FAILURE OF BROWN REPRESENTABILITY IN DERIVED
CATEGORIES
J. DANIEL CHRISTENSEN, BERNHARD KELLER, AND AMNON NEEMAN
dedicated to H. Lenzing on the occasion of his sixtieth birthday
Abstract.Let T be a triangulated category with coproducts, Tc T the full*
* sub-
category of compact objects in T. If T is the homotopy category of spectr*
*a, Adams
proved the following in [1]: All homological functors {Tc}op-! Ab are the*
* restrictions
of representable functors on T, and all natural transformations are the r*
*estrictions of
morphisms in T.
It has been something of a mystery, to what extent this generalises to *
*other trian-
gulated categories. In [35], it was proved that Adams' theorem remains tr*
*ue as long
as Tc is countable, but can fail in general. The failure exhibited was th*
*at there can be
natural transformations not arising from maps in T.
A puzzling open problem remained: Is every homological functor the rest*
*riction of
a representable functor on T? In a recent paper, Beligiannis [5] made som*
*e progress.
But in this article, we settle the problem. The answer is no. There are e*
*xamples of
derived categories T = D(R) of rings, and homological functors {Tc}op-! A*
*b which
are not restrictions of representables.
Contents
Introduction 1
1. Pure global dimension: module categories versus derived categories *
* 7
2. Failure of Brown representability *
*12
References 21
Introduction
The introduction is written for the reader who knows about derived categories*
*, but
is not necessarily familiar with previous articles by the authors and their clo*
*se friends.
We begin with a sketch of the work done in the last 10 years, generalising resu*
*lts from
homotopy theory to derived categories. The experts may want to skip this, and *
*go
directly to Notation 0.4 on page 4. After the very general survey, will come a*
* much
more focused one. We will give, in some detail, the history of the results on g*
*eneralising
___________
Key words and phrases. Brown representability, derived category, purity, pure*
* global dimension, hered-
itary ring.
1
2 J. DANIEL CHRISTENSEN, BERNHARD KELLER, AND AMNON NEEMAN
the theorem of Brown and Adams to derived categories. Then we will explain the *
*two
open problems, which we settle in this article. Finally, we will give the natu*
*re of our
counterexamples.
Let T be a triangulated category. The representable functors T(-; X) are all *
*homologi-
cal; that is, they take triangles to long exact sequences. Given a triangulated*
* subcategory
S T, we can restrict afrepresentableifunctor on T to a functor on S. We denot*
*e the
restriction by T(-; X)fiS. All such functors are clearly homological.
The most interesting version of this, is where T is a triangulated category w*
*ith co-
products, and S is the full subcategory Tc of all compact objects in T. We remi*
*nd the
reader:
Definition 0.1.An object c 2 T is called compact, if the functor T(c; -) commut*
*es with
coproducts.
We should perhaps remind the reader, that for T the homotopy category of spectr*
*a,
Tc T is the subcategory of finite spectra. And for T = D(R), the unbounded der*
*ived
category of right R-modules, Tc turns out to be the subcategory of perfect comp*
*lexes,
that is, complexes isomorphicftoifinite complexes of projective R-modules.
Since the functor T(-; X)fiTcplays a major role in what follows, we adopt a s*
*horthand
for it. We will write
fi
yX = T(-; X)fiTc:
The subject we will be studying began with a theorem of Adams [1].
Theorem 0.2. (Adams, 1971) Let T be the homotopy category of spectra, and Tc *
*the
subcategory of finite spectra. Then any homological functor {Tc}op-! Ab is isom*
*orphic
to yX, for some object X 2 T. Furthermore, any natural transformation of functo*
*rs
yX ---! yY
is induced by some (non-unique) map X -! Y .
Remark 0.3. This theorem is usually referred to as "Brown representability". *
*The
reason for this is that, 10 years earlier, Brown [13] proved a special case. I*
*n Brown's
theorem, there was a countability hypothesis on the functor.
Calling this theorem "Brown representability" is somewhat confusing, since in*
* the
same paper, Brown proved another result, somewhat related. He showed that, if *
*T is
the homotopy category of spectra, and H : Top -! Ab is a homological functor ta*
*k-
ing coproducts to products, then H is representable. There are two theorems her*
*e, one
about homological functors on T, and another about homological functors on the *
*sub-
category Tc. And both theorems usually go under the name Brown representability*
*. To
distinguish them, the theorem about functors on Tc is sometimes called Brown re*
*pre-
sentability for homology, while the theorem about functors on T goes by the nam*
*e Brown
representability for cohomology.
FAILURE OF BROWN REPRESENTABILITY 3
In hindsight, it seems natural to ask how these statements generalise to othe*
*r trian-
gulated categories. In particular, do they hold for T = D(R), the derived categ*
*ory of a
ring. But surprisingly, questions of this sort were not asked until the 1980's.
Even then, the first questions to be asked were: To what extent can results *
*about
rings R be generalised to homotopy theory. The first to suggest that this migh*
*t be
a fruitful pursuit was probably Waldhausen. Waldhausen proposed, that techniqu*
*es
from homological algebra_Hochschild homology and cohomology, trace maps, and cy*
*clic
versions of these_should all be done in the context of E1 ring spectra. The wo*
*rk that
followed, by Goodwillie, B"okstedt, Hsiang, Madsen and many others since, showe*
*d how
good the idea was.
The idea that translating results from homotopy theory to derived categories *
*could
be worthwhile came later. The first paper we are aware of is Hopkins' [19]; in*
* it, one
has a derived category version of the nilpotence theorem. But it was really on*
*ly in
B"okstedt-Neeman's [11] that the first attempt was made, to use homotopy theore*
*tic
techniques to solve standard problems on derived categories. In the 1990's, we *
*have seen
explosive growth in the subject. In [32] and [34], Neeman applied techniques co*
*ming from
homotopy theory to the study of, respectively, the localisation theorem in K-th*
*eory and
to Grothendieck duality. The articles by Rickard [39], Benson, Carlson and Rick*
*ard [7],
[8], [9], Benson and Krause [10], Krause [28], and Benson and Gnacadja [6], giv*
*e beautiful
applications to group cohomology. Keller [23], [25], [24] applies the techniqu*
*es to the
study of cyclic homology. And Voevodsky [45], [46] and [44], Suslin-Voevodsky [*
*42] and
Morel [30] and [31], have produced a string of results, which apply homotopy th*
*eory to
the study of motives.
