wheeler have interpreted the Green correspondence in this context [12].
(6) Hovey, Schwede and Shipley have worked in a more rigid context, studying*
* Quillen model
categories C0 such that the homotopy category Ho(C0) is triangulated [34*
*, 73, 75, 76].
(7) May and coauthors have studied the equivariant stable categories BG, oft*
*en using methods
that transfer easily to an axiomatic setting. This applies particularly*
* to their work on
duality, traces, and Picard groups [25, 52, 51, 48].
(8) Hovey and Palmieri and Strickland wrote a memoir [35] on axiomatic stabl*
*e homotopy
theory. We assumed much more than Neeman, and thus could obtain results*
* closer to
those previously known for B. In particular, we assume that C has a clos*
*ed symmetric
monoidal structure.
2. Axioms
We next discuss the various axioms that have been used. We start with a categ*
*ory C.
2.1. Basics. The category C should be triangulated, and should have coproducts *
*for all families
of objects (indexed by a set). These are core axioms, used by almost all author*
*s. Existence of
coproducts should be seen as an important test of the correctness of the techni*
*cal details of the
definition of C. Boardman's category itself came after several attempts to defi*
*ne a good category
of spectra, and it was the first to be triangulated and coproduct-complete; it *
*rapidly became clear
that Boardman's version was much more convenient than all the others. Similarly*
*, the earliest
AXIOMATIC STABLE HOMOTOPY _ A SURVEY 3
versions of DR incorporated various boundedness conditions, and so were not cop*
*roduct-complete.
Bokstedt and Neeman [13] adjusted the definitions to remove this problem, and t*
*his allowed much
smoother comparisons between DR and B.
We recall the definition of a triangulation:
Definition 2.1. A triangulation of an additive category C is an additive (suspe*
*nsion) functor
: C -!C giving an automorphism of C, together with a collection 4 of diagrams,*
* called distin-
guished triangles or cofibre sequences, of the form
X -!Y -!Z -! X
such that
1. Any diagram isomorphic to a cofibre sequence is a cofibre sequence.
2. Any diagram of the following form is a cofibre sequence:
0 -!X 1-!X -!0
3. The first of the following diagrams a cofibre sequence iff the second is*
* a cofibre sequence:
X f-!Y -g!Z h-! X
Y -g!Z h-! X --f--! Y.
4. For any map f :X -!Y , there is a cofibre sequence of the following form:
X f-!Y -!Z -! X
5. Suppose we have a diagram as shown below (with h missing), in which the *
*rows are cofibre
sequences and the rectangles commute. Then there exists a (nonunique) ma*
*p h making
the whole diagram commutative.
U ________V-________W-_______- U
| | || |
| | | |
|f | |h | f
| | | |
|? |? ||? |?
X ________Y-________Z-_______- X
6. Verdier's octahedral axiom holds: Suppose we have maps X -v!Y u-!Z, an*
*d cofibre
sequences (X, Y, U), (X, Z, V ) and (Y, Z, W ) as shown in the diagram. *
*(A circled arrow
U -!O X means a map U -! X.) Then there exist maps r and s as shown, *
*making
(U, V, W ) into a cofibre sequence, such that the following commutativit*
*ies hold:
au = rd es = ( v)b sa = f br = c
_____-V______
J
J] J
b c J a J
r J J s
Æ J J
X ___________-Zuv
J
OE J OE J J
c c vJ u J f JJ
J J J
J J^ J^ Æ
U ___________oe___________oecYW
d e
(If u and v are inclusions of CW spectra, this essentially just says tha*
*t (Z=X)=(Y=X) =
Z=Y . The diagram can be turned into an octahedron by lifting the outer*
* vertices and
drawing an extra line from W to U.)
4 N. P. STRICKLAND
Following the standard topological notation, we write [X, Y ] for the set C(X*
*, Y ) (of morphisms
in C from X to Y ). We also put [X, Y ]n = [ nX, Y ].
If C0 is a pointed Quillen model category, then the homotopy category C = Ho(*
*C0) automatically
has structure close to that described above, except that the functor : C -! C *
*need not be an
equivalence. Following Hovey, we say that C0 is a stable model category if is*
* an equivalence; if
so, one can show that C is triangulated [34, Chapter 7]. Similar results appear*
* in [51]. Schwede
and Shipley have shown [75] that most stable model categories are Quillen-equiv*
*alent to DA for
some ring spectrum A, and that the general case is only a little more general.
Various modifications and refinements of triangulated categories have been co*
*nsidered by Nee-
man [55], May [51], Franke [26] and probably others. It seems likely that these*
* all follow from the
existence of an underlying model category C0 as above. For the categories C occ*
*urring in practice,
it seems that there is always an underlying model category, and that any two na*
*tural choices are
Quillen equivalent. It thus seems reasonable to assume whenever convenient that*
* one is given a
category C0.
2.2. Smash products. In [35], we assume that our categories C come equipped wit*
*h a symmetric
monoidal product. We use notation coming from topology, and thus write X ^ Y fo*
*r the monoidal
product of X and Y , and S for the unit, so S ^ X = X = X ^ S. We also assume t*
*hat there are
adjoint function objects F (Y, Z), so [X, F (Y, Z)] ' [X ^ Y, Z], naturally in *
*all variables. We write
Sn = nS, so Sn ^ Sm = Sn+m and nX = Sn ^ X. We also put ßnX = [Sn, X].
This structure certainly exists in all the categories mentioned so far, excep*
*t for the category DA
when A is not commutative. However, the theory of blocks in group algebras deco*
*mposes StabkG
as a product of smaller categories, which need not have a symmetric monoidal st*
*ructure. There
may be similar examples related to BG.
It is natural to require that the smash product be compatible with the triang*
*ulation. In [35],
we wrote down the most obvious compatibility conditions:
(1) The smash product commutes with suspension, so (X ^ Y ) ' X ^ Y .
(2) The functors X ^ (-) and F (X, -) preserve cofibre triangles. The contra*
*variant functors
F (-, Y ) preserve cofibre triangles up to a sign change.
(3) The twist map S1 ^ S1 -!S1 ^ S1 is multiplication by -1.
