PICARD GROUPS, GROTHENDIECK RINGS, AND BURNSIDE
RINGS OF CATEGORIES
J. P. MAY
For Saunders Mac Lane, on his 90th birthday
Abstract.We discuss the Picard group, the Grothendieck ring, and the
Burnside ring of a symmetric monoidal category, and we consider examples
from algebra, homological algebra, topology, and algebraic geometry.
In October, 1999, a small conference was held at the University of Chicago in*
* honor of
Saunders Mac Lane's 90th birthday. I gave a talk there based on a paper that I *
*happened
to have started writing the month before. This is that paper, but with the pref*
*atory and
concluding remarks addressed to Mac Lane and the rest of the audience at the ta*
*lk.
Preface. According to Peter Freyd [12]: "Perhaps the purpose of categorical al-
gebra is to show that which is trivial is trivially trivial." That was written *
*early
on, in 1966. I prefer an update of that quote: "Perhaps the purpose of categori*
*cal
algebra is to show that which is formal is formally formal". It is by now abund*
*antly
clear that mathematics can be formal without being trivial. Categorical algebra
allows one to articulate analogies and to perceive unexpected relationships bet*
*ween
concepts in different branches of mathematics. For example, this talk will give*
* an
answer to the following riddle: "How is a finitely generated projective R-module
like a wedge summand of a finite G-CW spectrum?1"
1.Introduction
The classical Picard group Pic(R) of a commutative ring R is the group of iso-
morphism classes of R-modules invertible under the tensor product. This group
embeds in the group of units in the Grothendieck ring of finitely generated pro-
jective R-modules. By analogy, many other "Picard groups" have been defined
in algebraic geometry and algebraic topology. Most such groups are examples of
the Picard group Pic(C ) of a closed symmetric monoidal category C . The no-
tion of a symmetric monoidal category was formulated by Mac Lane [30] in 1963,
long before others were aware of the utility of such a common language for thin*
*k-
ing about categories with products (such as Cartesian products, tensor products,
smash products, etc.). The definition of Pic(C ) was pointed out by Hovey, Palm*
*ieri,
____________
Date: April 20, 2000.
1991 Mathematics Subject Classification. Primary 18D10, 18E30, 18G55, 55U35.
I thank Po Hu for spurring me to write up these observations, and I thank Ha*
*lvard Fausk and
Gaunce Lewis for careful readings of several drafts and many helpful comments. *
*I thank Madhav
Nori and Hyman Bass for help with the ring theory examples and Peter Freyd, Mic*
*hael Boardman,
and Neil Strickland for facts about cancellation phenomena in topology. I thank*
* Fabien Morel for
many interesting discussions of examples in algebraic geometry.
1R is a commutative ring; G is a compact Lie group.
1
2 J. P. MAY
and Strickland [20, p. 108]2, but there were many precursors. When C has finite
coproducts, Pic(C ) maps naturally to the group of units in the Grothendieck ri*
*ng
K(C ) of dualizable objects of C .
One of the goals of this paper is to advertise the general theory of duality *
*in
symmetric monoidal categories, which has not been fully exploited. We show that
there is an Euler characteristic homomorphism of rings O from K(C ) to the com-
mutative ring R(C ) of self-maps of the unit object of C . Moreover, O factors *
*as the
composite of a quotient homomorphism of rings K(C ) -! A(C ) and a monomor-
phism O : A(C ) -! R(C ), where A(C ) is a ring that we call the Burnside ring *
*of
C . When C is triangulated, O is additive on exact triangles, which makes A(C )
relatively computable. As an instance of a result that is formal but perhaps not
trivial, we prove the cited additivity in an Appendix. These definitions and ob*
*ser-
vations give a common way of thinking about some basic structure that arises in
several branches of mathematics.
The framework sheds light on and is in part motivated by equivariant stable
homotopy theory. If G is a compact Lie group and C = HoGS is the stable
homotopy category of G-spectra, then A(C ) is the Burnside ring A(G) and O :
A(C ) -! R(C ) is the standard isomorphism from A(G) to the zeroth equivariant
stable homotopy group of spheres. In a sequel [11], Fausk, Lewis, and I will ca*
*lculate
Pic(HoGS ) in terms of Pic(A(G)). I conjecture that O is also an isomorphism
when C is the A1-stable homotopy category of Morel and Voevodsky. Po Hu [25]
has made significant progress on the calculation of Pic(C ) in this case.
2.Duality and the definition of Picard groups
We shall build up structure on C as we need it, and we begin by assuming that
C is a closed symmetric monoidal category with unit object S, product ^, and
internal hom functor F . We will later assume that C has finite coproducts and *
*will
denote the coproduct by _. Our interest is in categories with far more structur*
*e,
such as the stable homotopy categories described axiomatically in [20].
The chosen notations will be congenial to the algebraic topologist, who will *
*think
of C as the stable homotopy category HoS with its smash products and function
spectra, the unit object being the sphere spectrum and coproducts being wedges.
There are many generalizations of this example in classical and equivariant sta*
*ble
homotopy theory, and many more in such modern refinements of stable homotopy
theory as [9].
The algebraist will prefer to think of C as the category MR of modules over a
commutative ring R under and Hom, with unit object R and coproduct . The
homological algebraist will prefer to replace MR by the derived category DR and
might want to generalize to differential graded modules over a differential gra*
*ded
commutative R-algebra (see e.g. [28]).
