K(N)LOCAL DUALITY FOR FINITE GROUPS AND GROUPOIDS
N. P. STRICKLAND
1.Introduction
The starting point of the investigations described here was our discovery of *
*a natural inner
product on the ring K(n)*BG, the n'th Morava Ktheory of the classifying space *
*of a finite group
G. If n = 1 and G is a pgroup then K(1)*BG is essentially the same as R(G)=p (*
*where R(G)
is the complex representation ring of G) and our inner product is just (V; W ) *
*= dimC(V W )G
(mod p). This is closely related to the classical inner product on R(G), given *
*by
= dimCHom G(V; W ) = (V; W *):
(For more general groups G, there is still a relationship with the classical pr*
*oduct but it is not
too close; see Section 11 for some pitfalls.)
It turns out to be useful to work with an inner product on the spectrum LG :=*
* LK(n)1 BG+
and then deduce consequences in Morava Ktheory (and other generalised cohomolo*
*gy theories)
by functorality. As background to this, in Section 2 we recall some results abo*
*ut inner products
on objects in arbitrary compact closed categories. Moreover, to elucidate the r*
*elationship between
the inner product and the ring structure on K(n)*BG, it is helpful to recall so*
*me facts about
Frobenius algebras and their relationship with topological quantum field theori*
*es, which we do in
Sections 3 and 4. In Section 5 we give a version of PoincareAtiyah duality for*
* manifolds which
illustrates these ideas nicely, and which has striking formal similarities with*
* our later treatment
of LG; indeed, one could probably set up a unifying categorical framework. We h*
*ave also found
that many aspects of our theory (for example homotopy pullbacks and free loop s*
*paces) can be
discussed more cleanly in terms of groupoids rather than groups. This is also c*
*onvenient for a
number of applications and calculations. Because of this, we give a fairly deta*
*iled treatment of
the homotopy theory of groupoids in Section 6. In Section 7 we discuss transfer*
*s for coverings
uptohomotopy, as outlined in [30, Remark 3.1]. In Section 8 we turn to the sp*
*ectra LG. In [17]
we used the GreenleesMay theory of generalised Tate spectra to exhibit an equi*
*valence LG '
F (LG; LK(n)S0). After comparing some definitions and feeding this into our mac*
*hinery, we find
that LG has a natural structure as a Frobenius object in the K(n)local stable *
*category, whenever
G is a finite groupoid. As part of the construction we define K(n)local transf*
*er maps for arbitrary
homomorphisms of finite groups, or functors of finite groupoids; these reduce t*
*o classical transfers
when the homomorphisms or functors are injective or faithful. In Section 9 we *
*deduce various
consequences for the generalised cohomology of BG; in the case where G is a fin*
*ite Abelian group,
we can be quite explicit. In Section 10, we deduce some further consequences i*
*n terms of the
HopkinsKuhnRavenel generalised character theory [14], which gives a complete *
*description of
Q E0BG for suitable cohomology theories E. Finally, in Section 11 we alert the*
* reader to some
possible pitfalls that can arise from overoptimism about the analogy with class*
*ical representation
theory.
2.Inner products
Let C be an additive compact closed category, in other words an additive clos*
*ed symmetric
monoidal category in which every object is dualisable. We write X ^Y for the sy*
*mmetric monoidal
product, and S for the unit object. We also write F (Y; Z) for the function ob*
*jects, so that
C(X; F (Y; Z)) ' C(X ^Y; Z). Finally, we write DX = F (X; S), so that D2X = X a*
*nd F (X; Y ) =
DX ^ Y .
___________
Date: November 17, 2000.
1
2 N. P. STRICKLAND
Definition 2.1.An inner product on an object X 2 C is a map b: X ^ X !S such t*
*hat
1.b is symmetric in the sense that b O o = b, where o :X ^ X !X ^ X is the t*
*wist map; and
2.the adjoint map b# :X !DX is an isomorphism.
Example 2.2.We could take C to be the category of finitely generated projective*
* modules over a
commutative ring R, with the usual closed symmetric monoidal structure so that *
*M ^N = M R N
and F (M; N) = Hom R(M; N) and DM = M* = Hom R(M; R). An inner product on M is *
*then a
symmetric Rbilinear pairing M x M !R that induces an isomorphism M ' M*. If R*
* is a field
then this just says that the pairing is nondegenerate. Note that we have no pos*
*itivity condition.
Remark 2.3. We see from [21, Theorem III.1.6] that a symmetric map b: X ^ X ! *
*S is an
inner product iff it is a duality of X with itself in the sense discussed there*
*, iff there is a map
c: S !X ^ X such that the following diagrams commute:
X _____wX1^^Xc^'X X ^ X ^ X u____X_c^1
' ' [
1 '') b^1 1^b [ [1
u u[[^
X X
Moreover, if b is an inner product then there is a unique map c as above, and i*
*t is symmetric; in
fact it is also the unique symmetric map making the left hand diagram commute.
Remark 2.4. The commutativity of the above diagrams can be expressed in terms o*
*f Penrose
diagrams [18] as follows:
________s c s_______
b = ________ = @@
c s_______ 1 ________s@b
Similarly, the symmetry of b and c gives the following equations:
@ @@ @ @@
@ s @ s @@s and s s @ s
o @ b = b @c = @c @o
@@ @@ @@ @
Definition 2.5.If X and Y are equipped with inner products and f :X ! Y then w*
*e write
ft:Y ! X for the unique map making the following diagram commute:
ft
Y _______Xw
 