However, May has given strong evidence that further conditions should be added.*
* To explain this,
consider a map f :X -!X. Under suitable finiteness conditions, this has a trace*
* ø(f): S -!S. If
we have a cofibre sequence
X0 -!X1 -!X2 -! X0
and compatible maps fi:Xi -!Xi, it is natural to hope that ø(f0) - ø(f1) + ø(f2*
*) = 0; this
is suggested by the theory of Lefschetz numbers, among other things. It turns *
*out that the
statement must be adjusted slightly: given f0 and f1, one can choose a compatib*
*le f2 such that
ø(f0) - ø(f1) + ø(f2) = 0, but this may not be the case for all compatible f2's*
*. This (and
various extensions) can be proved in Boardman's category, but the proof cannot *
*be transferred
to the axiomatic setting without adding some more conditions. In outline, consi*
*der two cofibre
sequences
X0 -!X1 -!X2 -! X0
Y0 -!Y1-!Y2 -! Y0.
From these we obtain a 4 x 4 diagram with vertices Xi^ Xj, in which all squares*
* commute,
except that one square anticommutes. By writing in the diagonal composite in ea*
*ch square, we
get 18 commutative triangles. Each commutative triangle fits in an octahedron, *
*as in Verdier's
axiom. The 18 resulting octahedra have many vertices and edges in common, and o*
*ne can hope
to add in some extra vertices and edges making everything fit together more coh*
*erently. May [51]
has formulated three axioms about this situation, and explained how they can be*
* checked when
C = Ho(C0) for some Quillen model category C0.
There are many interesting cases of noncommutative rings (or ring spectra) R *
*and R0for which
R 6' R0 but DR ' DR0; this is a natural extension of Morita theory. Examples c*
*ome from
Koszul duality, the Fourier-Mukai transform for sheaves on abelian varieties, t*
*ilting complexes
AXIOMATIC STABLE HOMOTOPY _ A SURVEY 5
in representation theory, and so on [68, 69, 14, 20, 74]. Nonetheless, it seems*
* that there are no
examples of this type where the categories involved have smash products and the*
* equivalence
respects them. We know of no rigorous results in this direction, however.
2.3. Generation. A key feature of Boardman's category is that every spectrum X *
*has a cell
structure, and (essentially equivalently) that if [S0, X]* = 0 then X = 0. More*
* generally, suppose
we have a set G of objects in a triangulated category C. We say that G generate*
*s C if there is no
proper triangulated subcategory C0 C closed under all coproducts such that G *
* C0. We also say
that G detects C if every object X 2 C with [A, X]* = 0 for all A 2 G actually *
*has X = 0. (By
taking C0= {A | [A, X]* = 0}, we see that generation implies detection, and the*
* converse holds
under suitable finiteness conditions.) Generating sets for the main examples ar*
*e as follows.
(1) {S0} generates B, and LES0 generates the subcategory of E-local spectra *
*[35, Section
3.5].
(2) If X is a finite spectrum of type n (in the usual chromatic sense) then *
*LK(n)X generates
the category of K(n)-local spectra [36, Theorem 7.3], and this is genera*
*lly a better choice
of generator than LK(n)S0.
(3) The G-spectra G=H+ (as H runs over conjugacy classes of closed subgroups*
*) generate BG.
(4) If R is a ring, then R generates DR. This also works for (strictly assoc*
*iative) ring spectra.
(5) The nonprojective simple kG-modules generate StabkG.
All authors in axiomatic stable homotopy theory assume that C is generated by*
* some set G of
objects, and impose some smallness conditions on G. The details vary between au*
*thors, however.
One popular condition is as follows. We say that an object A 2 C is small (or*
* compact) if the
natural map M M
[A, Xi] -![A, Xi]
i2I i
is an isomorphism for all families of objects {Xi}. For example:
(1) In B, the small objects are those of the form dX where d 2 Z and X is a*
* finite CW
complex.
(2) In the category of E(n)-local spectra, the small objects are those that *
*can be written as
a retract of dLE(n)X for some d 2 Z and some finite CW complex X.
(3) In the category of K(n)-local spectra, the small objects are those that *
*can be written as
a retract of dLK(n)X for some d 2 Z and some finite CW complex X of typ*
*e n.
(4) In BG, the small objects are those that can be written as a retract of *
*dX for some d 2 Z
and some finite G-CW complex X.
(5) In DR, the small objects are the finite complexes of finitely generated *
*projective modules.
(6) In StabkG, the small objects are the kG-modules M that are finite-dimens*
*ional over k.
(Most of these facts are proved in [35], for example.)
We say that C is compactly generated if there is a set G of small objects tha*
*t generates C. In
the terminology of [35], a stable homotopy category is algebraic iff it is comp*
*actly generated. This
is a very convenient condition, and is often satisfied; in particular, the gene*
*rators listed above for
B, BG, DR and StabkGare all small. In the case of the K(n)-local category, the *
*obvious generator
LK(n)S0 is not small, but LK(n)X is small whenever X is finite of type n, so th*
*e category is
nonetheless compactly generated. For a simpler example of the same phenomenon, *
*let C be the
p-completion of the category DZ; then the obvious generator is Zp (which is not*
* small), but the
object Z=p is also a generator, and is small. (There is a well-understood axio*
*matic framework
covering both of these examples: see [35, Section 3.3], and Section 7 of the pr*
*esent paper.) More
seriously, there are many known spectra E for which the category CE of E-local *
*spectra has no
nontrivial small objects, so in particular, CE is notWcompactly generated [60][*
*36, Appendix B]. For
example, this applies with E = BP or E = H or E = 0 n<1 K(n).
Another useful condition is dualisability. To formulate this, we need to assu*
*me that C has a
symmetric monoidal smash product, as in the previous section. We write DA = F (*
*A, S), where
S is the unit object for the smash product. We say that A is dualisable if the*
* natural map
DA ^ A -!F (A, A) is an isomorphism; this implies that we have DA ^ B = F (A, B*
*) for all B.
6 N. P. STRICKLAND
The category of dualisable objects is formally very similar to the category of *
*finite dimensional
vector spaces over a field.
In Boardman's category B, or in the derived category DR, it is known that an *
*object is dualisable
iff it is small. However, LK(n)S0 is dualisable but not small in the K(n)-local*
* category. A number
of interesting things are known about K(n)-locally dualisable spectra:
(1) A K(n)-local spectrum X is dualisable iff dimK(n)*K(n)*X < 1 (proved in *
*[36]).