Actually, in algebra, restriction to the commutative case is rather unnatural*
*. A
more elaborate definitional framework, working with suitable monoidal, not just
symmetric monoidal, categories would allow for Picard groups of bimodules over
associative algebras and their derived analogues. The latter have been introduc*
*ed
and studied by Miyachi and Yekutieli [35, 45] and by Rouquier and Zimmermann
[41], as a follow-up to Rickard's work on tilting complexes [39, 40]. The deri*
*ved
Picard group of a commutative k-algebra A in those papers is not the same as our
____________
2Page 108 is the last page of [20]: this paper can be viewed as a continuati*
*on of that one.
PICARD GROUPS, GROTHENDIECK RINGS, AND BURNSIDE RINGS OF CATEGORIES 3
Pic(DA ) since the former is defined in terms of A-bimodules, whereas Pic(DA ) *
*is
defined in terms of left A-modules3.
The algebraic geometer will think of C as the category sh(X) of sheaves of
modules over a scheme X under the tensor product and internal Hom, with unit
object the structure sheaf OX . A more recent example in algebraic geometry is
the A1-stable homotopy category of Morel and Voevodsky [37], which is closely
analogous to the initial examples from stable homotopy theory in topology and is
one of our motivating examples.
The notion of a "strongly dualizable" (or "finite") object in C was defined in
[29, III.1.1]; we shall abbreviate by calling such objects "dualizable". An ea*
*rly
definition of this type was given by Dold and Puppe [8], but essentially the sa*
*me
definition also appears in the literature of algebraic geometry [3] and their a*
*re many
precursors. The simplest of the many equivalent forms of the definition is as f*
*ollows.
In any closed symmetric monoidal category, we have unit and counit isomorphisms
S ^ X ~=X and X ~=F (S; X) and a pairing
(2.1) ^ : F (X; Y ) ^ F (X0; Y 0) -! F (X ^ X0; Y ^ Y 0):
Define
(2.2) : F (X; Y ) ^ Z -! F (X; Y ^ Z)
by replacing Z by F (S; Z) and applying the pairing (2.1). Define the dual of X*
* to
be DX = F (X; S).
Definition 2.3.An object X of C is dualizable if the canonical map
: DX ^ X -! F (X; X)
is an isomorphism in C . When X is dualizable, we define the "coevaluation map"
j : S -! X ^ DX to be the composite
-1 fl
S _____//F (X; X)___//DX ^ X_____//X ^ DX;
where is adjoint to the identity map of X and fl is the natural commutativity
isomorphism given by the symmetric monoidal structure. Note that we have an
evaluation map " : DX ^ X -! S for any object X.
The following examples already answer our riddle: finitely generated projecti*
*ve
R-modules and wedge summands of finite G-CW spectra are the dualizable objects
in their ambient symmetric monoidal categories.
Example 2.4. Let R be a commutative ring. It is an exercise to show that an R-
module M is dualizable if and only if M is finitely generated and projective. I*
*ndeed,
if is an isomorphism, then the resulting description of the identity map M -! M
gives a recipe for presenting M as a direct summand of a finitely generated free
R-module, and the converse is even easier.
Example 2.5. (i) A spectrum X (in the sense of algebraic topology) is dualizable
in HoS if and only if it is a wedge summand of a finite CW spectrum [34, XVI.7.*
*4].
The cited result proves this more generally for G-spectra in the equivariant st*
*able
homotopy category HoGS for any compact Lie group G. In fact, a wedge sum-
mand of a finite CW spectrum is itself a finite CW spectrum (e.g. [12, 4.5]), b*
*ut
that is not true equivariantly.
____________
3These may be viewed as "central A-bimodules," whose left and right actions *
*agree.
4 J. P. MAY
(ii) The characterization in (i) is axiomatized by [20, 2.1.3], which gives the*
* anal-
ogous conclusion in any "unital algebraic stable homotopy category". Such a cat-
egory has a set G of dualizable small generators, and an object X is dualizable
if and only if it is in the thick subcategory generated by G , namely the small*
*est
subcategory of C that is closed under cofibrations and retracts and contains G .
The following characterizations of dualizable objects are proven in [29, III.*
*1.6];
other characterizations are given in [20, 2.1.3].
Theorem 2.6. Fix objects X and Y of C . The following are equivalent.
(i)X is dualizable and Y is isomorphic to DX.
(ii)There are maps j : S -! X ^Y and " : Y ^X -! S such that the composites
j^id id^"
X ~=S ^ X ____//_X ^ Y ^ X___//_X ^ S ~=X
and
id^j "^id
Y ~=Y ^ S ____//_Y ^ X ^ Y___//_S ^ Y ~=Y
are identity maps.
(iii)There is a map j : S -! X ^ Y such that the composite
(-)^Y (id^j)*
C (W ^ X; Z)_____//C (W ^ X ^ Y; Z ^ Y_)_//_C (W; Z ^ Y )
is a bijection for all objects W and Z of C .
(iv)There is a map " : Y ^ X -! S such that the composite
(-)^X (id^")*
C (W; Z ^ Y )___//_C (W ^ X; Z ^ Y ^_X)__//C (W ^ X; Z)
is a bijection for all objects W and Z of C .
Here the adjoint "": Y -! DX of a map " satisfying (ii) or (iv) is an isomorp*
*hism
under which the given map " corresponds to the canonical evaluation map " :
DX ^ X -! S. We also have the following observations [29, IIx1].