Along with the applications, came the study of the degree to which the theore*
*ms
extend. Homotopy theorists, over a period of 30 years, developed certain tools *
*to han-
dle the category of spectra. It became interesting to know, which parts of the*
*se tools
work, in the new and greater generality. This has also led to a series of paper*
*s. Hovey,
Palmieri and Strickland [20] set up a convenient axiomatic formalism. Without *
*go-
ing into detail, we remind the reader of the work of Beligiannis [5], Christens*
*en [14],
Christensen-Strickland [15], Franke [16], Keller [22], Krause [27], [26], [28],*
* Krause and
Reichenbach [29], and Neeman [35], [36] and [37].
This skimpy historical survey was intended to explain why people have studied*
* whether
Brown representability generalises to derived categories. As we mentioned in Re*
*mark 0.3,
the term Brown representability is used to cover two theorems. Brown representa*
*bility
for cohomology is a characterisation of representable functors Top -! Ab, while*
* Brown
representability for homology is a more complicated statement about functors {T*
*c}op-!
Ab. Of these, the generalisation of Brown representability for cohomology is ve*
*ry well
understood. The best and most recent results were obtained independently by Fra*
*nke [16]
and Neeman [37], and one of the remarkable aspects of their theorems, is that t*
*hey prove
new results even in homotopy theory. The theorems tell us, that Brown represent*
*ability
4 J. DANIEL CHRISTENSEN, BERNHARD KELLER, AND AMNON NEEMAN
for cohomology generalises to the categories of E-acyclic spectra and E-local s*
*pectra,
for any homology theory E.
This paper addresses the less well understood problem, of Brown representabil*
*ity for
homology. In the remainder of the Introduction, we will do two things. First, w*
*e will
go through the history of this problem in detail, explaining what was already k*
*nown.
Then, we will outline the counterexamples produced in this article. But before *
*we start,
we need to establish some notation.
Notation 0.4. All rings will be associative, with unit. All R-modules will be *
*right,
unitary modules. The ring R is called hereditary if its global dimension is at*
* most 1.
The triangulated category T = D(R) will be the unbounded derived category of ri*
*ght
R-modules. The category Tc is, as above, the full subcategory of compact object*
*s in T.
We will denote the category of right R-modules by the symbol Mod- R. The sub-
category of finitely presented R-modules will be denoted mod-R . The category *
*of all
additive functors {Tc}op-! Ab will be denoted Mod-Tc, while the category of all*
* additive
functors {mod- R}op-! Ab will bear the name Mod (mod- R).
When speaking of objects of the category Mod- Tc, that is, of functors {Tc}op*
*-! Ab,
we frequently wish to single out the ones that are homological, that is, take t*
*riangles to
long exact sequences. We will feel free to interchangeably use the adjectives "*
*homolog-
ical", "exact" or "flat". We remind the reader that an object of Mod- Tcis exac*
*t if and
only if it is a filtered colimit of representable functors. Furthermore, the re*
*presentable
functors are projective. (We remind the reader that the term "representable" m*
*eans
functors of the form yC, with C compact. In the literature, by abuse of notatio*
*n, people
sometimes call all functors yX representable.)
We also need to remember the notion of purity for R-modules. A short exact se*
*quence
of R-modules
0 - --! A - --! B - --! C - --! 0
is called pure exact, if it remains exact when tensored with an arbitrary left *
*R-module.
An R-module P is called pure projective, if the functor Hom (P; -) takes pure e*
*xact
sequences to exact sequences. Any coproduct of finitely presented modules is pu*
*re pro-
jective. The pure projective dimension of an R-module M is defined to be the le*
*ngth of
its shortest pure resolution by pure projectives.
A module I is called pure injective, if the functor Hom (-; I) takes pure exa*
*ct sequences
to exact sequences. The pure injective dimension of a module I is the length o*
*f the
shortest pure resolution by pure injectives. The pure global dimension of R, d*
*enoted
pgldimR, is the supremum over all M, of the pure projective dimension of M. Th*
*is
equals the supremum of the pure injective dimensions. We refer the reader to [2*
*1] for a
more thorough discussion, with proofs.
Finally, recallfourishorthand: for X 2 T, we write yX for the exact=homologic*
*al=flat
functor T(-; X)fiTc. It is also convenient to make a definition which is not so*
* standard:
FAILURE OF BROWN REPRESENTABILITY 5
Definition 0.5.(Beligiannis [5]) The pure global dimension of T, denoted pgldi*
*m T,
is defined to be the supremum, over all X 2 T, of the projective dimension in M*
*od- Tcof
the object yX.
It is useful to have also the following proposition:
Proposition 0.6.(This is Proposition 11.2 of Beligiannis [5], the proof is base*
*d on an
idea by Jensen, which appeared in a paper by Simson [41], Theorem 2.7 on page 9*
*6.)
The pure global dimension of T is also the supremum over all homological=exact *
*functors
F , of the projective dimension of F . Note that, as we will discover in this a*
*rticle, there
can be more F 's then yX's.
Let T be a triangulated category with coproducts, and Tc T the full subcateg*
*ory of
compact objects. We adopt the following notation:
[BRO]: The category T satisfies [BRO] if every exact {Tc}op-! Ab is of the f*
*orm
yX, for some X 2 T.
[BRM]: The category T satisfies [BRM] if every natural transformation yX -! *
*yX0
is induced by a map X -! X0.
The theorem of Adams (see 0.2) says, that if T is the homotopy category of spec*
*tra,
then both [BRO] and [BRM] hold in T. In [35], Neeman found a necessary and suff*
*icient
condition for this to generalise, to arbitrary compactly generated T's. For thi*
*s article, in
the statements that follow, assume T = D(R) is the derived category of a ring R.
Theorem 0.7. [35] The following are equivalent:
(i)Both [BRM] and [BRO] hold in T
(ii)pgldimT 1.
The direction (i)=)(ii) was also observed in [15]. Beligiannis, using his Propo*
*sition 0.6
above, recently showed:
Theorem 0.8. [5, Theorem 11.8] [BRM]=)[BRO].
Neeman [35] also showed that when R is countable, [BRM] (and therefore also [BR*
*O])
holds.
Keller produced the first example, where [BRM] fails. It may be found in Nee-
man's [35]. The example hinges on the following observation. If [BRM] holds, th*
*en by
Theorem 0.7, we have pgldim T 1. That is, for any object X 2 T, yX has project*
*ive
dimension at most 1. If R is a noetherian ring, this means that the cohomology *
*modules
HiY have pure projective dimension at most 1. For a counterexample, one needs o*
*nly
produce an object Y 2 T = D(R), so that its cohomology is of pure projective di*
*mension
2.