(2) If X is a finite complex, then LK(n)Q1 X is easily seen to be dualisable.
(3) If X is a connected space with | k>0ßkX| < 1, then it is probably true *
*that K(n)*X
is finite-dimensional and so LK(n) 1 X is dualisable. The results in the*
* literature involve
some additional conditions, however; for example, the claim is true if X*
* = BG with G
finite [65] (in which case LK(n) 1 BG turns out to be self-dual [77]), o*
*r if X is a double
loop space [31].
In [35], we assume that our generators are dualisable, but not that they are *
*small. This theory
has the advantage that any localisation of a category satisfying our axioms, ag*
*ain satisfies our
axioms. The disadvantages are:
(i)We need a smash product to formulate the definition of dualisability, an*
*d this is absent or
unnatural in many examples, such as DR when R is not commutative.
(ii)If C = Ho(C0) for some Quillen model category C0, then there are natural*
* conditions on
objects in C0 guaranteeing that they are small in C. This is not the cas*
*e for dualisability:
we need special geometric arguments to show that G=H+ is dualisable in B*
*G, for example.
(iii)There are naturally occurring cases where the generators are small but *
*not dualisable,
for example the category of G-spectra based on an incomplete universe, a*
*nd possibly also
derived categories for nonaffine schemes.
In [63], Neeman introduces the notion of a well-generated triangulated catego*
*ry; the definition
is explained and simplified in [42]. To explain the nature of this concept, we *
*recall some generali-
sations of Quillen's small object argument. Quillen originally considered a cat*
*egory E closed under
limits and colimits, and looked for objects A that were small in the sense that*
* the functor E(A, -)
preserves filtered colimits. Later, it was realised that one can fix a large ca*
*rdinal ~ and say that
A is ~-small if E(A, -) preserves colimits of sequences indexed by ordinals lar*
*ger than ~. In many
categories, every object is ~-small for some ~, and in many applications relate*
*d to localisation,
this is an adequate substitute for smallness. Neeman works with triangulated ca*
*tegories, which
typically do not have colimits for most diagrams. Thus, the above cannot be app*
*lied directly, but
a somewhat more elaborate argument leads to Neeman's well-generated categories.*
* It is shown
in [62] that the derived category of any Grothendieck abelian category is well-*
*generated.
2.4. Representability. A cohomology functor on C is a contravariant functor fro*
*m C to the
category Ab of abelian groups, that converts coproducts to products and cofibre*
* sequences to exact
sequences. For any fixed object Z, it is well-known that the representable func*
*tor X 7! [X, Z]
is a cohomology functor. We say that the representability theorem holds for C i*
*f the converse is
true, so that every cohomology theory on C is representable. Brown proved the r*
*epresentability
theorem for B, and the same proof works for any compactly generated triangulate*
*d category. By
much more elaborate arguments, Neeman has extended this to all well-generated t*
*riangulated
categories [63], and similar results have been obtained by Krause [44, 45] and *
*Franke [27].
It is also easy to see that if the representability theorem holds for C, then*
* it holds for any
localisation of C.
In [35], we take the representability theorem as an axiom; this gives a conve*
*nient way to treat
compactly generated categories and their localisations in parallel. All other a*
*uthors assume axioms
that turn out to imply the representability theorem.
2.5. Extra axioms. We now discuss some possible additional assumptions. No auth*
*or takes any
these as a standard axiom, but they define special classes of examples with use*
*fully simplified
behaviour.
(a) Let C be an abelian category in which all monomorphisms split (and thus *
*all epimorphisms
split), and suppose we have an equivalence : C -!C. We can give C a tri*
*angulation by
AXIOMATIC STABLE HOMOTOPY _ A SURVEY 7
declaring that all all triangles of the form
A B f-!B C g-!C A h-! A B
(with f(a, b) = (b, 0) and so on) are cofibre sequences. We call this the*
* abelian case. Under
mild finiteness conditions, one can show that C is the category of graded*
* A*-modules, for
some graded ring A* that is a finite product of graded division rings. Th*
*is situation is
discussed in [35, Section 8].
Examples include the category of rational G-spectra for any finite grou*
*p G, or the
category StabkGwhen |G| is invertible in k.
(b) Suppose that C has a symmetric monoidal structure, and that C is generate*
*d by the single
object S (the unit for the smash product). We then have a graded-commutat*
*ive ring ß*S
defined by ßnS = [ nS, S]. If this is noetherian, we say that we are in t*
*he noetherian
case; this is discussed in [35, Section 6]. Examples include DR where R i*
*s commutative
and noetherian, and DkG.
(c)Suppose again that C has a symmetric monoidal structure, and that C is ge*
*nerated by the
single object S. If ßnS = 0 for n < 0, then we are in the connective case*
*; this is discussed
in [35, Section 7].
We will make a number of remarks about the noetherian case below. Beyond that*
*, we refer the
reader to [35] for further discussion.
3.Functors on small objects
Let F be the category of small objects in C, and consider the category A = [F*
*op, Ab] of
additive contravariant functors from F to the category of abelian groups. This *
*is a bicomplete
abelian category satisfying the AB5 condition (filtered colimits are exact). Th*
*e functor : C -!C
induces a functor : A -!A. If C has a good symmetric monoidal structure, then *
*so does A. If
the objects of F are strongly dualisable, then we have F ' Fop and so A ' [F, A*
*b]. One can
think of F as a "ring with many objects", and regard A as its module category.
There is a Yoneda functor h: C -!A sending X to the functor hX (A) = [A, X] (*
*for A 2 F).
The structure of A and the behaviour of h have proved to be very useful in the *
*study of C, at
least when C is compactly generated. If C is merely well-generated, then Neeman*
* has developed
a partially parallel theory based on more complicated functor categories [63, C*
*hapter 6]. In the
compactly generated case, Beligiannis [2] has considered categories of the form*
* [Gop, Ab] where G
is an arbitrary triangulated subcategory of F.
Let E A be the category of exact functors: those that send cofibre sequence*
*s in F to exact
sequences in Ab. It is standard that hX 2 E for all X. In good cases, E is the *
*category of objects
of finite projective (or injective) dimension in A, and h: C -!E is close to be*
*ing an equivalence;
see Section 9 for more discussion.