Proposition 2.7.If X and Y are dualizable, then DX and X ^ Y are dualizable
and the canonical map ae : X -! DDX is an isomorphism. Moreover, the map
of (2.2) is an isomorphism if either X or Z is dualizable, and the map ^ of (2.*
*1) is
an isomorphism if both X and X0 are dualizable or if both X and Y are dualizabl*
*e.
We have the following definition and observation [20, A.2.8].
Definition 2.8.An object X of C is invertible if there is an object Y and an
isomorphism X ^ Y ~=S.
Lemma 2.9. If X is invertible with inverse Y , then X and Y satisfy the equiva*
*lent
conditions of Theorem 2.6.
Proof.Since the functor (-) ^ Y on C is an equivalence of categories, any isomo*
*r-_
phism j : S -! X ^ Y satisfies condition (iii) of Theorem 2.6. |*
*__|
Following [20, A.2.7], we make the following definition. Henceforward, we ass*
*ume
that there is only a set of isomorphism classes of dualizable objects in C .
Definition 2.10.Define the Picard group Pic(C ) to be the set of isomorphism
classes [X] of invertible objects X with product and inverses defined by
[X][Y ] = [X ^ Y ] and [X]-1 = [DX]:
As is easily seen, Pic(C ) is a well-defined Abelian group with identity elemen*
*t [S].
PICARD GROUPS, GROTHENDIECK RINGS, AND BURNSIDE RINGS OF CATEGORIES 5
Example 2.11. By Lemma 2.9 and Example 2.4, an invertible R-module is finitely
generated and projective. By [2, x5.4], it follows that M is invertible if and *
*only if
it is finitely generated projective of rank one. This shows that Pic(MR ) coinc*
*ides
with Pic(R) as defined classically. In fact, for any scheme X, our Pic(sh(X)) *
*is
isomorphic to Pic(X) as defined classically [16, II.6.12]; see [10].
Example 2.12. The Picard groups of the derived categories DR and of the anal-
ogous derived categories of sheaves of modules have been calculated by Halvard
Fausk [10].
Example 2.13. The Picard group Pic(HoS ) of the stable homotopy category
is just Z, the sphere spectra being the only invertible spectra [18, 44]. One *
*can
construct localizations of HoS with respect to homology theories, and the probl*
*em
of computing the resulting Picard groups is non-trivial. The Picard groups of K*
*(n)-
local spectra are studied in [18, 23], and the Picard groups of E(n)-local spec*
*tra
are studied in [21].
We shall return to the study of Pic(C ) for a general stable homotopy category
C in [11], where Pic(HoGS ) is computed. The category HoGS is constructed so
as to invert the one-point compactifications SV of real representations V , but*
* we
shall see that inverting the SV has the effect of inverting other G-spectra as *
*well.
Example 2.14. Hu [24] has begun the study of Pic(C ) when C is the A1-stable
homotopy category of Morel and Voevodsky [37] by finding a surprising variety
of exotic invertible elements of C . Here again, C is constructed so as to inv*
*ert
certain canonical spheres, and Hu's examples show that many other varieties are
also inverted. A complete computation is not yet in sight.
3.The Grothendieck and unit endomorphism rings of C
We now bring Grothendieck rings into the picture, and we add the assumption
that C has finite coproducts. We write * for the coproduct of the empty set of
objects; it is an initial object of C .
Definition 3.1.Define K(C ), or better K0(C ), to be the Grothendieck ring as-
sociated to the semi-ring Iso(C ) of isomorphism classes of dualizable objects *
*of
C , with _ as addition and ^ as multiplication; [*] and [S] are the 0 and 1. L*
*et
ff : Iso(C ) -! K(C ) be the canonical map of semi-rings.
The following definition and observation explain when ff is injective.
Definition 3.2.Dualizable objects X and Y are stably isomorphic if there is a
dualizable object Z and an isomorphism X _ Z ~=Y _ Z. The category C satisfies
the cancellation property if stably isomorphic dualizable objects are isomorphi*
*c.
Remark 3.3.In the topological examples, the notion of stable isomorphism must
not be confused with the totally different notion of stable homotopy equivalenc*
*e.
When C is the stable homotopy category, the cancellation property and the struc-
ture of K(C ) have been studied extensively by Freyd [12, 13, 14, 15] and Margo*
*lis
[33]. Cancellation does not hold in general, but only due to mixing of primes. *
*Can-
cellation does hold for the stable homotopy category after localization or comp*
*letion
at a prime p, as a consequence of a unique decomposition theorem expressing any
finite p-local or p-complete spectrum as a finite wedge of indecomposable p-loc*
*al
or p-complete spectra. An inspection of the proofs shows that these results rem*
*ain
valid for the stable homotopy category of G-spectra for any compact Lie group G.
6 J. P. MAY
Proposition 3.4.Dualizable objects X and Y are stably isomorphic if and only if
ff[X] = ff[Y ], hence ff : Iso(C ) -! K(C ) is an injection if and only if C sa*
*tisfies
the cancellation property.
Corollary 3.5.ff[X] is a unit of K(C ) if and only if there is a dualizable obj*
*ect
Y such that X ^ Y is stably isomorphic to S.
Let Rx denote the group of units of a commutative ring R.
Proposition 3.6.ff restricts to a homomorphism fi : Pic(C ) -! K(C )x , and fi
is a monomorphism if stably isomorphic invertible objects are isomorphic.
The last condition is much weaker than the general cancellation property. For
example, cancellation usually does not hold in MR , but, as pointed out to me by
Madhav Nori, it is known to hold on invertible R-modules.