#  #
bY bX
 
u u
DY _____wDX:Df
This can also be characterised by the equation
bY O (f ^ 1) = bX O (1 ^ ft): X ^ Y ! S
or equivalently, the following equality between Penrose diagrams.
X PP f X P P
P sP P P
P PPsi = PPPis
i i i i isi i
ii bY ii i bX
Y Y ft
It is clear that ftt= f and that 1t= 1 and (gf)t= ftgt whenever this makes sens*
*e. We call ft
the transpose of f.
K(N)LOCAL DUALITY FOR FINITE GROUPS AND GROUPOIDS *
* 3
Remark 2.6. Suppose that X and Y have inner products bX and bY . We then define
bX^Y = (X ^ Y ^ X ^ Y 1^o^1!X ^ X ^ Y ^ Y bX^bY!S):
It is easy to check that this is an inner product on X ^ Y . Similarly, if C is*
* an additive category
(with direct sums written as X _ Y ) and ^ is bilinear then there is an obvious*
* way to put an inner
product on X _ Y . By abuse of language, we call these inner products bX ^ bY a*
*nd bX _ bY . If we
use these inner products, we find that (f ^ g)t= ft^ gt and (f _ g)t= ft_ gt.
3. Frobenius objects
Definition 3.1.Let C be a symmetric monoidal category. A Frobenius object in C *
*is an object
A 2 C equipped with maps S j!A  A ^ A and S fflA !A ^ A such that
(a)(A; j; ) is a commutative and associative ring object.
(b)(A; ffl; ) is a commutative and associative coring object.
(c)(The "interchange axiom") The following diagram commutes:
A ^ A_________Aw
 