The functor h: C -! A always preserves coproducts and sends cofibre sequences*
* to exact se-
quences. In other words, it is an A-valued homology theory on C. A morphism u: *
*X -!Y in C is
said to be phantom if hu: hX -!hY is zero.
4.Types of subcategories
Let R be a commutative noetherian ring, and let zar(R) denote the space of pr*
*ime ideals in
R, with the Zariski topology. (We do not use the notation spec(R), to avoid con*
*flicting uses of
the word "spectrum".) It turns out [30, 56] that we can recover zar(R) from the*
* category DR,
in several slightly different ways. More precisely, one can recover the lattice*
* of radical ideals in
R, which is well-known to be anti-isomorphic to the lattice of closed subsets o*
*f zar(R), and this
lattice determines zar(R) itself [37]. The key is to study various lattices of *
*subcategories of DR.
Using parallel constructions in the category StabkG, we can recover the space z*
*ar(H*(G; Fp)). In
the case of Boardman's category, this study makes contact with the chromatic ap*
*proach to stable
homotopy theory, and the nilpotence theorems of Devinatz, Hopkins and Smith. Th*
*is has many
important applications that are not visible in the purely algebraic examples. F*
*or example, suppose
we want to prove that all finite spectra X have some property P (X). Suppose we*
* can show that
8 N. P. STRICKLAND
o Whenever we have a cofibre sequence X -!Y -! Z in which two terms have pr*
*operty P ,
then the third also has property P
o Whenever P (X _ Y ) holds, so do P (X) and P (Y )
o There exists a finite spectrum X such that H*(X; Q) 6= 0 and P (X) holds.
Then one can show using the subcategory classification theorems that P (X) hold*
*s for all X.
The basic definitions are as follows.
Definition 4.1. Let D be a full subcategory of C. For simplicity, we assume tha*
*t any object in
C that is isomorphic to an object in D, is itself in D. Let A be an arbitrary c*
*ollection (possibly a
proper class) of objects in C.
(a) D is thick if
(i)The zero object lies in D.
(ii)Any retract of any object in D, again lies in D.
(iii)Whenever X -!Y -! Z is a cofibre sequence with two terms in D, the *
*third term is
also in D.
(b) If C has a symmetric monoidal structure, we say that D is an ideal if X *
*^ Y 2 D whenever
X 2 D. Dually, we say that D is a coideal if F (Y, Z) 2 D whenever Z 2 D.
(c) D is a localising subcategory if it is thick, and closed under (possibly*
* infinite) coproducts.
Dually, D is a colocalising subcategory if it is thick, and closed under*
* (possibly infinite)
products.
(d) A (co)localising (co)ideal is a (co)localising subcategory that is also *
*a (co)ideal.
(e) A bilocalising subcategory is a subcategory that is both a localising su*
*bcategory and a
colocalising subcategory. A biideal is a subcategory that is both an ide*
*al and a coideal.
If C is monoidal and the unit object S 2 C is small and generates C, then eve*
*ry (co)localising
subcategory is a (co)ideal. This holds in the following cases:
(1) C = B (but not C = BG for general G)
(2) C = DR, where R is commutative
(3) C = StabkG, where k has characteristic p and G is a p-group.
If we have a functor F between triangulated categories that preserves cofibre*
* sequences, then
ker(F ) := {X | F X = 0} is evidently a thick subcategory. Similarly, if F is *
*a functor from a
triangulated category to an abelian category, and F converts cofibre sequences *
*to exact sequences,
then ker(F ) will again be a thick subcategory. Under various auxiliary conditi*
*ons, we can conclude
that ker(F ) is a (co)localising subcategory or a (co)localising ideal.
Little is known about classification of subcategories that are not ideals. Th*
*e examples studied
in [8, Section 6] suggest that there is no simple and general picture.
On the other hand, there are good classification results for many of our cent*
*ral examples; the
main method of proof will be discussed in Section 8. To state the results, it *
*is convenient to
introduce one more definition. Given an object A 2 F, we write thickid for *
*the smallest
thick ideal containing A. We then say that a thick ideal I is finitely generate*
*d if it has the form
thickid for some A (this makes senseWbecause the thick ideal generated by A1*
*, . .,.Ar is also
generated by the single object A = iAi). A classification of finitely generate*
*d thick ideals (or
thick subcategories) in F extends in a fairly obvious way to give a classificat*
*ion of all ideals (or
thick subcategories) in F.
(a) Let R be a commutative noetherian ring, and put C = DR. Then the localis*
*ing subcate-
gories of C biject with the colocalising subcategories, and with the sub*
*sets of zar(R). The
finitely generated thick subcategories of F biject with closed subsets o*
*f zar(R). On the
other hand, Neeman has considered the nonnoetherian ring
R = k[x2, x3, x4, . .].=(x22, x33, x44, . .).,
where k is a field. This has only one prime ideal, but DR has an enormou*
*s collection of
localising subcategories [61].
(b) Let k be a field, let G be a finite group, and put C = StabkG. It is pro*
*ved in [8] that the
finitely generated thick ideals in F biject with the closed subsets of t*
*he projective scheme
AXIOMATIC STABLE HOMOTOPY _ A SURVEY 9
proj(H*(G; k)). One can also show that the (co)localising (co)ideals in C*
* biject with all
subsets of proj(H*(G; k)).
(c)In the category of E(n)-local spectra, the (co)localising subcategories i*
*n C biject with
subsets of {0, 1, . .,.n}, and the thick subcategories of F biject with t*
*he subsets of the
form {m, m + 1, . .,.n} for some m. All the relevant subcategories are (*
*co)ideals [36,
Theorem 6.14].
(d) Now let C be the category of K(n)-local spectra. Then 0 and C are the onl*
*y localising
subcategories of C, and also the only colocalising subcategories of C [36*
*, Theorem 7.5].
Similarly, 0 and F are the only thick subcategories of F.
(e)Finally, let C be the category of p-local spectra. The thick subcategorie*
*s of F are then the
categories Fn := {X | K(m)*X = 0 for allm < n}, where 0 n 1; this was*
* proved
in [32]. The theory of Bousfield classes gives many known examples of (c*
*o)localising
subcategories and inclusions between them. However, almost nothing is kno*
*wn about the
collection of all localising subcategories (which might even be a proper *
*class).