Proposition 3.7.Stably isomorphic invertible modules M and N over a commu-
tative ring R are isomorphic.
Proof.Adding a suitable finitely generated projective module to a given isomor-
phism if necessary, we have M F ~=N F for some finitely generated free __
R-module F . Applying the determinant functor gives an isomorphism M ~=N. |__|
We have the following commutative diagram, in which the horizontal arrows are
inclusions:
Pic(C )____//Iso(C )
fi|| |ff|
fflffl| |fflffl
K(C )x ____//_K(C ):
Proposition 3.8.Let C = MR for a commutative ring R. Then the diagram just
displayed is a pullback in which fi is a monomorphism.
Proof.Here K(C ) = K0(R). To show that the diagram is a pullback, we must
show that if P is a finitely generated projective R-module such that ff[P ] is *
*a unit,
then P is invertible. There are finitely generated projective R-modules P 0and Q
such that (P P 0) Q ~=R Q. This implies that the localization of P P 0at
any prime ideal is free of rank one, so that P P 0has rank one. But then P P *
*0, __
hence also P , is invertible. Proposition 3.7 gives that fi is a monomorphism. *
* |__|
The proofs above don't generalize, but the results might.
Problem 3.9. Find general conditions on C that ensure that the diagram above
is a pullback in which fi is a monomorphism.
Now assume further that the category C is additive, so that _ is its biproduc*
*t;
it follows that the functor ^ is bilinear. We bring another ring into the pictu*
*re, the
unit endomorphism ring R(C ).
Definition 3.10.Define R(C ) to be the commutative ring C (S; S) of endomor-
phisms of S, with multiplication given by the ^ product of maps or, equivalentl*
*y,
by composition of maps. Then C (X; Y ) is an R(C )-module and composition is
R(C )-bilinear, so that C is enriched over MR(C).
PICARD GROUPS, GROTHENDIECK RINGS, AND BURNSIDE RINGS OF CATEGORIES 7
Definition 3.11.Define a functor ss0 : C -! MR(C)by letting ss0(X) = C (S; X),
so that ss0(S) = R(C ), and observe that ss0 is a lax symmetric monoidal functor
under the natural map
OE : ss0(X) R(C)ss0(Y ) -! ss0(X ^ Y )
induced by ^. Say that X is a K"unneth object of C if X is dualizable and OE is*
* an
isomorphism when Y = DX.
The adjoint of ss0(") O OE : ss0(DX) R(C)ss0(X) -! ss0(S) is a natural map
ffi : ss0(DX) -! D(ss0(X)) of R(C )-modules. By [29, III.1.9], we have the foll*
*owing
result relating K"unneth objects of C to dualizable R(C )-modules.
Proposition 3.12.Let X be a K"unneth object of C . Then ss0(X) is a finitely
generated projective R(C )-module, ffi : ss0(DX) -! D(ss0(X)) is an isomorphism,
and OE : ss0(X) R(C)ss0(Y ) -! ss0(X ^ Y ) is an isomorphism for all objects Y .
We shall return to the study of K"unneth objects and the functor ss0 in [11],*
* where
the relationship between K"unneth objects of C and finitely generated projective
R(C )-modules is made considerably more precise.
In many of our examples, we have been considering morphisms of degree zero in
triangulated categories. The notion of a K"unneth object is sensitive to the gr*
*ading.
Definitions 3.10 and 3.11 make sense for graded morphisms in C . Here R(C ) is a
graded commutative ring, the theory of triangulated categories giving rise to t*
*he
usual signs in the commutativity relation and of course we replace the notation
ss0(X) by ss*(X) in Definition 3.11.
Example 3.13. (i) In the derived category DR with morphisms of degree zero,
where R(DR ) = R, nR is not a K"unneth object unless n = 0. However, in the
derived category D*Rof R-chain complexes and Z-graded morphisms, where again
R(D*R) = R (= Ext*R(R; R)), all nR, n 2 Z, are K"unneth objects.
(ii) Similarly, in the stable homotopy category Ho S with morphisms of degree
zero, where R(Ho S ) = Z, Sn is not a K"unneth object unless n = 0. In the stab*
*le
homotopy category Ho*S with Z-graded morphisms, where R(Ho *S ) = ss*(S), all
Sn, n 2 Z, are K"unneth objects.
(iii) The equivariant stable homotopy category Ho GS admits both a Z-graded
version Ho*GS and an RO(G)-graded version HooGS . Just as nonequivariantly,
all Sn, n 2 Z, are K"unneth objects in Ho*GS . For ff = V - W 2 RO0(G), there
is a sphere G-spectrum Sff= SV -W . If dim V H - dimW H = n for all (closed)
subgroups H of G and some integer n independent of H, then results of tom Dieck
and Petrie [5, 7] imply that Sffis also a K"unneth object in Ho*GS ; see [11]. *
*All
Sffare K"unneth objects in HooGS , where R(Ho oGS ) = ssGo(S). Here the signs in
the graded commutativity must be interpreted as units in ssG0(S).
4.Euler characteristics and the Burnside ring
In the previous example, ssG0(S), which by definition is the ring of endomorp*
*hisms
of the sphere G-spectrum in HoGS , is isomorphic to the Burnside ring A(G). When
G is finite, A(G) is the Grothendieck ring of the semi-ring of finite G-sets, a*
*nd this
isomorphism was first observed by Segal [42]. For a general compact Lie group G,
tom Dieck defined A(G) and proved this isomorphism [4, 5]. The variant of tom
Dieck's argument presented in [29] readily generalizes to give a definition of *
*A(C )
and a monomorphism A(C ) -! R(C ) for any stable homotopy category C .