This leaves two very obvious questions:
Q1: What is the precise relation between the pure global dimension of R, den*
*oted
pgldim R, and the pure global dimension of T, denoted pgldimT? Just how clo*
*sely
are the two related to [BRM] and [BRO]?
6 J. DANIEL CHRISTENSEN, BERNHARD KELLER, AND AMNON NEEMAN
Q2: Does [BRO] hold in general?
Until this article, the only progress was a theorem of Beligiannis:
Theorem 0.9. [5, Remark 11.12] [BRO] holds, whenever pgldim T 2.
In this article, we settle both [Q1] and [Q2]. Regarding [Q1], we have the fol*
*lowing
proposition, which is not too surprising:
Proposition 1.4(i) Suppose that R is a coherent ring and all finitely presente*
*d R-
modules are of finite projective dimension. (This hypothesis holds when R is no*
*etherian
of finite global dimension.) Then pgldim R pgldimD(R).
Weaker versions were known before, and the inequality was after all at the basi*
*s of Keller's
counterexample to [BRM]. The really new result we show here is that, for some R*
*, the
inequality can be strict; Example 1.5 gives such an R. The idea of the countere*
*xample is
to produce two rings R and S, of different pure global dimensions, but with D(R*
*) ~=D(S).
Then pgldim D(R) = pgldim D(S) must be at least the maximum, and strictly bigger
than the minimum, of pgldim R and pgldim S. These rings are finite-dimensional *
*non-
commutative k-algebras described by means of quivers.
Even more surprisingly, we show that in general the answer to [Q2] is negativ*
*e: [BRO]
can fail. It fails for the rings R and S mentioned above when the cardinality *
*of k is
at least @2, for the ring k[x; y] when |k| @3, and also for the ring T = kof
polynomials in two non-commuting variables when |k| @2. [What do you think of
including something like the next sentence?] In particular, since it is consist*
*ent
with ZFC that |C| = @3, it is impossible to prove [BRO] using ZFC when R = C[x;*
* y].
The proof that these are counterexamples is presented in Section 2. Our method *
*is to
find an exact sequence
0 - --! yA - --! F - --! yB - --! 0
in Mod- Tc, and show that F is not isomorphic to yY for any Y . The idea is to *
*study
the extension group Ext1(yB; yA). We get a handle on this group using several s*
*pectral
sequences.
What is mysterious here, is that given a homological F , we cannot directly t*
*ell whether
it is of the form yX. We have no criterion to distinguish yX's from other homol*
*ogical
functors. In fact, Beligiannis' Proposition 0.6 tells us, that given any homol*
*ogical F ,
there exists a yX of projective dimension greater than or equal to that of F ; *
*projec-
tive dimension will not distinguish yX's from other homological functors. What *
*we do
amounts to finding a trick, to get around this problem.
For general rings, this is all we can say. We can give a refinement of the re*
*sults for
hereditary rings R; recall that R is hereditary if and only if its global dimen*
*sion is 1.
Theorem 0.10. Let R be a hereditary ring. Then
(i)[BRM] holds in T if and only if the pure global dimension of R is at most 1*
*; and
(ii)[BRO] holds in T if and only if the pure global dimension of R is at most *
*2.
FAILURE OF BROWN REPRESENTABILITY 7
Proof.(i) holds by Neeman's theorem 0.7, combined with the equality we prove in*
* Propo-
sition 1.4: for hereditary rings
pgldim R = pgldimD(R):
For (ii), note that Beligiannis' result (Theorem 0.9) tells us, that [BRO] ho*
*lds if
pgldimD(R) 2. The converse comes from our counterexamples of Section 2. Propos*
*i-
tion 2.9 says that if N is an R-module of injective dimension 1 and PExt3(M; N*
*) 6= 0,
then [BRO] fails for T = D(R). Thus if R is hereditary but of pure global dime*
*nsion
3, [BRO] must fail. (Here we have used the easy fact that every hereditary ri*
*ng is
__
coherent.) |__|
Acknowledgements. The authors would like to thank Apostolos Beligiannis, Thom*
*as
Br"ustle, Henning Krause and Michel Van den Bergh for helpful conversations. Th*
*e first
and second authors thank the third author, and the Centre for Mathematics and i*
*ts Ap-
plications at the Australian National University, for providing a friendly and *
*productive
setting while this work was carried out.
1. Pure global dimension: module categories versus derived categories
Let R be a ring. We denote by T the unbounded derived category D(R) of the ca*
*tegory
of (right) R-modules, and by Tc the full subcategory of compact objects. Recal*
*l that
a complex is a compact object of T iff it is quasi-isomorphic to a bounded comp*
*lex of
finitely generated projective R-modules. Here and elsewhere, we identify the ca*
*tegory
Mod-R of R-modules with the subcategory of T consisting of complexes concentrat*
*ed in
degree 0.
Lemma 1.1. The following are equivalent:
(i)R is coherent and each finitely presented R-module is of finite projective *
*dimension.
(ii)Each finitely presented R-module is compact when viewed as an object of D(*
*R).
(iii)A complex X is compact iff each HnX is finitely presented and HnX ~=0 for *
*all
but finitely many n.
Remark 1.2. In particular, the conditions of the lemma are satisfied if R is n*
*oetherian
and of finite global dimension. They are also satisfied by any hereditary ring,*
* that is,
any ring of global dimension at most 1.
Proof.We will prove (i)()(ii), and then that (i)+(ii)()(iii). But first, we re*
*mind
the reader that a ring is coherent iff the kernel of every map between finitely*
* generated
projective modules is finitely presented. We will also use the easy fact that a*
* module is
a compact object of D(R) iff it admits a finite resolution by finitely generate*
*d projective
objects.
Assume (i) holds. Let M be a finitely presented module. Since R is coherent, *
*M admits
a resolution by finitely generated projective modules. Since M is of finite pr*
*ojective
8 J. DANIEL CHRISTENSEN, BERNHARD KELLER, AND AMNON NEEMAN
dimension, this resolution may be chosen to be finite. So M is compact in D(R).*
* That
is, (ii) follows.
Suppose that (ii) holds. Then each finitely presented module admits a finite *
*resolution
by finitely generated projectives, and so in particular has finite projective d*
*imension.