The classification results in Examples (a) and (b) have a partial generalisatio*
*n that applies in the
noetherian case. A strong but technically complex statement is proved in [35, S*
*ection 6.3]; given
some additional hypotheses (conjecturally always satisfied) this implies the ev*
*ident analog of (a)
and (b).
So far we have only discussed results about ideals in F; we next consider res*
*ults about (co)localising
subcategories or (co)ideals in C. On the one hand, given D C we can certainly*
* consider the thick
subcategory D \ F F; if we have a good understanding of F then this will be a*
* useful invariant,
but rather a coarse one. On the other hand, given a thick subcategory A F we *
*can consider
the category
A? := {X | [A, X]* = 0 for allA 2 A},
which is easily seen to be a bilocalising subcategory. The telescope conjectur*
*e for C is closely
related to the statement that every bilocalising subcategory is of the form A? *
*for some A. This
is known to hold in DR when R is noetherian and commutative, and also in StabkG*
*when k has
characteristic p and G is a p-group. It is believed to be false in Boardman's c*
*ategory, although
many years of study have still not produced a watertight argument.
5.Quotient categories and Bousfield localisation
Let C be a triangulated category. Given a thick subcategory D, we can look fo*
*r a triangulated
category C0 and an exact functor Q: C -! C0 that sends all objects in D to zero*
*. It turns out
that there is an initial example of such a functor, whose target we call C=D. T*
*o be more precise,
we say that a map s: X -! Y in C is a D-equivalence if the cofibre of s lies in*
* D. The class of
D-equivalences has a number of useful properties:
o Any isomorphism is a D-equivalence.
o Given morphisms X s-!Y -t!Y , if any two of {s, t, ts} are D-equivalence*
*s then so is the
third.
o Given maps X f-!Y- sZ in which s is a D-equivalence, there is a commutat*
*ive square
g
W ____//_Z
t|| |s|
fflffl|fflffl|
X __f__//Y
in which t is a D-equivalence.
We then define C=D as follows: the objects are the same as in C, and the morphi*
*sms from X to
Y are equivalence classes of öf rmal fractions" gt-1, where g and t fit in a di*
*agram of the shape
X- t W -g!Y , and t is a D-equivalence. The properties listed above allow us to*
* compose and
manipulate fractions in a natural way.
Krause has considered some more delicate notions of quotient categories, whic*
*h are important
in the study of smashing localisations; but we will not discuss these here.
10 N. P. STRICKLAND
As is well-known, there is a potential problem with the above construction, w*
*hich is of great
importance in some applications. We always assume implicitly that the morphism *
*sets C(X, Y )
are genuine sets; but as defined above, (C=D)(X, Y ) might be a proper class. T*
*here are a number
of techniques that can be used in different circumstances to show that this pro*
*blem does not arise.
As far as I know, no one has looked systematically for examples where proper cl*
*asses do arise; it
is possible that (some version of) our standing axioms are enough to prevent th*
*is.
If C=D has small Hom sets, then for any Y 2 C we have a functor from C to Ab *
*given by
X 7! (C=D)(X, Y ). The Representability Theorem shows that this is representabl*
*e, so we have
an object LX 2 C and an isomorphism C(X, LY ) ' (C=D)(X, Y ), naturally in X. A*
* standard
argument shows that L can be regarded as a functor C=D -! C, right adjoint to t*
*he quotient
functor C -!C=D. If we put
D? = {Y | C(X, Y ) = 0 for allX 2 D},
we find that L actually gives an equivalence C=D ' D? . We also use the letter *
*L for the composite
functor C -!C=D -!D? C; in this guise, it is left adjoint to the inclusion of*
* D? in C.
The functors L: C -!C arising in this way can be characterised by certain wel*
*l-known properties:
they are exact functors, equipped with a natural map iX :X -!LX such that LiX :*
*LX -!L2X
is an equivalence, and i*X:[LX, LY ] -![X, LY ] is an isomorphism for all Y . W*
*e call such a pair
(L, i) a Bousfield localisation functor. We can recover D as the category ker(L*
*) = {X | LX = 0}.
The above discussion shows that quotients are really localisations. Of course*
*, the converse is
also true: to invert a class of maps E is the same as to quotient out the local*
*ising subcategory
generated by the cofibres of the maps in E.
In [35], it is assumed that C is symmetric monoidal and that D is a localisin*
*g ideal. In this
case, the quotient category C=D (or equivalently, the category D? ) inherits a *
*symmetric monoidal
structure.
Given any localisation functor L, there is another functor C and natural tran*
*sformations
CX -qX-!X -iX-!LX -dX-! CX giving a cofibre sequence for all X. The theory can*
* be set up
in such a way that C and L play precisely dual r^oles. In any case, the pair (L*
*, i) determines (C, q)
(up to an obvious notion of equivalence) and vice versa.
Given two localisation functors (L, i) and (L0, i0), there is at most one mor*
*phism u: L -!L0
with ui = i0. We write L L0 if such a morphism exists. This gives a partial*
* order on the
collection of isomorphism classes of localisation functors. It is not known whe*
*ther this collection
is a set or a proper class.
6.Versions of the Bousfield lattice
In this section, we assume that C has a symmetric monoidal structure. Without*
* such a structure,
one could set up a formal theory along the same lines, but it seems hard to ana*
*lyze any examples
explicitly.
We can now define various partially ordered sets i; some of them may actuall*
*y be proper
classes, but we will suppress this from the terminology. An optimistic conjectu*
*re would be that
they are all the same; this is known to be true in the noetherian case. The gen*
*eral case appears to
be open (related work of Gutierrez and Casacuberta turns out not to provide a c*
*ounterexample).
Definition 6.1. o0is the class of all colocalising coideals (ordered by in*
*clusion). For any
class A of objects in C, we write colocid for the intersection of all*
* colocalising ideals
containing A. The poset 0 is actually a lattice, with meet operation D *
*^ D0= D \ D0,
and join D _ D0= colocid.
o 1 is the class of all localising ideals, ordered by reverse inclusion. *
*This is a lattice by a
dual argument.
o For any class A of objects in C, we put
A? = {X | F (A, X) = 0 for allA 2 A}
?A = {X | F (X, A) = 0 for allA 2 A}.