8 J. P. MAY
We first define traces and Euler characteristics, and for this we do not requ*
*ire
our closed symmetric monoidal category C to have coproducts.
Definition 4.1.Define the Euler characteristic O(X) 2 R(C ) of a dualizable ob-
ject X to be the map
j fl "
S _____//X ^ DX_____//DX ^ X_____//S:
More generally, define the trace T (f) of an endomorphism f : X -! X to be the
map
j fl id^f "
S_____//X ^ DX_____//DX ^ X ____//_DX ^ X____//_S:
Traces and Euler characteristics are suitably natural in C , by [29, III.7.7].
Proposition 4.2.Let : C -! C 0be a strong symmetric monoidal functor be-
tween closed symmetric monoidal categories with unit objects S and S0. For an
endomorphism f of a dualizable object X of C , T (f) : S0 -! S0 agrees with
T (f) on S0~=S. In particular, O(X) agrees with O(X).
A still more general definition of trace maps is possible and useful [29, III*
*.7.1].
One can study analogues of the Lefschetz fixed point theorem starting from these
trace maps, but we shall restrict attention to the Euler characteristic. In alg*
*ebraic
settings, the same notion is sometimes referred to as the rank [1, 17, 43], and*
* here
again it is unnatural to restrict to the commutative case.
Euler characteristics enjoy the following basic properties. We again assume t*
*hat
C is additive.
Proposition 4.3.O(X _ Y ) = O(X) + O(Y ), O(X ^ Y ) = O(X)O(Y ), O(*) = 0,
O(S) = 1, and O(DX) = O(X).
Proof.The easy proofs are explicit or implicit in [8, 4.7] or [29, IIIx7]. As p*
*ointed
out to me by Gaunce Lewis and Halvard Fausk, O(DX) = O(X) since the following
diagram is seen to commute by use of the first diagram in the proof of [29, III*
*.1.2]:
j fl
S ___________//X ^ DX_______//DXf^fX
ffff
j || ffid^aefffffffff "||
fflffl|ssffffff fflffl|
DX ^ DDX __fl//_DDX ^ DX _"_____//_S:
|___|
Remark 4.4.Suppose that X has a diagonal map : X -! X^X and a projection
ss : X -! S such that (id^ss) O : X -! X ^ S ~=X is the identity map. Then
O(X) = ss O o, where the "transfer" o is defined to be the composite
j fl id^ "^id
S _____//X ^ DX_____//DX ^ X_____//DX ^ X ^ X_____//X:
In the equivariant context, this factorization has proven to be a powerful comp*
*u-
tational tool.
The additivity on coproducts implies that O(X) = O(Y ) if X and Y are stably
isomorphic. This allows the following definition.
PICARD GROUPS, GROTHENDIECK RINGS, AND BURNSIDE RINGS OF CATEGORIES 9
Definition 4.5.Define O : K(C ) -! R(C ) to be the ring homomorphism ob-
tained by universality from the semi-ring homomorphism O : Iso(C ) -! R(C ) that
sends [X] to O(X). Define the Burnside ring A(C ) to be the quotient ring of K(*
*C )
obtained by identifying two elements if they have the same Euler characteristic;
equivalently, A(C ) is the image of . Write O : A(C ) -! R(C ) for the resulting
monomorphism of rings.
Proposition 4.6.For a commutative ring R, A(MR ) is the subring of R generated
by its idempotent elements.
Proof.Up to terminology, this is stated without proof by Bass [1, 2.11]. Fausk *
*and
Bass showed me the following quick argument. By Hattori [17, Ex. 6], if P is a
finitely generated projective R-module of rank n, then O(P ) is multiplication *
*by
n. (Hattori assumes that R is Noetherian, but he doesn't use that hypothesis). *
*If
Spec(R) is connected, then every finitely generated projective R-module is of r*
*ank
n for some n [2, IIx5.3] and the result follows. By consideration of products o*
*f rings,
this implies the result when Spec(R) has finitely many open and closed componen*
*ts,
as always holds if R is finitely generated. By Proposition 4.2, Euler character*
*istics
are natural with respect to homomorphisms of rings. We may identify R with the
colimit of its finitely generated subrings, and K0(R) is the colimit of K0 appl*
*ied_to
these subrings. The general case follows. |__|
We assume henceforward that C is a triangulated category with triangulation
compatible with ^, in the sense made precise by [20, A.2]. In this case, additi*
*ve
inverses are already present in the image of Iso(C ) -! R(C ), which therefore
coincides with A(C ). That is, A(C ) is a quotient ring of the semi-ring Iso(C*
* ).
Note that X is dualizable if and only if X is dualizable.
Lemma 4.7. O(nX) = (-1)nO(X); in particular, O(X) = -O(X).
Proof.With Sn = nS, we have nX ~= X ^ Sn. The result follows from the
multiplicativity formula for O and the fact that O(Sn) is the transposition map
fl : S ~=Sn ^ S-n -! S-n ^ Sn ~=S;
which is multiplication by (-1)n [20, p. 105]. |_*
*__|
Now the fact that O(DX) = O(X) implies the following observation.
Lemma 4.8. Every unit [X] of the ring A(C ) satisfies [X]2 = 1.
We must still explain why we call A(C ) the Burnside ring of C .