Now let K be the kernel of a map f : P1 -! P0 between finitely generated projec*
*tives.
Let C be the cokernel of f. In D(R), we have the canonical triangle
K -! P -! C -! 2K;
where P is the complex P1 -! P0. By assumption, P and C are compact. Hence K
is compact. So it admits a finite resolution by finitely generated projective o*
*bjects. In
particular, it is finitely presented. Thus R is coherent; (ii) holds.
Thus far, we have proved (i)()(ii). Assume these equivalent conditions hold; *
*we wish
to prove (iii). Let X be a compact object in D(R). It is isomorphic to a finite*
* complex
of finitely generated projective modules. By (i), R is coherent; hence HnX is *
*finitely
presented for all n. And since the complex X is finite, HnX ~=0 for all but fin*
*itely many
n.
Suppose now that HnX is finitely presented for all n, and that HnX ~=0 for al*
*l but
finitely many n. The t-structure on D(R) gives us triangles
Xn ---! X ---! X>n - --! Xn
and these allow us to assemble X from its homology. Now HnX is finitely present*
*ed for
all n, and by (ii) it is compact. This forces X, an iterated extension of compa*
*ct objects,
to also be compact. We conclude that (iii) holds.
*
* __
Finally, (iii)=)(ii) is immediate. *
* |__|
Recall that the functor y : T ! Mod-Tc sends an object X 2 T to the functor
yX = T (-; X)|Tc:
For i 2 Z and F 2 Mod-Tc, we define the ithhomology of F by
HiF = F (-iR):
The functor Hi : Mod- Tc-! Mod- R extends the homology functor on T in the sense
that we have a canonical isomorphism HiO y = Hi.
As in the case of a module category, a sequence
0 -! F1 -! F2 -! F3 -! 0
of Mod- Tcis called pure exact if the sequence
0 -! Hom (G; F1) -! Hom (G; F2) -! Hom (G; F3) -! 0
is exact for each finitely presented functor G. (In particular, the sequence is*
* then exact.)
Lemma 1.3. Suppose that the conditions of Lemma 1.1 hold.
FAILURE OF BROWN REPRESENTABILITY 9
(i)The functor y : Mod-R -! Mod-Tc commutes with filtered colimits. It takes *
*pure
projective R-modules to projective objects of Mod- Tc. It transforms pure *
*exact
sequences of R-modules into pure exact sequences in Mod- Tc.
(ii)For each i 2 Z, the functor Hi commutes with filtered colimits. It takes p*
*rojective
objects of Mod-Tc to pure projective R-modules. It transforms pure exact se*
*quences
of Mod- Tcinto pure exact sequences of R-modules.
Proof.(i) Let M be a filtered system of R-modules. Clearly, if P = iR for some*
* i 2 Z,
the canonical map
colim-!T (P; M ) -! T (P; colim-!M )
is bijective. Since both sides are cohomological functors of P , this map is st*
*ill bijective
if P is any compact object of T, since Tc is the thick subcategory generated by*
* R. This
means that y takes colim-!M to colim-!yM .
Each pure projective R-module is a direct factor of a coproduct of finitely p*
*resented
modules. Since the functor y commutes with coproducts, it is enough to show tha*
*t yM
is projective if M is finitely presented. But in this case, M is compact in T,*
* by our
assumption on the ring R. So yM is projective since it is even representable.
Now let
0 -! L -! M -! N -! 0
be a pure exact sequence of R-modules. Clearly, if N is finitely presented, the*
* sequence
splits. An arbitrary module N is a filtered colimit of finitely presented modul*
*es. Thus
the sequence is a filtered colimit of split sequences. Since the functor y comm*
*utes with
filtered colimits, the image of the sequence is also a filtered colimit of spli*
*t sequences.
Thus it is pure.
(ii) By definition, the functor Hi is evaluation at -iR. Thus it commutes wit*
*h co-
limits. The projective objects of Mod-Tc are direct factors of coproducts of re*
*presentable
functors, and the functor Hi commutes with coproducts. So it is enough to show *
*that
HiyP = HiP is pure projective for P 2 Tc. This is clear since HiP is finitely p*
*resented,
by our assumption on the ring R.
Let
0 -! F1 -! F2 -! F3 -! 0
be a pure exact sequence of Mod-Tc. Clearly if F3 is finitely presented, the se*
*quence splits.
In the general case, F3 is a filtered colimit of a system of finitely presented*
* functors. So
the sequence is a filtered colimit of split sequences. Since the functor Hi com*
*mutes with
*
* __
filtered colimits, this implies the last assertion. *
* |__|
The pure global dimension of the derived category D(R) = T is by definition [*
*5] the
supremum of the projective dimensions of the functors yX, X 2 T. We write pgldi*
*m for
10 J. DANIEL CHRISTENSEN, BERNHARD KELLER, AND AMNON NEEMAN
`pure global dimension'. Part (ii) of the following lemma is due to Beligiannis*
* [5, Prop.
12.8].
Proposition 1.4.Suppose that the conditions of Lemma 1.1 hold.
(i)Let M be an R-module. Then the projective dimension of yM equals the pure
projective dimension of M. Hence we have
pgldim R pgldimD(R):
(ii)Suppose that R is hereditary. Then we have
pgldim R = pgldimD(R):
Proof.(i) The first part of the preceding lemma shows that the functor y takes *
*pure pro-
jective resolutions of a module M to projective resolutions of yM. Hence the pr*
*ojective
dimension of yM is no more than the pure projective dimension of M. Conversely,*
* let
: :-:! Q1 -! Q0 -! yM -! 0
be a projective resolution of yM. If M is finitely presented, the resolution i*
*s nullho-
motopic. An arbitrary M is still a filtered colimit of finitely presented modul*
*es. So for
arbitrary M the resolution is a filtered colimit of nullhomotopic complexes. Th*
*us it is a
pure exact sequence. By the second part of the above lemma, its image under H0 *
*is a
pure projective resolution of H0yM = M. Thus the pure projective dimension of M*
* is
no more than the projective dimension of yM.
(ii) By part (i), it suffices to prove that pgldim R pgldim D(R). Let X 2 D(*
*R).
Since R is hereditary, the object X is isomorphic in D(R) to the coproduct of t*
*he
-iHiX, i 2 Z; (cf. [33], the paragraph at the top of page 19 and bottom of page*
* 20).