AXIOMATIC STABLE HOMOTOPY _ A SURVEY 11
It is easy to see that A? 2 0 and ?A 2 1. We say that a colocalising co*
*ideal D is closed
if it has the form A? for some A, or equivalently if D = (?D)?; we write *
* 2 for the set
of closed colocalising coideals, so 2 0.
o Dually, we say that a localising ideal E is closed if it has the form ?A *
*for some A, or
equivalently if E = ?(E? ). We write 3 for the set of closed localising*
* ideals, so that
3 1.
o There are order-preserving maps 0 -! 3 and 1 -! 2, given by D 7! ?D and*
* E 7! E? .
A purely formal argument shows that these give an isomorphism 2 ' 3.
o We say that a colocalising coideal D is reflective if the inclusion D -!C*
* has a left adjoint;
one can show that this implies that D is closed, so the coreflective coid*
*eals give a subset
4 2. Dually, we say that a localising ideal E is coreflective if the *
*inclusion has a right
adjoint, and these ideals give a subset 5 3. The bijection 2 ' 3 re*
*stricts to give
a bijection 4 ' 5. If D and E correspond under this bijection, then the*
*re is a pair of
functors (L, C) as in Section 5, with
D = image(L) = ker(C) = ker(L)? = image(C)?
E = image(C) = ker(L) = ?ker(L) = ?image(C).
It follows that 4 and 5 are equivalent to the poset of localisation fun*
*ctors L for which
ker(L) is an ideal, or to the poset of colocalisation functors C for whic*
*h ker(C) is a coideal.
o We say that a localising ideal E is principal if E = locid<{E}> for some *
*object E. Note that
if E = locid<{Ei}i2I>W(where I is a set, not a proper class) then we also*
* have E = locid<{E}>
where E = i2IEi, so E is principal. In Boardman's category, it is known *
*that principal
ideals are coreflective, so they form a subset 6 5. It is not clear i*
*n what generality
this argument works. If E = locid<{E}> then the corresponding localisati*
*on functor is
called stable E-nullification, and written P *E. (Confusingly, it was cal*
*led colocalisation
in Bousfield's original papers.)
o We say that a localising ideal E is a Bousfield class if it has the form *
*E = = {X | E ^
X = 0} for some E 2 C. We write 7 for the collection of all Bousfield c*
*lasses. In
Boardman's category, this is contained in 6; it is not clear how far thi*
*s fact can be
generalised. It is also known that 7 is a set rather than a proper class*
* [64, 22]. To see
this, for any finite spectrum A and any element x 2 E*A we let annA(x) be*
* the set of
maps f :A -!B in F such that (E*f)(x) = 0. We then write
<> = {annA(x) | A 2 F , x 2 E*A},
and call this the Ohkawa class of E. As F has small Hom sets and only a s*
*et of isomorphism
classes, we see that there is only a set of possible Ohkawa classes. One *
*can check that
<> determines , so there is only a set of Bousfield classes.
7.Special types of localisation
In this section, we assume that C is compactly generated.
Definition 7.1. A Bousfield localisation functor L: C -! C is smashing if image*
*(L) (which is
automatically a colocalising subcategory) is closed under coproducts (and so is*
* also a localising
subcategory). In the monoidal case, this implies that there is a natural equiva*
*lence LS ^ X -!
LX, and the corresponding colocalisation functor C also satisfies CX = CS ^ X. *
*The category
D = image(L) = ker(C) = ker(L)? = image(C)? is then both a localising ideal and*
* a colocalising
coideal. We put U = ?D and V = D? , and then bCX = F (LS, X) and bLX = F (CS, X*
*). It turns
out that bLis a localisation functor, and bCis the corresponding colocalisation*
*. Moreover, we have
U = image(C) = ker(L) =
V = image(bL) = ker(Cb)
D = image(Cb) = ker(bL) = image(L) = ker(C)
= U? = ?V = .
12 N. P. STRICKLAND
It follows that CCb = 0 = bLL, so bLC ' bLand CLb' C. This implies that bL:U -*
*! V and
C :V -!U are mutually inverse equivalences.
Apart from the finite localisations discussed below, the most important examp*
*les are the local-
isations with respect to the Johnson-Wilson spectra E(n). It is a highly nontri*
*vial theorem [67,
Chapter 8] that these are smashing.
Krause [40] has shown that L is determined by the set ann(L|F ) of morphisms *
*u: A -!B in F
for which Lu = 0. This means in particular that there is only a set of smashing*
* localisations.
Definition 7.2. A finite localisation is a localisation functor L: C -!C where *
*ker(L) = loc for
some thick subcategory A F. Functors of this type are always smashing [53][35*
*, Section 3.3].
One formulation of the telescope conjecture for C is the statement that every s*
*mashing localisation
is a finite localisation. This is known to be true in many noetherian cases, bu*
*t believed to be false
in Boardman's category. Keller [38] has provided a counterexample in DR for a c*
*ertain ring R
(but his framework of definitions is slightly different from ours, and we have *
*not pinned down the
precise relationship).
An important example of finite localisation is as follows. Let R be a noether*
*ian ring, and put
C = DR. Fix an ideal I, and let A consist of the objects X 2 C for which ß*X is*
* an I-torsion
module. Here the category U = ker(L) consists of the I-torsion objects in C, a*
*nd V = ker(Cb)
consists of I-complete objects in a suitable sense. Thus, the equivalence U ' V*
* shows that the
torsion category and the complete category are essentially the same. All this i*
*s closely related to
the theory of local (co)homology [29]. See [21] and [35, Section 3.3] for other*
* perspectives.
Definition 7.3. Suppose that C is symmetric monoidal, and is generated by the u*
*nit object
S. Given a set T of homogeneous elements in the graded ring ß*S, we let A deno*
*te the thick
subcategory of F generated by the cofibres of the maps in T , and then let L be*
* the corresponding
finite localisation functor. One can show that ß*LX = (ß*X)[T -1]. Functors of *
*this type are called
algebraic localisations; in the special case where T Z, they are called arith*
*metic localisations.
8. Nilpotence
Our understanding of Boardman's category relies heavily on the nilpotence the*
*orem of Devinatz,
Hopkins and Smith [19] and its consequences [32, 67]. We next explain the forma*
*l parts of this
story that are amenable to axiomatic generalisation [78][35, Section 5]. We wil*
*l assume here that
C is compactly generated and has a symmetric monoidal structure, and that all o*
*bjects of F are
strongly dualisable.