Example 4.9. Let G be a compact Lie group and let C = HoGS be the stable
homotopy category of G-spectra. Then, by definition, R(C ) = ssG0(S), where S is
the sphere G-spectrum. By [29, V.2.12], we can define the Burnside ring of G by
A(G) = A(C ). When G is finite, A(G) is isomorphic to the classical Burnside ri*
*ng
of finite G-sets, as we shall explain in Example 4.17.
Now [29, V.2.11] gives the following version of the cited isomorphism of Segal
[42] and tom Dieck [4, 5].
Theorem 4.10. Let C = HoGS . Then
O : A(G) = A(C ) -! R(C ) = ssG0(S)
is an isomorphism of rings.
10 J. P. MAY
We offer the following conjecture.
Conjecture 4.11. The analogue of Theorem 4.10 holds for the A1-stable homotopy
category C of Morel and Voevodsky (for a given ground field k). Precisely, we h*
*ave
defined a monomorphism of rings O : A(C ) -! R(C ), and we conjecture that it is
an isomorphism.
Remark 4.12.When char k 6= 2, Morel [36] has conjectured that R(C ) is iso-
morphic to the Grothendieck-Witt ring GW (k), and he has constructed a split
monomorphism GW (k) -! R(C ). He has also proven4 that this monomorphism
factors through A(C ). Thus, if his conjecture is true, then so is ours.
Of course, A(C ) always gives a lower bound on the size of R(C ). The force of
the definition of the Burnside ring comes from the following result, which makes
A(C ) a reasonably computable object. This result is proven in [29, III.7.10] w*
*hen
C = HoGS . We give a general proof in the Appendix. The argument there requires
additional hypotheses on C , but these hypotheses are satisfied in practice.
Theorem 4.13. Let X -! Y -! Z -! X be an exact triangle. Then
O(Y ) = O(X) + O(Z):
Example 4.14. When C = HoS , the theorem implies that O is just the classical
Euler characteristic on finite CW spectra.
In the triangulated context, we have another candidate for the Grothendieck r*
*ing
of the category C .
Definition 4.15.Define K0(C ) to be the quotient of K(C ) by the subgroup gener-
ated by the elements [Y ]-[X]-[Z] for all exact triangles X -! Y -! Z -! X.
The compatibility of ^ with the triangulation ensures that the cited subgroup i*
*s an
ideal, so that K0(C ) is a quotient ring of K(C ).
Corollary 4.16.The quotient map K(C ) -! A(C ) factors through K0(C ).
Example 4.17. Let G be a compact Lie group and C = HoGS . Write [G=H]
for the element of K0(C ) or A(C ) represented by the suspension G-spectrum of
G=H+ , where H is a closed subgroup of G and the + denotes adjunction of a disj*
*oint
basepoint. We take one H from each conjugacy class of subgroups. There are wedge
summands of finite G-CW spectra that are not themselves finite G-CW spectra;
their isomorphism classes, together with the [G=H], generate K0(Ho GS ). The
[G=H] generate a subring, which is isomorphic to the Euler ring U(G) introduced
by tom Dieck [5, x5.4]. When G is finite, U(G) ~= A(G). However, a transfer
argument using Remark 4.4 shows that O(1 G=H+ ) = 0 unless H has finite index
in its normalizer. Some further argument shows that A(G) is the free Abelian gr*
*oup
generated by the remaining [G/H]; see [29, III.8.3, V.2.6]. It is remarkable th*
*at the
cited wedge summands make no contribution: as we have defined it, A(G) is a
quotient of K0(C ), but it turns out to be a quotient of U(G); see [29, V.2.12]*
*. It is
unclear whether or not such a simplification occurs more generally in the conte*
*xt
of the unital algebraic stable homotopy categories described in Example 2.5(ii).
____________
4Private communication.
PICARD GROUPS, GROTHENDIECK RINGS, AND BURNSIDE RINGS OF CATEGORIES 11
Conclusion. This paper is a very modest example of a kind of mathematics new to
the last half of the 20th century. A great deal of modern mathematics would qui*
*te
literally be unthinkable without the language of categories, functors, and natu*
*ral
transformations that was introduced by Eilenberg and MacLane in 1945. It was
perhaps inevitable that some such language would have appeared eventually. It w*
*as
certainly not inevitable that such an early systematization would have proven s*
*o re-
markably durable and appropriate; it is hard to imagine that this language will*
* ever
be supplanted. Its introduction heralded the present golden age of mathematics.
Appendix A. The additivity of the Euler characteristic
Let C be a triangulated closed symmetric monoidal category and consider an
exact triangle
(A.1) X __f_//_Yg__//Z_h_//_X:
We wish to prove that O(Y ) = O(X) + O(Z). This is proven in [29, III.7.10] when
C = HoGS . While the proof there makes use of the underlying point-set level
category of G-spectra, what is required on the point-set level in the cited pro*
*of is
structure that is present in all known examples. It is unclear to me whether or*
* not
the argument in [29] can be elaborated to apply within the axiomatic framework
for stable homotopy theory given in [20]. Certainly such an argument would be
quite delicate, since it would involve proving simultaneous compatibilities bet*
*ween
maps that, a priori, are not uniquely defined.
At the request of Fabien Morel, I will give an adaptation of the proof in [29*
*] that
applies in an axiomatic framework. However, I will assume a restrictive point-s*
*et
level axiomatic framework that encodes much more precise data than was available
in any known category of G-spectra when [29] was written. The more precise data
simplifies the proof, and modern technology (e.g. [9, 22, 25, 26, 32, 31]) show*
*s that
the axioms are realized in the cases of interest in topology and algebraic geom*
*etry.