Hence the projective dimension of yX is no greater than the supremum of the pro*
*jective
*
* __
dimensions of the yHiX. These are bounded by pgldimR thanks to part (i). *
* |__|
Example 1.5. Let k be an uncountable field of cardinality @t. We will exhibit*
* a k-
algebra R such that the inequality
pgldim R pgldimD(R)
is strict. Our example is based on the observation that there are algebras with*
* equivalent
derived categories but widely differing pure global dimensions. More precisely,*
* we will
exhibit a finite-dimensional k-algebra R with pgldim R = 0 such that D(R) is tr*
*iangle
equivalent to D(S) for a finite-dimensional hereditary k-algebra S whose pure g*
*lobal
dimension is t + 1. Thus we have
pgldim R < pgldimS = pgldimD(S) = pgldimD(R);
where we have used part (ii) of the above proposition for the first equality.
A consequence of this is that [BRM] fails for D(R) even though R has pure glo*
*bal
dimension 0.
FAILURE OF BROWN REPRESENTABILITY 11
We will define the algebras R and S using the language of quivers with relati*
*ons
(cf. [40], [18], [3]). Here is all we need: A quiver is an oriented graph. It i*
*s thus given by
a set Q0 of points, a set Q1 of arrows, and two maps s; t : Q1 -! Q0 associatin*
*g with
each arrow its source and its target. A simple example is the quiver
"A10: 1 ff1-!2 ff2-!3 -! : :-:! 8 ff8-!9 ff9-!10:
A path in a quiver Q is a sequence (y|fir|fir-1| : :|:fi1|x) of composable arro*
*ws fii with
s(fi1) = x, s(fii) = t(fii-1), 2 i r, t(fir) = y. In particular, for each poi*
*nt x 2 Q0, we
have the lazy path (x|x). It is neutral for the obvious composition of paths. T*
*he quiver
algebra kQ has as its basis all paths of Q. The product of two basis elements *
*equals
the composition of the two paths if they are composable and 0 otherwise. For ex*
*ample,
the quiver algebra of Q = "A10is isomorphic to the algebra of lower triangular *
*10 x 10
matrices.
The construction of the quiver algebra kQ is motivated by the (easy) fact tha*
*t the
category of left kQ-modules is equivalent to the category of all diagrams of ve*
*ctor spaces
of the shape given by Q. It is not hard to show that each quiver algebra is her*
*editary.
It is finite-dimensional over k iff the quiver has no oriented cycles.
Gabriel [17] showed that the quiver algebra of a finite quiver has only a fin*
*ite number
of k-finite-dimensional indecomposable modules (up to isomorphism) iff the unde*
*rlying
graph of the quiver is a disjoint union of Dynkin diagrams of type A, D, E.
The above example has underlying graph of Dynkin type A10 and thus its quiver
algebra has only a finite number of finite-dimensional indecomposable modules.
An ideal I of a finite quiver Q is admissible if for some N we have
(kQ1)N I (kQ1)2;
where (kQ1) is the two-sided ideal generated by all paths of length 1. A quiver*
* Q with
relations R is a quiver Q with a set R of generators for an admissible ideal I *
*of kQ. The
algebra kQ=I is then the algebra associated with (Q; R). Its category of left m*
*odules is
equivalent to the category of diagrams of vector spaces of shape Q obeying the *
*relations
in R. The algebra kQ=I is finite-dimensional (since I contains all paths of len*
*gth at least
N), hence artinian and noetherian. By induction on the number of points one can*
* show
that if the quiver Q contains no oriented cycle, then the algebra kQ=I is of fi*
*nite global
dimension.
One can show that every finite-dimensional algebra over an algebraically clos*
*ed field
is Morita equivalent to the algebra associated with a quiver with relations and*
* that the
quiver is unique (up to isomorphism).
Now we let R be the finite-dimensional k-algebra associated with the above qu*
*iver "A10
and the relation ff8ff7: :f:f1 (no ff9!). The algebra R is a quotient of kA"10a*
*nd thus it
admits only a finite number of indecomposable finite-dimensional modules. By a *
*result
of Auslander [2] and Tachikawa [43], this is equivalent to pgldimR = 0.
12 J. DANIEL CHRISTENSEN, BERNHARD KELLER, AND AMNON NEEMAN
Let S be the quiver algebra of the quiver
2 -! 3 -! 4 -! 5 -! 6 -! 7 -! 8 -! 9 -! 10
E : #
10:
Thus S is finite-dimensional over k and hereditary. Moreover, by [12], there i*
*s a full
exact embedding of the category Mod- kof modules over the ring of polyno*
*mials
in two non-commuting variables into Mod- S, which means that the algebra S is w*
*ild.
By a result of Baer-Lenzing [4], it follows that pgldimS = t + 1, if we assume *
*that k is
uncountable of cardinality @t.
Finally, we need to show that R and S have equivalent derived categories. Ind*
*eed, the
algebra R admits a tilting complex with endomorphism ring S so that the equival*
*ence
follows from Rickard's Morita theorem for derived categories [38]. To describe *
*the tilting
complex, let Pi= eiR be the projective R-module associated with the idempotent *
*ei=
(i|i) (the lazy path). The tilting complex is then the sum of the complexes
P1 -! P2; P1 -! P3; : :;: P1 -! P8; P1 -! 0; 0 -! P9; 0 -! P10;
where the first term of each complex is in degree 0.
2. Failure of Brown representability
In this section, R will be a ring satisfying the equivalent conditions of Lem*
*ma 1.1. In
particular, all the theorems hold if R is a noetherian ring of finite global di*
*mension, or
if R is hereditary. We begin by reminding ourselves of a standard spectral sequ*
*ence.
Lemma 2.1. Let A be an abelian category satisfying AB5, and with enough projec*
*tives.
Suppose that X and Y are objects of A and that X = colim-!X expresses X as a f*
*iltered
colimit of objects X 2 A. Then there is a spectral sequence, converging to Ext*
*i+j(X; Y ),
whose E2 term is
lim-iExtj(X ; Y ):
Proof.There is a standard chain complex which computes the derived functors of *
*colim-!.
Since the abelian category A satisfies AB5, the derived functors of filtered co*
*limits vanish,
and we deduce an exact sequence in A
M M
. .-.--! X ---! X ---! X - --! 0:
!
This gives us a resolution of X in A, and the spectral sequence is just the spe*
*ctral
*
* __
sequence of the functor Ext*(-; Y ) applied to this resolution. *
* |__|
In the following, we write mod-R for the category of finitely presented R-mod*
*ules and
Mod (mod- R) for the category of contravariant additive functors from mod-R to *
*Ab. The
FAILURE OF BROWN REPRESENTABILITY 13
object
fifi
Mod- R - ; M fi
mod-R
of Mod (mod- R) will be denoted zM.