We say that an object I 2 F equipped with a map i: I -!S is an ideal if the m*
*ap i^1: I^S=I -!
S=I is null. (Here S=I denotes the cofibre of i.) We write I J if the map I -*
*! S -! S=J is
zero. It turns out that if I -i!S and J -j!S are ideals, then so is I ^ J -i^j*
*-!S; we will just
write IJ for this, making the set of isomorphism classes of ideals into a commu*
*tative monoid. We
say_that I and J are radically equivalent if for large n we have In J and Jn *
* I. We write
Id(S) for the set of radical equivalence classes of ideals. Given any A 2 F, th*
*e fibre of the unit
map S -!F (A, A) is an ideal, and we write ann(A) for its equivalence_class. On*
*e can show that
the rule ann(A) $ thickid gives a well-defined bijection between Id(S) and t*
*he set of finitely
generated thick ideals in F.
Next, we say that a map u: A -!B in F is smash-nilpotent if the m'th smash po*
*wer u(m):A(m)-!
B(m)is zero for m 0. One checks that Im J for some m iff the composite I -!*
*S -!S=J is
smash-nilpotent.
Now suppose we are given a set N and a collection of objects K(n) 2 C for eac*
*h n 2 N. For
any object X 2SC we put supp(X) = {n | K(n) ^ X 6= 0}. Similarly, given a thick*
* ideal A F we
put supp(A) = A2A supp(A).
We say that the K(n)'s detect ideals if whenever A, B 2 F and supp(A) supp(*
*B), we have
thickid thickid**. This implies that the map A -! supp(A) gives an embed*
*ding of the
lattice of thick ideals in the lattice of subsets of N. (Except in the noetheri*
*an case, we know of
no general method to determine the image of this map.)
AXIOMATIC STABLE HOMOTOPY _ A SURVEY 13
We next explain two versions of what it might mean for the K(n)'s to detect n*
*ilpotence. It is a
key theorem that if the K(n)'s detect nilpotence, then they also detect ideals.
For the most algebraically natural version, we need auxiliary hypotheses. Fir*
*st, we assume that
each K(n) has a commutative ring structure, and that every nonzero homogeneous *
*element in the
coefficient ring K(n)* is invertible (so K(n)* is a graded field). We also assu*
*me that the resulting
Künneth maps
K(n)*(X) K(n)*K(n)*(Y ) -!K(n)*(X ^ Y )
are isomorphisms for all X and Y (this is not automatic unless C is generated b*
*y {S}). Thus,
we can regard K(n) as giving a monoidal functor from F to the category of finit*
*e-dimensional
vector spaces over K(n)*. We say that a map u: A -!B in F is K(*)-null if the i*
*nduced map
K(n)*A -! K(n)*B is zero for all n. We say that the K(n)'s detect smash-nilpot*
*ence if every
K(*)-null map is_smash-nilpotent. Assuming this, a straightforward argument (b*
*ased on our
discussion of Id(S)) shows that the K(n)'s detect ideals.
In a general stable homotopy category C, it is very hard to produce ring obje*
*cts K(n) such
that K(n)* is a graded field. However, one can use another line of argument wit*
*h rather different
hypotheses. First, we say that the K(n)'s detect rings ifWfor every nonzero rin*
*g object R we have
K(n) ^ R 6= 0 for some n. (This will obviously hold if n = **~~ in the Bo*
*usfield lattice.)
Suppose in addition that whenever A 2 F and K(n) ^ A 6= 0 we have = *
*. We
claim that the K(n)'s detect ideals. To see this, consider a thick ideal A F,*
* and let L be the
finite localisation functor with ker(L) = loc~~~~ (and thus ker(L) \ F = A). Giv*
*en X 2 F with
supp(X) supp(A), we must show that X 2 A. It turns out to be equivalent to sa*
*y that the ring
object R = F (X, X) ^ LS is zero, so it will suffice to show that K(n) ^ R = 0 *
*for all n, and this
is easy.
The main examples are as follows.
(a)In the motivating example [19, 32, 67], C is the category of p-local spec*
*tra, N is N [ {1},
and K(n) is the n'th Morava K-theory (which is well-known to be a field t*
*heory, up to a
slight adjustment of the definition at the prime 2). The proof that these*
* theories detect
nilpotence is a tour de force of stable homotopy theory, using methods ve*
*ry far from those
surveyed in this paper. It follows that they also detect smash nilpotence*
*. It is deduced
in [32] that they detect nilpotence in various other senses, and that the*
*y detect rings. Part
of this argument can be axiomatised (at least in a connective stable homo*
*topy category)
but we shall not attempt that here.
(b) Let G be a finite group, and let C be the category of p-local G-spectra. *
*We then let N be
the set of pairs (H, n), where H is a (representative of a) conjugacy cla*
*ss of subgroups of
G, and n 2 N [ {1}. We take K(H, n) to be the representing object for the*
* cohomology
theory X 7! K(n)* H X, where H is the geometric fixed point functor, and*
* K(n) is
the usual nonequivariant Morava K-theory. These representing objects can *
*be made quite
explicit, but we shall not give the details. It follows quite easily from*
* the previous example
that they detect smash-nilpotence, and thus that they detect ideals [78].
(c)Let R be a noetherian ring, and put C = DR. We then take N to be the set *
*of prime
idealsWin R, and let K(p) be the field of fractions of R=p. Here it is no*
*t hard to show that
p = ~~~~ and that is a minimal Bousfield class; it follows t*
*hat these objects
detect nilpotence, and also that they detect ideals [56][35, Section 6].
(d) Now consider the case C = StabkG, where k is a field of characteristic p *
*and G is a finite
p-group. Take N to be the set of homogeneous prime ideals in H*(G; k). Ne*
*xt, fix an
algebraically closed field L of infinite transcendence degree over k. For*
* any p 2 N, the
theory of "shifted subgroups" gives an algebra A LG isomorphic to L[u]=*
*up and an
object K(p) 2 C such that K(p) ^ M = 0 iff L kM is free asWa module over*
* A. It follows
easily from the infinite version of Dade's Lemma [7] that p = ~~~~.*
* Morever, we
see from [7, Theorem 10.8] that = whenever K(p) ^ M 6= *
*0, so the
K(p)'s detect ideals.