Thus we assume that C is the homotopy category HoS obtained by inverting
the weak equivalences of a closed symmetric monoidal Quillen model category S .
In order to have canonical cylinders, cones, and suspensions of the usual sort,*
* we
assume that S is enriched over the category of based topological spaces, based
simplicial sets, or chain complexes of modules over a commutative ring R. Then
S is complete and cocomplete and has tensors X ^ A with based spaces, based
simplicial sets, or chain complexes A. We define X = X ^ S1 (where S1 is R
concentrated in degree 1 in the last case). For a map f : X -! Y , we can define
the cofiber Cf = Y [f CX, where CX = X ^ I (I being the usual chain complex
that defines chain homotopies in the last case). We then have canonical maps
g : Y - ! Cf and h : Cf -! X. We assume that the exact triangles of C are
triangles isomorphic in C to canonical triangles (= cofiber sequences)
(A.2) X __f__//Y_g__//Cf_h_//_X
in S . Thus we may assume that Z = Cf in (A.1). By cofibrant approximation
and one of the factorization axioms, every such cofiber sequence is isomorphic *
*in C
to one in which f is a cofibration between cofibrant objects, and then the cano*
*nical
map Cf -! Y=X is a weak equivalence. We assume that f in (A.1) is of this form.
We assume that the closed symmetric monoidal structure of C arises from the
closed symmetric monoidal structure of S . To ensure this, it is natural to ass*
*ume
12 J. P. MAY
that each pair of functors (X ^ (-); F (-; Y )) on S is an enriched Quillen adj*
*oint
pair (see e.g. [19, 27, 38]). It therefore induces an adjoint pair on C . Moreo*
*ver,
X ^ (-) commutes with tensors and therefore preserves cofiber sequences. We
also assume that the functor X ^ (-) preserves weak equivalences when X is cofi-
brant. The expert will recognize that this formidable looking set of hypotheses
simply encodes standard information about the good modern categories of spectra
in topology and algebraic geometry.
The following result, which is [29, 1.4], applies directly to C and does not *
*require
our added hypotheses since its proof only involves diagram chases in C .
Proposition A.3. Let X be a dualizable object of C and Y be any object. Define
ffi = ffi(X; Y ) to be the following composite:
DY ^ X _fl_//_X ^ DYae^id//_DDX ^ DY^__//D(DX ^ Y ):
Then ffi is an isomorphism in C . Moreoever, when Y = X, ffi is a canonical sel*
*f-
duality isomorphism for DX ^ X and the following diagram commutes:
j fl "
S______//X ^ DX_______//DX ^ X_______________________//_S
j|~=| ffi|| ~=j||
fflffl| fflffl| fflffl|
DS ________D"________//D(DX ^ X) _Dfl//_D(X ^ DX)_Dj_//_DS:
Therefore O(X) coincides with the composite
-1 "
(A.4) S ~=DS _D"__//D(DX ^ X)ffi_//DX ^ X____//_S:
The main point of our introduction of S is that ffi is already defined in S ,
where it is natural in the variables X and Y . This allows us to obtain canonic*
*al
and compatible maps on cofibers induced by maps of the variables. We shall use
this idea to construct the following commutative diagram in C , which we refer *
*to as
the "main diagram". Chasing the outside of the diagram, using the identification
of O(X) in (A.4), we read off the desired formula O(Y ) = O(X) + O(Z).
To simplify the identification of entries in the main diagram, we generally u*
*se
quotient notation for cofibers in various exact triangles induced from the give*
*n exact
triangle (A.1) by combinations of taking smash products and duals. In fact, we
shall carry out the construction using S , where we can use (functorial) cofibr*
*ant
approximation and factorizations to replace cofibers by equivalent quotients. *
*In
order to focus on the main ideas, we shall tacitly leave to the reader the peda*
*ntic
details of passage from S to C in the arguments to follow.
PICARD GROUPS, GROTHENDIECK RINGS, AND BURNSIDE RINGS OF CATEGORIES 13
S ~=DS T_____________D"_____________//_TD(DY|^|Y7)7
| TTTTT oooo |
| TTTTT ooo |
| DOE TTTT)) oooo Dj ||
(D";D")|| D(_DY_^Y_DZ^X) |
| jjj | |
| jjjj | |
| jj(Dff;Dfi)jjj | |
fflffl|uujjj | ||
D(DX ^ X) _ D(DZ ^ Z) ffi-1| |
| | |
| | |
| | |
| fflffl| |
ffi-1_ffi-1| -1(DX^Z_ ) ffi-1|
| jj DY|^Y55 |
| (;)jjjjj | 5 |
| jjjjj | 55 |
fflffl|uujjj | 55 |
| 55 |
(DX ^ X) _ (DZT^TZ) k| 55 |
| TTTTff+fi | 55 |
| TTTTT | i5 |
| TTTTT fflffl| 55 |
| T))DY ^Y 55 |
"+" | _____DZ^X 55 |
| jjjjj ggPPP 55 |
| OEjjjjj PPPjP 55 |
| jjjj PPPP 55|
fflffl|uujjjjjj P afflffl|eae
S oo_______________"_______________DY ^ Y
In any closed symmetric monoidal category, the dual Df of a map f : X -! Y
is characterized by the commutative diagram
id^f
DY ^ X _____//DY ^ Y
Df^id|| "||
fflffl| fflffl|
DX ^ X __"____//_S:
Applying this to the map g : Y -! Z in S , we see that the composite
" O (Dg ^ f) : DZ ^ X -! DY ^ Y -! S
coincides with "O(id^g)O(id^f) = "O(id^gf), which is the trivial map. Therefore
" factors through a canonical induced map OE : (DY ^Y )=(DZ^X) -! S in C . This
can be interpreted literally after using cofibrant approximation and factorizat*
*ion
to replace Dg ^ f by a cofibration between cofibrant objects, or one can interp*
*ret
the quotient as the relevant cofiber and proceed directly in S . The essential *
*point
is that OE is canonically determined by ", with no choice of homotopy involved.