Lemma 2.2. Let R be a ring, and let M be a filtered diagram of R-modules with*
* colimit
M. Then
(i)yM = colim-!yM in Mod- Tc.
(ii)zM = colim-!zM in Mod (mod- R).
Proof.(i) was proved in Lemma 1.3 (i). The second statement is more familiar in*
* the
equivalent form, which states that Mod-R (K; M) = colim-!Mod-R(K; M ) for any f*
*initely
presented K. This is not hard to prove.
__
|__|
Remark 2.3. Let R be a ring and let M be an R-module. Consider the filtered di*
*agram
of finitely presented modules M equipped with a map to M. Then one can show th*
*at
M is the colimit of this diagram. This is the setting in which we will apply Le*
*mma 2.2.
The following Lemma is well known; the proof may be found, for example, in Th*
*eo-
rem 2.8 of Simson's [41]. We include a sketch of the proof for the reader's con*
*venience.
Lemma 2.4. Let R be a ring satisfying the conditions of Lemma 1.1, and let M b*
*e an
R-module. As mentioned in Remark 2.3, M is the filtered colimit of all finitely*
* presented
modules M mapping to M.
(i)Let F be an object of Mod- Tc. That is, F is a functor {Tc}op-! Ab. Then the
group Exti(yM; F ) of extensions in Mod- Tcis isomorphic to lim-iF (M ).
(ii)Let F be an object of Mod (mod- R). That is, F is a functor {mod- R}op *
*-!
Ab. Then the group Exti(zM; F ) of extensions in Mod (mod- R) is isomorphi*
*c to
-limiF (M ).
Proof.(i): By Lemma 2.2, yM is the colimit of yM in Mod- Tc. Lemma 2.1 then te*
*lls
us that we get a spectral sequence with E2 term
lim-iExtj(yM ; F )
converging to the group Exti+j(yM; F ) of extensions in Mod- Tc. The functor y*
*M is
representable, since by our hypothesis on R the module M is compact. Thus yM *
*is
projective, the Extj terms vanish unless j = 0, the spectral sequence collapses*
*, and the
desired isomorphism follows.
*
* __
The proof of (ii) is similar. *
* |__|
Remark 2.5. In part (i) of Lemma 2.4, we computed the extensions of yM by F . *
*This
interests us most in the case where F = yjN, with N an R-module. In this case, *
*the
14 J. DANIEL CHRISTENSEN, BERNHARD KELLER, AND AMNON NEEMAN
computation tells us that we have isomorphisms
Ext i(yM; yjN) = lim-iT(M ; jN) = lim-iExtjR(M ; N):
In part (ii) of Lemma 2.4, we computed the extensions of zM by F . This inter*
*ests us
most in the case where F = zN, with N an R-module. In this case, the computation
tells us that we have an isomorphism
Exti(zM; zN) = lim-iHomR(M ; N):
Moreover the group Exti(zM; zN) of extensions in Mod (mod- R) can be identified*
* with
the group PExti(M; N); see [21]. We deduce that
PExt i(M; N) = lim-iHomR(M ; N):
Corollary 2.6.If M and N are R-modules and j > 0, then every map yjM -! yN
vanishes. Moreover, maps yM -! yN are in one-to-one correspondence with maps of
R-modules M -! N.
Proof.For j > 0, we must show that any map yM -! y-jN vanishes. But by
Remark 2.5, the group of such maps is
lim-0Ext-jR(M ; N);
which vanishes because there are no extensions of negative degree.
The group of maps yM -! yN is exactly
-lim0Ext0R(M ; N);
__
which is Hom R(M; N). |__|
Lemma 2.7. Let F be an object in Mod- Tc, that is, a contravariant additive fu*
*nctor
from Tc to Ab. Suppose there exists an integer j > 0, R-modules M and N, and a *
*short
exact sequence in Mod- Tc
0 - --! yjN --ff-!F --fi-!yM ---! 0:
Then this sequence is unique up to isomorphism.
Proof.The integer j and the modules M and N are clearly determined by the homol*
*ogy
of F . In Corollary 2.6 we saw that any map yjN -! yM vanishes. Therefore, given
any map fl : yjN -! F , the composite
yjN - -fl-!F - fi--!yM
vanishes, and hence fl must factor through ff. Dually, any map F -! yM must fac*
*tor
*
* __
through fi. This shows that the given exact sequence is unique. *
* |__|
FAILURE OF BROWN REPRESENTABILITY 15
Lemma 2.8. Let F be an object of Mod- Tc, and suppose there exists an integer *
*j > 0,
R-modules M and N, and a short exact sequence in Mod- Tc
0 - --! yjN --ff-!F --fi-!yM ---! 0:
The functor F will be of the form yY if and only if the short exact sequence co*
*mes from
a triangle. That is, if and only if there exists a triangle in T
jN ---! Y ---! M --@-! j+1N
with @ a phantom map, so that the sequence
0 ---! yjN - ff--!F - fi--!yM ---! 0
is obtained by restricting the representable functors to Tc.
We remind the reader that a map W -! X in T is called phantom if the composite
C -! W -! X is zero for each compact object C and each map C -! W .
Proof.The implication (= is trivial. If the triangle exists and is isomorphic t*
*o the short
exact sequence of functors on Tc, then F is the restriction of a representable *
*functor on
T. We wish to prove =). We suppose therefore that the short exact sequence of f*
*unctors
is given, and that F is the restriction of a representable. We want to produce *
*a triangle.
The short exact sequence
0 ---! yjN - ff--!F - fi--!yM ---! 0
permits us easily to compute F (nR), for all n 2 Z. We have
8
< M ifn = 0
F (nR) = N ifn = j
:
0 otherwise.
But if F = yY , then F (nR) = H-n (Y ). The above computes for us the cohomology
of Y , as an object in D(R) = T.