14 N. P. STRICKLAND
9. Brown representability
For all authors, it is either an axiom or a theorem that cohomology functors *
*defined on the
whole category C are representable. It follows easily that the Yoneda functor i*
*s an equivalence
between C and the category of cohomology functors defined on C. This is a very*
* satisfactory
result, with many applications (existence of infinite products, existence of Bo*
*usfield localisations,
Brown-Comenetz duality, and so on).
It is desirable to extend this result to various subcategories D C. If the*
*re is an exact
localisation functor L: C -!D (as in Section 5), then this is easy. In some oth*
*er cases, it can be
proved using Neeman's theory of well-generated categories [63, Chapter 8].
Similarly, it would be helpful to have a dual theorem. This should say that *
*any product-
preserving exact covariant functor C -!Ab has the form Y 7! [X, Y ] for some re*
*presenting object
X. This has also been proved by Neeman [59, 63], under some additional hypothes*
*es.
Next, let F C be the subcategory of small objects, and suppose that F gener*
*ates C. Given
a cohomology functor H :Fop -!Ab, it is natural to ask whether there is an obje*
*ct Z 2 C and a
natural isomorphism HX = [X, Z] for X 2 F. It is equivalent to ask whether H ca*
*n be extended
to a cohomology functor defined on all of C.
We first observe that in the case C = B, this reduces to a more familiar ques*
*tion. In that
context, the Spanier-Whitehead duality functor D :X 7! F (X, S) gives an equiva*
*lence Fop ' F,
so the covariant functor H0 = H O D :F -! Ab is homological. A natural isomorph*
*ism HX =
[X, Z] (for all X 2 F) is thus the same as a H0X = ß0(X ^ Z), and Brown's homol*
*ogical
representability theorem (in the version proved by Adams) says that such an iso*
*morphism can
always be found. Adams's proof used some countability arguments, and implicitly*
* relied on the
existence of an underlying model category, so it could not directly be transfer*
*red to our axiomatic
setting. Margolis [49] and Neeman [58] independently gave reformulations that d*
*o not use model
categories. Neeman also showed, however, that the countability hypothesis is es*
*sential.
To explain this and related results (mostly distilled from [58, 17, 2]), it i*
*s convenient to use
the category A = [Fop, Ab], the subcategory E A of exact functors, and the Yo*
*neda functor
h: C -! E, as discussed in Section 3. Brown's theorem says that when C = B, th*
*e functor
h: C -!E is full and essentially surjective. The work of Margolis and Neeman sa*
*ys that the same
holds whenever
(a) C is compactly generated; and
(b) F has only countably many isomorphism classes, and F(A, B) is countable *
*for all A, B 2
F.
Now consider the case C = D(k[x, y]), where k is a field. Neeman has shown th*
*at if |k| @2
then h is not full, and if |k| @3 then h is not essentially surjective. These*
* examples are obtained
from more general and more complicated statements of two different types. Firs*
*tly, there are
results relating properties of h to homological algebra in A; secondly, there a*
*re relations between
homological algebra in A and in the category MR of R-modules, in the case where*
* C = DR.
For the first step, we define pgldim(C) to be the supremum of the projective *
*dimensions in A
of all the objects in E. Even though h need not be essentially surjective, this*
* is known to be the
same as the supremum of the projective dimensions of objects in the image of h.*
* It is also known
that
(a) pgldim(C) 1 iff h is full.
(b) If pgldim(C) 2, then h is essentially surjective.
(c) Thus, if h is full, then it is essentially surjective.
For the second step, we recall some additional definitions. An R-module P is *
*said to be pure-
projective if it is a retract of a (possibly infinite) direct sum of finitely p*
*resented modules. The
pure-projective dimension of a module M is the minimum possible length for a pu*
*re projective
resolution of M. The pure global dimension of R (written pgldim(R)) is the supr*
*emum of the pure-
projective dimensions of all R-modules. The ring R is said to be hereditary if *
*every submodule of
a projective module is again projective. It is known that pgldim(R) pgldim(DR*
*); the inequality
AXIOMATIC STABLE HOMOTOPY _ A SURVEY 15
is an equality when R is hereditary, but can be strict in more general cases. U*
*sing this and some
additional arguments, one proves the following result:
Theorem 9.1. Suppose that C = DR, where R is hereditary. Then the functor h: C *
*-!A is
(a) full iff pgldim(R) 1
(b) essentially surjective iff pgldim(R) 2.
Benson and Gnacadja [9, 10] have proved similar results for the case C = Stab*
*kG, involving
questions of purity for kG-modules. In particular, they show that the following*
* are equivalent:
(a) h is full and essentially surjective
(b) pgldim(kG) 1
(c) Either k is countable, or the Sylow p-subgroup of G is cyclic (where p i*
*s the characteristic
of k).
They also give a number of intricate examples related to these results.
Now consider the case where h is full and essentially surjective, as with the*
* original case of
Boardman's category of spectra. We then say that C is a Brown category. This ha*
*s a number of
useful consequences [35, 18, 2]. Firstly, for F 2 A, the following are equivale*
*nt:
(a) F has finite projective dimension in A
(b) F has projective dimension at most one
(c) F has finite injective dimension
(d) F has injective dimension at most one
(e) F 2 E
(f)F is in the image of h.
Next, we say that a map v in C is phantom if h(v) = 0. In a Brown category, the*
* composite of
any two phantom maps is zero, so the phantoms form a square-zero ideal. (Benson*
* [5] has shown
that this can fail when C is not a Brown category; in particular, it fails when*
* C = StabkG, k is an
uncountable field of characteristic p, and the p-rank of G is at least two.) So*
*me further properties
of phantom maps are studied in [16].
Finally, consider a diagram X :I -! C, where I is a filtered category. A weak*
* colimit for the
diagram consists of a object U and compatible maps Xi-! U for i 2 I, such that *
*the induced map
[U, Y ] -!lim -[Xi, Y ] is surjective for all Y 2 C. Such a weak colimit is min*
*imal if the induced
I
map lim-![Z, Xi] -![Z, U] is a bijection for all Z 2 F. In a Brown category, it*
* is known that
i
(a) Every filtered diagram of small objects has a minimal weak colimit, whic*
*h is a retract of
any other weak colimit.
(b) Every object can be expressed as the minimal weak colimit of a filtered *
*diagram of small
objects.
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Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, UK
E-mail address: N.P.Strickland@sheffield.ac.uk
~~