^1 
 
u u
A ^ A ^ A_____Aw^1A:^
The point for us will be that for any finite groupoid G, the spectrum LG := L*
*K(n)1 BG+ has
a natural structure as a Frobenius object in the K(n)local stable category (Th*
*eorem 8.7).
The last axiom can be restated as the following equality of Penrose diagrams:
__s_________
@ @
@@s___s = @
@ @
@@ __________s_@@
Remark 3.2. If (A; j; ffl; ; ) is a Frobenius object in C then (A; ffl; j; ; *
*) is evidentally a Frobe
nius object in Cop.
Convention 3.3.For the rest of this paper, we use the following conventions for*
* Penrose dia
grams. Unless otherwise specified, each diagram will involve only a single obj*
*ect A, for which
some subset of the maps , , j, ffl will have been defined. We also automatical*
*ly have a twist
map o :A ^ A !A ^ A.
o Any unlabelled node with two lines in and one line out is implicitly labell*
*ed with .
o A node with one line in and no lines out is implicitly labelled .
o A node with no lines in and one line out is implicitly labelled j.
o A node with one line in and no lines out is implicitly labelled ffl.
o A node with two lines in and two lines out is implicitly labelled o.
Another interesting point of view is that Frobenius objects are equivalent to*
* topological quan
tum field theories (TQFT's). In more detail, let S be the 1 + 1dimensional cob*
*ordism category,
whose objects are closed 1manifolds and whose morphisms are cobordisms. Some c*
*are is needed
to set the details up properly: a good account is [1], although apparently the *
*results involved
were "folk theorems" long before this. The category S has a symmetric monoidal *
*structure given
by disjoint unions. The circle S1 is a Frobenius object in S: the maps j and ff*
*l are the disc D2
regarded as a morphism ; ! S1 and S1 !; respectively, and the maps and are*
* the "pair
of pants" regarded as a morphism S1 q S1 ! S1 and S1 ! S1 q S1 respectively. *
* It follows
easily from [1, Proposition 12] that this is a universal example of a symmetric*
* monoidal category
equipped with a Frobenius object. For further analysis of the category S, see [*
*7, 31].
4 N. P. STRICKLAND
Remark 3.4. Using the Frobenius structure on S1, a Penrose diagram as in Conven*
*tion 3.3 gives
rise to a morphism in S. This has the following appealing geometric interpreta*
*tion. We first
perform the replacement
@ @
@ @
@ s ________@@ @
@  @
@ @
@@ @@
(It makes no real difference whether we introduce an under crossing or an over *
*crossing.) This
converts the Penrose diagram to a graph embedded in [0; 1] x R2. The boundary *
*of a regular
neighbourhood of this graph is a surface which we can think of as a cobordism *
*between \
({0} x R2) and \ ({1} x R2) and thus as a morphism in S. For example, the Penr*
*ose diagram
A
A
A
A
@@ AA @ s
@ s_s_sA @@s___s
@@s @
@ @@
s_______s
becomes the following cobordism:
Definition 3.5.Let C be a compact closed category, and let A be an object of C *
*equipped with
a commutative and associative product : A ^ A !A. (We do not assume that there*
* is a unit.)
A Frobenius form on A is a map ffl: A !S such that the map b = ffl is an inner*
* product.
K(N)LOCAL DUALITY FOR FINITE GROUPS AND GROUPOIDS *
* 5
Example 3.6.The most familiar example in topology is that if M is a closed mani*
*fold with
fundamental class [M] 2 H*(M; F2) then the equation ffl(u) = is a Frob*
*enius form on
H*(M; F2) (regarded as an ungraded module over F2). This can of course be gener*
*alised to other
coefficients at the price of a few words about orientations and gradings. For a*
* geometrised version
of this, see Section 5.
Example 3.7.Another elementary example is to let k be a field and G a finite Ab*
*elian group.
We can then define a map ffl: k[G] !k sending [1] to 1 and [g] to 0 for g 6= 1*
*. This is easily seen
to be a Frobenius form.
Example 3.8.Let C be the category of finitely generated free Abelian groups. Le*
*t G be a finite
group, let R = R(G) be its complex representation ring, and define ffl: R !Z b*
*y ffl[W ] = dimCW G.
It is easy to see that this is a Frobenius form, and that the associated inner *
*product is ([U]; [W ]) =
dimC(U W )G as considered in the introduction. To generalise this to finite gr*
*oupoids, let V be
the category of finite dimensional complex vector spaces. A representation of G*
* means a functor
G ! V . The set R+(G) = ss0[G; V ] of isomorphism classes of representations *
*has a natural
structure as a semiring, and we let R(G) denote its group completion. If W is a*
* representation
then we write W G:= lim W 2 V and t[W ] = dimCW Gas before. One can easily ded*
*uce from
G
the classical case that this is a Frobenius form on R(G).
Lemma 3.9. If (A; j; ffl; ; ) is a Frobenius object in a compact closed catego*
*ry C, then ffl is a
Frobenius form.
Proof.Put b = ffl: A ^ A !S; we need to show that this is an inner product. Pu*
*t c = j :S !
A ^ A; it will suffice to check the identities in Remark 2.3. The symmetry cond*
*itions are clear, so
we just need the two compatibility conditions for b and c. One of them is prove*
*d as follows:
cs_______ s_s_______ s
@ = @ = @@s___s = ______
______@@s_b ________s_s@@ @@s 1
The first equation is just the definition of b and c, the second is the interch*
*ange axiom, and the
third uses the (co)unit properties of j and ffl. The other compatibility condit*
*ion_follows because b
and c are symmetric. *
*__
We now prove a converse to the above result.
Proposition 3.10.Let C be a compact closed category, and let A be an object equ*