There is a t-structure truncation on D(R), giving a triangle
Y -1 ---! Y ---! Y 0 --@-! Y -1 ;
and our homology computation shows that Y -1 and Y 0 each have only one non-ze*
*ro
cohomology group. The triangle is therefore of the form
jN - --! Y - --! M --@-! j+1N:
We deduce an exact sequence
yjN - --! yY ---! yM:
Now recall that yY = F , and that by the proof of Lemma 2.7, any map yjN -! F
factors through ff, and any map F - ! yM factors through fi. The exact sequence
16 J. DANIEL CHRISTENSEN, BERNHARD KELLER, AND AMNON NEEMAN
coming from the triangle therefore factors through
yjN
?
f?y
0 - --! yjN --ff-!F --fi-!yM ---! 0:
??
y g
yM
By Corollary 2.6, the morphisms f and g in the diagram above come from maps of
modules N -! N and M -! M. Evaluating the functors at R and jR, we compute
that both f and g are canonical isomorphisms. Hence the triangle gives rise to *
*the short
*
* __
exact sequence of functors, and @ must be a phantom map. *
*|__|
Next comes a spectral sequence argument. To help the reader, we will first d*
*o the
easy, baby case.
Proposition 2.9.Let R be a ring satisfying the conditions of Lemma 1.1. Let N b*
*e an
R-module with injective dimension at most 1 and pure injective dimension at lea*
*st 3.
Then in Mod- Tcthere exists a homological functor F : {Tc}op-! Ab which is not *
*the
restriction of any representable. That is, there exists no Y with yY = F .
Example 2.10. Let k be a field and R the algebra of the quiver E of Example 1.*
*5.
Then R is finite-dimensional over k and hereditary, since it is the quiver alge*
*bra of a
finite quiver. So all R-modules are of injective dimension at most 1. Assume *
*that k
is uncountable of cardinality @t. Then by [4], the pure global dimension of R *
*equals
t + 1. Thus when t 2 there does exist an R-module satisfying the assumptions o*
*f the
proposition.
Similarly, the ring T = kof polynomials in two non-commuting variables*
* is an
example when t 2.
To obtain examples where R is commutative, we will need to use Theorem 2.11, *
*which
is a refined version of the above proposition.
Proof.Because N is of pure injective dimension at least 3, there exists a modul*
*e M and
integer n 3, so that PExtn(M; N) 6= 0. If n > 3, choose a pure exact sequence
0 - --! M0 - --! P - --! M ---! 0;
with P pure projective. Then PExtn(M; N) = PExtn-1(M0; N). By a sequence of such
dimension shifts, we may find an M so that
PExt3(M; N) 6= 0:
By Remark 2.3, we may express M as a filtered colimit of finitely presented m*
*odules
M . By Lemma 2.1, applied this time to the category of R-modules, there is a sp*
*ectral
FAILURE OF BROWN REPRESENTABILITY 17
sequence with E2 term lim-iExtjR(M ; N) converging to Exti+jR(M; N). We will n*
*ow
compute in this spectral sequence.
In Lemma 2.4, we computed that
-lim3Ext0R(M ; N) = PExt3(M; N);
and by the above, this does not vanish. On the other hand, we know that Ext3R(M*
*; N) =
0, since by hypothesis N is of injective dimension at most 1. It follows that o*
*ne of the
differentials in the spectral sequence into the term
lim-3Ext0R(M ; N)
must be non-zero.
But there are only two differentials into this term, one from lim-1Ext1and on*
*e from
lim-0Ext2. The latter vanishes, since by hypothesis N is of injective dimension*
* at most
1. It follows that
lim-1Ext1R(M ; N) 6= 0:
But in Lemma 2.4 we showed that this is the group of extensions, in Mod- Tc,
0 ---! yN ---! F ---! yM - --! 0:
The group does not vanish so we may choose a non-trivial extension. Since F is*
* the
extension of two homological functors, F must be homological. Now we will show *
*that
F cannot be isomorphic to a functor yY .
Lemma 2.8 tells us that if F is isomorphic to yY , then there is a triangle i*
*n T
N - --! Y ---! M --@-! 2N
so that the exact sequence of functors above is isomorphic to the one obtained *
*from the
triangle. But the map @ : M -! 2N is an element of
Ext2(M; N) = 0;
and therefore the triangle splits. The exact sequence of functors is not split*
*, and we
*
* __
conclude that F cannot be isomorphic to any yY . *
*|__|
The next Theorem is the more macho computation with the same spectral sequenc*
*e.
Theorem 2.11. Let R be a ring satisfying the conditions of Lemma 1.1. Let N be*
* an
R-module with injective dimension at most n and pure injective dimension at lea*
*st n+2.
Then in Mod- Tcthere exists a homological functor F : {Tc}op-! Ab, which is not*
* the
restriction of any representable. That is, there exists no Y with yY = F .
Proof.Let N be a module satisfying the hypotheses. As in the proof of Propositi*
*on 2.9,
we may choose a module M with PExt n+2(M; N) 6= 0: We may also express M as a
filtered colimit of finitely presented modules M .
18 J. DANIEL CHRISTENSEN, BERNHARD KELLER, AND AMNON NEEMAN
Lemma 2.1 gives us a spectral sequence, whose E2 term is
lim-iExtjR(M ; N);
which converges to Exti+jR(M; N). Once again, we have that
lim-n+2Ext0R(M ; N) = PExtn+2(M; N);
and this does not vanish, by the choice of M. But Extn+2R(M; N) = 0, since N i*
*s of
injective dimension at most n, so there must be a non-zero differential into th*
*e term
-limn+2Ext0R(M ; N):
Now observe that
lim-0Extn+1R(M ; N) = 0;
since N is of injective dimension at most n. It follows that for some i with 1 *
* i n,
there is a non-zero differential in the spectral sequence, from
lim-iExtn+1-iR(M ; N)
to the term lim-n+2Ext0R(M ; N) 6= 0.
Now recall the construction of our spectral sequence, from Lemma 2.1. Since M*
* is the
filtered colimit of M , there is an exact resolution of M
M M
. . .---! M ---! M - --! M ---! 0:
!
This resolution is a pure exact resolution by pure projectives. (It is pure exa*
*ct because
it remains exact in the category Mod (mod- R). And direct sums of finitely pre*
*sented
modules M are pure projective.) By Lemma 1.3, it becomes an exact resolution *
*by
projectives in the category Mod- Tc.