Taking j in the main diagram to be the evident canonical map, the bottom triang*
*le
commutes by definition, and the top triangle is its dual.
The maps ff and fi in the main diagram are specified as the diagonal arrows in
the following commutative diagrams in C , which are induced from the characteri*
*stic
14 J. P. MAY
diagrams for Df and Dg:
DY_^X__id^f//_DY_^Y DZ^Y__Dg^id//_DY_^Y
DZ^X sDZ^X99 DZ^X DZ^X99
ss ttt
Df^id'|| sffss OE|| id^g|'| ttt OE||
fflffl|sss fflffl| fflffl|fitttfflffl|
DX ^ X ___"___//S DZ ^ Z ___"__//_S:
That is, ff and fi are obtained by inverting the arrows marked '; these arrows *
*are
weak equivalences since taking duals and smashing with (cofibrant) objects pres*
*erve
cofiber sequences and weak equivalences. It is clear from these diagrams that t*
*he
bottom left triangle in the main diagram commutes, and the top left triangle is*
* its
dual.
The maps and in the main diagram are defined by the diagram
DX^-1Z__~ -1 _DX^Z_ oo___-1(DX^Z_ ) _____//-1(DX^Z_ ) ~=DX^Z__
DX^-1Y = (DX^Y ) k DY ^Y SS DY ^Z DY ^Z
kkk SSSSS |
id^-1h |'| kkkkkkk SSSSS ' |Dh^id
fflffl|uukkkk S))S fflffl|
DX ^ X DZ ^ Z;
in which the horizontal arrows are canonical maps in cofiber sequences.
The map i in the main diagram is part of an evident cofiber sequence and we
define k to be j O i, so that the bottom right triangle commutes. To understand*
* the
isosceles triangle with base k, consider a second exact triangle
f0 0g0 0h0 0
X0_____//Y____//Z____//X :
We are thinking of the example
-1h
(A.5) DZ _Dg__//DYDf__//DXD___//_D-1Z:
Taking f and f0to be cofibrations between cofibrant objects, we have an equival*
*ence
Z0^ Z ' (Y 0=Y ) ^ (Y=X) ~=(Y 0^ Y )=((Y 0^ X) [ (X0^ Y )):
Comparing the exact triangle
0^h
(Y 0^ X) [ (X0^ Y_)___//Y 0^hY___//Z0^_Z___//((Y 0^ X) [ (X0^ Y ))
to the exact triangle
0^Z k Y 0^Y h0^hZ0^Z Z0^Z
-1 Z____Y_0^Y//___X0^X_//___X0^X//____Y 0^Y
0^Z 0 0
we see that -1 Z____Yi0^Ys equivalent to (Y ^ X) [ (X ^ Y ) and that, under th*
*is
equivalence, the map k corresponds to the evident composite
(Y 0^ X) [ (X0^ Y )____//(Y_0^X)[(X0^Y_)X0^X//_Y_0^YX0^X:
The term in the middle can be identified with the wedge of
Y_0^_X_' Z0^ X and X0^_Y__' X0^ Z:
X0^ X X0^ X
Applying this observation to the exact triangles (A.1) and (A.5) and tracing th*
*rough
the main diagram, this gives the commutativity of the isosceles triangle with b*
*ase
k; that is, k is the sum of the composites ff O and fi O mapping through DX ^*
* X
and DZ ^ Z.
PICARD GROUPS, GROTHENDIECK RINGS, AND BURNSIDE RINGS OF CATEGORIES 15
It remains to consider those parts of the main diagram that involve maps ffi-*
*1.
The bottom square in the following diagram commutes in S , by naturality, and
there results a canonical comparison of fibers ffi making the top square commut*
*e in
C ; that square is the trapezoid at the right of our main diagram.
-1(DX^Z_DY)^Y_ffi//_D(_DY_^Y_DZ^X)
i|| Dj||
fflffl|ffi fflffl|
DY ^ Y _______//D(DY ^ Y )
Df^g|| D(Dg^f)||
fflffl|ffi fflffl|
DX ^ Z _______//D(DZ ^ X):
The remaining parallelogram in the main diagram is essentially a naturality dia-
gram. In view of the definitions of ff, fi, , and , it is convenient to constr*
*uct
auxiliary maps
ffi : -1(_DX_^_Z_DX)^-Y! D(DY_^_X__DZ)^ Xand ffi : -1(DX_^_Z_DY)^-Z! D(_DZ_^_Y*
*_DZ)^ X
by means of diagrams just like the previous one. An essential point is that the*
*se
maps ffi are both compatible with the map ffi of fibers in that diagram; the na*
*turality
of the original map ffi in the underlying category S makes it easy to verify th*
*is.
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Department of Mathematics, The University of Chicago, Chicago, IL 60637
E-mail address: may@math.uchicago.edu