*ipped with a
commutative and associative product and a Frobenius form ffl. Then there are u*
*nique maps j;
making (A; j; ffl; ; ) into a Frobenius object.
Proof.By hypothesis b = ffl is an inner product on A, and trivially the canonic*
*al isomorphism
S ^ S = S is an inner product on S. We can thus define j := fflt:S !A, so j is*
* the unique map
such that b O (1 ^ j) = ffl, or in other words the unique map giving the follow*
*ing equality of Penrose
diagrams:
______s___s _____s_
=
j s
We claim that j is a unit for , or in other words that we have the following eq*
*uality:
______s___ ______
= 1
s
To prove this, we observe that for any two maps f; g :B !A we have f = g if an*
*d only if
b O (1 ^ f) = b O (1 ^ g): A ^ B !S:
6 N. P. STRICKLAND
In view of this, the claim follows from the following diagram, in which the fir*
*st equality comes
from the associativity of and the second from the defining property of j.
____________s_s ____s_____s_s ____s_s
______s_ = =
s s
We next equip A^A with the inner product b^b and define := t. As (A; j; ) is *
*a commutative
and associative monoid object, it is easy to deduce that (A; ffl; ) = (A; jt; *
*t) is a commutative
and associative comonoid object. Thus, to prove that A is a Frobenius object, w*
*e need only check
the interchange axiom.
It follows directly from the definition that is the unique map giving the f*
*ollowing equality:
________s___s __
@
____s = ___@ss@s_
@
______@@s___s
Using the perfectness of b, we see that two maps f; g :B !A ^ A are equal if a*
*nd only if we have
(b ^ b)(1 ^ f ^ 1) = (b ^ b)(1 ^ g ^ 1): A ^ B ^ A !A ^ A ^ A ^ A:
In view of this, the interchange axiom is equivalent to the following equation:
________s_s _________s_s_
__s_____
@@s_s = @
@ ______s_@@
______@@s_s _______@@s_s_@@
This equation can be proved as follows:
__________s_s ____s_____s ________s_s
__s_____ __s @_s
@ = @ = @ = @@s_s
______s_@@ @ @ @
________@@s_s @@s__@@_s_s __s__@@s_s_@@ ______@@s_s
The first equality uses associativity of , the second uses the defining propert*
*y of , and the third
uses the same two ideas backwards.
We still need to check that j and are the unique maps giving a Frobenius st*
*ructure. For j
this is easy, because the unit for a commutative and associative product is alw*
*ays unique. For
, suppose that OE: A ! A ^ A is another map giving a Frobenius structure. We *
*then have the
following equations:
________s_s s __
@@s_s @ s @
____s = O@E = @ = ___@s@s_s
O@E @ @
______@@s_s _______@@_s_s _____@@s_s_
The first equality is the interchange axiom, the second is the counit property *
*of ffl, and the third_is
the associativity of . This shows that OE has the defining property of , so OE*
* = as required. __
Scholium 3.11.Let (A; j; ffl; ; ) be a Frobenius object. Give A the inner prod*
*uct b = ffl and
give A ^ A the inner product b ^ b. Then j :S !A is adjoint to ffl: A !S and *
* :A !A ^ A is
adjoint to : A ^ A !A.
Proof.This is implicit in the proof of the proposition. *
* ___
Scholium 3.12.The map ffl: A !S is the unique one such that (ffl ^ 1) j = j :S*
* !A.
K(N)LOCAL DUALITY FOR FINITE GROUPS AND GROUPOIDS *
* 7
Proof.We saw in the proof of the proposition that j is the unique map giving th*
*e following equality
of Penrose diagrams. ffl ffl
______s___s _____s_
=
j s
The claim follows by working in the opposite category and using Remark 3.2. *
* ___
Remark 3.13. Let A and B be Frobenius objects, and suppose that f :A ! B is a *
*ring map
with respect to j and . We can use this to make B into an Amodule. We claim th*
*at ft:B !A
is automatically a map of Amodule objects. We will give the proof in the cate*
*gory of vector
spaces over a field; it can easily be made diagrammatic. The claim is that ft(f*
*(a)b) = a ft(b). It
suffices to prove that bA(a0; ft(f(a)b)) = bA(a0; a ft(b)). The left hand side *
*is bB(f(a0); f(a)b) =
fflB(f(a)f(a0)b) = fflB(f(aa0)b). The right hand side is
fflA(aa0ft(b)) = bA(aa0; ft(b)) = bB(f(aa0); b) = fflB(f(aa0)b);
as required.
4. The trace form
We now construct an interesting map A !S which may or may not be a Frobenius*
* form.
Definition 4.1.Let A be an arbitrary commutative ring object in an additive com*
*pact closed
category C. We can then transpose the multiplication map : A ^ A !A to get ae:*
* A !DA ^ A
and compose with the evaluation map DA ^ A = F (A; S) ^ A !S to get a map :A *
*!S. This
is called the trace form.
Remark 4.2. If K is a ring and C is the category of finitely generated free mod*
*ules over K then
(a) is just the trace of the multiplicationbya map. Now suppose that K is a p*
*erfect field. One
can check that is a Frobenius form if and only if A has no nilpotents, if and *
*only if A is a finite
product of finite extension fields of K (this is wellknown and can mostly be e*
*xtracted from [11,
Section I.1], for example).
Proposition 4.3.Let A be a Frobenius object in an additive compact closed categ*
*ory C. Then
the trace form is given by = b = ffl :A ! S. Moreover, if ff = c = j :S *
*! A then
= b(ff ^ 1).
Proof.The adjunction between the functors A ^ () and F (A; ) = DA ^ () is gi*
*ven by two
maps unit:S !DA ^ A and eval:DA ^ A !S. It follows from the basic theory of p*
*airings and
duality [21, Chapter III] that the following diagrams commute:
A ^ A[ S [_____wAc^ A
 [ [ 
#  [b [  #
b ^1 unit b ^1
 [[] [[]
u u
DA ^ A _____Sweval DA ^ A
It follows that the following diagram commutes:
1^ b
A_____wAc^^A1^ A_______Aw^ A__________wS
  