To simplify the notation, we will write the above resolution as
. .-.--! P2 ---! P1 - --! P0 ---! M - --! 0:
Let Ki stand for the image of the map Pi- ! Pi-1. In Lemma 2.4 we showed that
lim-iExtn+1-iR(M ; N)
is the group of extensions
Exti(yM; yn+1-iN):
But since the pure exact sequence
0 ---! Ki-1 ---! Pi-2 ---! . . .---! P0 ---! M ---! 0
remains exact in Mod- Tc, and the middle modules map to projectives in Mod- Tc,*
* we
deduce that the above extension group is isomorphic to
Ext1(yKi-1; yn+1-iN):
FAILURE OF BROWN REPRESENTABILITY 19
In other words, an element of the group
lim-iExtn+1-iR(M ; N)
may be thought of as a short exact sequence in Mod- Tc
0 - --! yn+1-iN ---! F ---! yKi-1 ---! 0:
We know that in the spectral sequence, for some 1 i n, there is a non-zero di*
*fferential
lim-iExtn+1-iR(M ; N) E - fl--! -limn+2Ext0R(M ; N);
for a subgroup E lim-iExtn+1-iR(M ; N). What we will now show is that, if fl(x*
*) 6= 0,
then x corresponds to an exact sequence
0 ---! yn+1-iN - --! F ---! yKi-1 ---! 0
where F is not isomorphic to yY . Expressing the same thing slightly differentl*
*y, we will
show that if x 2 lim-iExtn+1-iR(M ; N) comes from an exact sequence of functors*
* with
F = yY , then fl(x) = 0.
Suppose therefore that we are given a short exact sequence in Mod- Tc
0 ---! yn+1-iN - --! yY - --! yKi-1 - --! 0:
We need to show that in the spectral sequence, the differential fl annihilates *
*x. By
Lemma 2.8, the exact sequence of functors comes from a triangle
n+1-iN ---! Y - --! Ki-1 --@-! n+2-iN
with @ a phantom map. From the definition of the modules Ki, we have a pure exa*
*ct
sequence of R-modules
0 - --! Ki ---! Pi-1 - --! Ki-1 ---! 0:
This exact sequence gives a triangle in T = D(R). The fact that @ : Ki-1- ! n+2*
*-iN
is phantom tells us that the composite
Pi-1 ---! Ki-1 --@-! n+2-iN
must vanish, since Pi-1 is a coproduct of compact objects. But then the map @ m*
*ust
factor as
Ki-1 - --! Ki - --! n+2-iN:
Thus if an element x 2 lim-iExtn+1-iR(M ; N) comes from a short exact sequence
0 ---! yn+1-iN - --! F ---! yKi-1 ---! 0
with F ' yY , then Y is determined by a class
y 2 Extn+1-iR(Ki; N):
20 J. DANIEL CHRISTENSEN, BERNHARD KELLER, AND AMNON NEEMAN
In conclusion, we deduce the following. Let us define K0 = 0. We have a map o*
*f chain
complexes
. .-.--! Pi ---! Pi-1 - --! . .-.--! P1 ---! P0 ---! 0
?? ? ? ?
y ?y ?y ?y
. .-.0--!Ki --0-! Ki-1 - -0-! . .-.0--!K1 --0-! K0 ---! 0 .
Hence there is a map of spectral sequences in hypercohomology. On the E2 term, *
*it is
ExtjR(Ki; N)---! lim-iExtj(M ; N):
The whole point is that the spectral sequence on the left degenerates at E1, si*
*nce it comes
from a complex with zero differentials. We have shown that if x 2 lim-iExtn+1-i*
*(M ; N)
corresponds to an extension
0 ---! yn+1-iN - --! F ---! yKi-1 ---! 0
with F ' yY , then x is the image of some y from the trivial spectral sequence.*
* Therefore,
*
* __
all differentials out of x vanish. *
* |__|
Example 2.12. Let k be an uncountable field of cardinality @t. Then by [4], th*
*e poly-
nomial ring R = k[x; y] is of pure global dimension t + 1. On the other hand, *
*it is of
global dimension 2. Hence there do exist modules N over R = k[x; y], satisfyin*
*g the
assumptions of the theorem when t is at least 3.
The following lemma explains why the functors F : {Tc}op -! Mod- kthat we con-
struct always take values in infinite-dimensional vector spaces. The idea of th*
*e double
dual used in the proof is due to M. Van den Bergh.
Lemma 2.13. Let k be a field and
F : {Tc}op-! mod-k
an exact functor which takes its values in the category mod-k of finite-dimensi*
*onal vector
spaces. Then F is of the form yX for some X 2 T.
Proof.Denote by D the functor which takes a vector space to its dual. Then the *
*functor
G = D O F is exact and covariant. Let
G": T -! Mod-k
be the Kan extension of G to T. Thus, for Y 2 T, we have
"G(Y ) = colimGC;
-!
where the colimit is taken over the category of arrows C -! Y from a compact C *
*to
Y . A moment's thought will convince the reader that G"is exact and commutes w*
*ith
coproducts. Hence DOG"is exact and takes coproducts to products. By Brown's the*
*orem,
it is representable: We have
D O "G= T (-; X)
FAILURE OF BROWN REPRESENTABILITY 21
for some X 2 T. We claim that yX = F . Indeed, the restriction of D O "Gto Tc*
* is
isomorphic to D O D O F , and this functor is isomorphic to F because F C is fi*
*nite-
*
*__
dimensional for all C 2 Tc. |*
*__|
References
1.J. Frank Adams, A variant of E. H. Brown's representability theorem, Topolog*
*y 10 (1971), 185-198.
2.M. Auslander, Representation theory of Artin algebras II, Comm. Algebra 1 (1*
*974), 269-310.
3.M. Auslander, I. Reiten, and S. Smalo, Representation theory of Artin algebr*
*as, Cambridge Studies
in Advanced Mathematics, vol. 36, Cambridge University Press, 1995 (English).
4.D. Baer and H. Lenzing, A homological approach to representations of algebra*
* I: The wild case, J.
Pure Appl. Algebra 24 (1982), 227-233.
5.A. Beligiannis, Relative homological algebra and purity in triangulated cate*
*gories, (1999), Preprint.
6.D. Benson and G. Gnacadja, Phantom maps and purity in modular representation*
* theory I, to appear
in Fund. Math.
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Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, *
*Balti-
more, MD, USA
E-mail address: jdc@math.jhu.edu
Universite Paris 7, UFR de Mathematiques, Institut de Mathematiques, UMR 7586*
* du
CNRS, Case 7012, 2, place Jussieu, 75251 Paris Cedex 05, FRANCE
E-mail address: keller@math.jussieu.fr
Center for Mathematics and its Applications, School of Mathematical Sciences,*
* John
Dedman Building, The Australian National University, Canberra, ACT 0200, AUSTRA*
*LIA
E-mail address: Amnon.Neeman@anu.edu.au