    
 #  #  
b^1^1 b ^1  
 u u 
A _______wDAu^nAi^tA^1____DAw^1A^_________wSeval
On the bottom row, the composite of the first two maps is ae so the whole compo*
*site is just .
Thus, = b(1 ^ )(c ^ 1). To complete the proof, it is easiest to think in terms*
* of TQFT's. Let
M be a torus with a small open disc removed. We leave it to the reader to check*
* that ff = c is
represented by M, considered as a cobordism from ; to S1. Moreover, the maps b(*
*1^)(c^1), b
8 N. P. STRICKLAND
and b(ff ^ 1) are all represented by M considered as a cobordism from S1 to ;. *
*The_proposition
follows. *
*__
5.Manifolds
We next show how to use manifolds to construct Frobenius objects in suitable *
*categories of
module spectra. This is of course just a reformulation of AtiyahPoincare dual*
*ity, but it is a
nice illustration of the theory of Frobenius objects. It is also strikingly for*
*mally similar to the
constructions in the K(n)local stable category which we discuss later.
Let M be the category of evendimensional closed manifolds M equipped with a *
*complex
structure on the stable normal bundle, or equivalently a complex orientation on*
* the map from
M to the onepoint manifold; we refer to Quillen's work [25] for a careful disc*
*ussion of what this
means.
Next, let MP denoteWthe Thom spectrum of the tautological virtual complex bun*
*dle over
Z x BU, so that MP = n2Z2nMU and 2MP ' MP . More generally, if V is a complex
bundle over a space X then there is a canonical Thom class uV :XV ! MP which c*
*ombines with
the usual diagonal map XV ! X+ ^XV to give a canonical equivalence MP ^XV ' MP*
* ^1 X+.
With a little care, this also goes through for virtual bundles.
The spectrum MP can be constructed as an E1 ring spectrum, and thus as a stri*
*ctly commu
tative ring spectrum (or "Salgebra") in the EKMM category [9]. We can thus def*
*ine a category of
MP modules in the strict sense, and the associated derived category D = DMP . *
*(There are also
other approaches to our results using less technology.) The category D is a uni*
*tal algebraic stable
homotopy category in the sense of [16]; in particular it is a closed symmetric *
*monoidal category.
We write F for the thick subcategory of D generated by MP , which is the same a*
*s the category
of small or strongly dualisable objects [16, Theorem 2.1.3(d)]. This is clearly*
* a compact closed
category.
Define T :M !F by T (M) = MP ^ 1 M+. This is clearly a covariant functor tha*
*t converts
products to smash products and disjoint unions to wedges.
Now suppose we have a smooth map f :M ! N of closed manifolds. Let j :M ! R*
*k be a
smooth map such that (j; f): M !Rkx N is a closed embedding, with normal bundl*
*e (j;f)say.
This is stably equivalent to k + M  f*N . The PontrjaginThom construction app*
*lied*to the
embedding (j; f) gives a map kN+ !M(j;f)and thus a stable map f!:1 N+ !MM f *
*N .
Now suppose that M and N have specified complex orientations, so they are obj*
*ects of M.
Then the virtual bundle*f = M f*N has a canonical complex structure, so there *
*is a canonical
equivalence MP ^MM f N ' T (M). Thus, by smashing f!with MP we get a map Uf :*
*T (N) !
T (M). One can check that this construction gives a contravariant functor U :M *
*! D, which
again converts products to smash products and disjoint unions to wedges. If f i*
*s a diffeomorphism,
one checks easily that U(f) = T (f)1. We also have the following "Mackey prope*
*rty". Suppose
we have a commutative square in M:
f
K ______Lw
 
g  h
 
u u
M _____N:wk
Suppose also that the square is a pullback and the maps h and k are transverse *
*to each other, so
that when x 2 K with hf(x) = kg(x) = y say, the map of tangent spaces
(Dh; Dk): Tf(x)L Tg(x)M !TyN
is surjective. We then have U(h)T (k) = T (f)U(g), as one sees directly from th*
*e geometry.
For any manifold M 2 M, there is a unique map ffl: M !1, where 1 is the one*
*point manifold.
We also have a diagonal map :M !M x M. We allow ourselves to write ffl and *
* for T (ffl) and
T ( ), and we also write j = U(ffl) and = U( ).
K(N)LOCAL DUALITY FOR FINITE GROUPS AND GROUPOIDS *
* 9
Proposition 5.1.The above maps make T (M) into a Frobenius object in M. If we *
*use the
resulting inner product, then for any map f :M !N in M we have T (f)t= U(f).
Proof.If we make M into a symmetric monoidal category using the cartesian produ*
*ct, it is clear
that ffl and make M into a comonoid object. As T and U are monoidal functors,*
* the first covariant
and the second contravariant, we see that 1 M+ = T (M) = U(M) is a monoid objec*
*t under
and j, and a comonoid object under and ffl. For the interchange axiom = (1 *
*^ )( ^ 1), we
note that the following diagram is a transverse pullback and apply the Mackey p*
*roperty.
M _________wM x M
 
 
 1x
 
u u
M x M _____wM xxM1x M:
Similarly, to prove that T (f)t= U(f), we note that the following square is a t*
*ransverse pullback:
f
M _________Nw
 
(1;f) 
 
u u
M x N _____NwxfN:x1
We then apply the Mackey property, noting that (1; f) = (1 x f) M ; this gives *
*the following
commutative diagram:
T(f)^1
T (M) ^ T (N)_______________wT (N) ^ T (N)
 
1^U(f) N
 
u u
T (M) ^ T (M)____wTM(M)________wTT(N):(f)
We then compose with fflN , noting that fflN T (f) = fflM and fflM M = bM an*
*d fflN N = bN_._We
conclude that bN (T (f) ^ 1) = bM (1 ^ U(f)), so T (f)t= U(f) as claimed. *
* __
6.Groupoids
__
Let G denote the category of groupoids and functors between them, and let G b*
*e the quotient
category in which two functors are identified if there is a natural isomorphism*
* between them. We
say that a groupoid G is finite if there are only finitely many isomorphism cla*
*sses of objects,_and
G(a; b) is finite for any a; b 2 G. We write Gffor the category of finite group*
*oids, and Gf for the
obvious quotient category. __
We next exhibit an equvalence between Gand a certain homotopy category of spa*
*ces. As usual in
homotopy theory, it will be convenient to work with compactly generated weakly *
*Hausdorff spaces
(so we have Cartesian closure). Let B be the_category of such spaces X for whic*
*h ssk(X; x) = 0
for all k > 1 and all x 2 X. We also write B for_the associated homotopy catego*
*ry (in which weak
equivalences are inverted), and we let Bf and Bf be the subcategories whose obj*
*ects are those
X 2 B for which ss0X is finite and ss1(X; x) is finite for any basepoint x.
Milgram's classifying space construction gives a functor B :G ! B. One can *
*also define a
functor 1:B !G: the set of objects of 1(X) is X, and the set of morphisms from*
* x to y is the
set of paths from x to y modulo homotopy relative to the endpoints. Both G and *
*B have finite
products and coproducts, and both our functors preserve them. __ __ __ *
*__
It is easy to check that these constructions give equivalences G ' B and Gf' *
*Bf.
Any (finite) group G can be regarded as a (finite) groupoid with one_object. *
* If G and H
are groups then G(G; H) is the set of homomorphisms from G to H, and G(G; H) is*
* the set of
conjugacy classes of such homomorphisms.
10 N. P. STRICKLAND
Conversely, if G is a finite groupoid then we can choose a family {ai}i2Icon*
*taining precisely
one object`of G from_each isomorphism class and then let Hi be the group G(ai;*
* ai). We find
that G ' iHiin G. Thus, all our questions about groupoids can be reduced to q*
*uestions about
groups by some unnatural choices. Our next lemma sharpens this slightly.
Definition 6.1.A groupoid G is discrete if all its maps are identity maps, and*
* indiscrete if there
is precisely one map from a to a0for all a; a02 G.
Remark 6.2. The category of discrete groupoids is equivalent to that of sets, *
*as is the category
of indiscrete groupoids. The classifying space of a discrete groupoid is disc*
*rete, and that of a
nonempty indiscrete groupoid is contractible.
Lemma 6.3. Any nonempty connected groupoid is isomorphic to A x H for some`non*
*empty in
discrete groupoid A and some group H. Thus, any groupoid is isomorphic to IA*
*ix Hifor some
family of nonempty indiscrete groupoids Aiand groups Hi.
Proof.Let G be a connected groupoid. Choose an object x 2 G and let H be the g*
*roup G(x; x).
Let A be the indiscrete groupoid with obj(A) = obj(G), and for each a 2 A choo*
*se a map
ka:x !a in G. Put B = A x H, so obj(B) = obj(G) and B(a; a0) = H for all a; a*
*0. Composition
is given by multiplication in H. Define u: B !G by u(a) = a on objects, and
1 h k 0
ua;a0(h) = (a ka!x !x a!a0)
on morphisms. This is easily seen to be functorial and to be an isomorphism. *
* __
The generalisation to the disconnected case is immediate. *
* __
6.1. Model category structure. We now complete an exercise assigned by Anderso*
*n [2] to his
readers, by verifying that his definitions (reproduced below) do indeed make t*
*he category G into a
closed model category in the sense of Quillen [23] (see also [8] for an exposi*
*tion and survey of more
recent literature). As well as being useful for our applications, this seems p*
*edagogically valuable,
as the verification of the axioms is simpler than in most other examples. The *
*homotopy theory
of the category of all small categories has been extensively studied (see [22]*
* for example), but the
case of groupoids is easier so it makes sense to treat it independently.
Definition 6.4.We say that a functor u: G !H of groupoids is
(a)a weak equivalence if it is full, faithful and essentially surjective (in *
*other words, an equiva
lence of categories);
(b)a cofibration if it is injective on objects; and
(c)a fibration if for all a 2 G, b 2 H and h: u(a) ! b there exists g :a ! *
*a0in G such that
u(a0) = b and u(g) = h.
As usual, an acyclic fibration means a fibration that is also an equivalence, *
*and similarly for acyclic
cofibrations.
Remark 6.5. Let u: G !H be a homomorphism of groups. Then u is automatically *
*a cofibra
tion of groupoids, and it is a fibration iff it is surjective. It is an equiva*
*lence of groupoids iff it is
an isomorphism.
Remark 6.6. Let v :X ! Y be a map of sets. If we regard X and Y as discrete *
*categories
then v is automatically a fibration. It is a cofibration iff it is injective, *
*and an equivalence iff it
is bijective. If we regard X and Y as indiscrete categories then v is automati*
*cally an equivalence
(unless ; = X 6= Y ). It is a cofibration iff it is injective, and a fibration*
* iff it is surjective.
Theorem 6.7.The above definitions make G into a closed model category.
Proof.We need to verify the following axioms, numbered as in [8]:
MC1: G has finite limits and colimits.
MC2: If we have functors G u!H v!K and two of u, v and vu are weak equivalenc*
*es then so is
the third.
K(N)LOCAL DUALITY FOR FINITE GROUPS AND GROUPOIDS *
* 11
MC3: Every retract of a weak equivalence is a weak equivalence, and similarly f*
*or fibrations and
cofibrations.
MC4: Cofibrations have the left lifting property for acyclic fibrations, and ac*
*yclic cofibrations have
the left lifting property for all fibrations.
MC5: Any functor u has factorisations u = pi = qj where i and j are cofibration*
*s, p and q are
fibrations, and i and q are equivalences.
MC1: This from the fact that G is the category of models for a leftexact sk*
*etch [3, Section
4.4]. More concretely, for limits we just have obj(lim Gi) = lim obj(Gi) and*
* mor(lim Gi) =
` i ` i *
* ` i
limmor(Gi). Similarly, for coproducts we have obj( iGi) = iobj(Gi) and mo*
*r( iGi) =
` i
imor(Gi). Coequalisers are more complicated and best handled by the adjoint f*
*unctor theorem.
MC2: This is easy.
MC3: Let v be an equivalence and let u: G !H be a retract of v. Then ss0(u)*
* is a retract of
ss0(v), so ss0(u) is a bijection and so u is essentially surjective. If a; b 2*
* G then ua;b:G(a; b) !
H(ua; ub) is a retract of a map of the form vc;dand thus is a bijection, so u *
*is full and faithful.
Thus u is an equivalence as required.
It is clear that a retract of a cofibration is a cofibration.
For fibrations, let 1 be the terminal groupoid. Let I be the groupoid with o*
*bjects {0; 1} and
two nonidentity morphisms u: 0 ! 1 and u1: 1 ! 0. Let i: 1 ! I be the inc*
*lusion of {0}.
Then fibrations are precisely the maps with the right lifting property for i, *
*and it follows that a
retract of a fibration is a fibration.
MC4: Consider a commutative square as follows, in which i is a cofibration a*
*nd p is a fibration.
G _____wKu
 
i p
 
u u
H _____L:wv
Because p is a fibration, it is easy to see that the image of p is replete: if*
* d 2 L is isomorphic to
pc then d has the form pc0for some c02 K.
Suppose that p is an equivalence; we must construct a functor w: H ! K such*
* that pw = v
and wi = u. As p is essentially surjective and the image is replete, we see th*
*at obj(p) is surjective.
By assumption i is a cofibration so obj(i) is injective. Define a map w: obj(H*
*) ! obj(K) by
putting w(i(a)) = u(a) for a 2 obj(A) and choosing w(b) to be any preimage und*
*er p of v(b) if
b 62 image(i). Clearly pw = v and wi = u on objects. Given b; b02 H we define *
*wb;b0to be the
composite
vb;b0 0 0 p1wb;wb0 0
H(b; b0) !L(vb; vb ) = L(pwb; pwb ) ! K(wb; wb ):
One can check that this makes w a functor with pw = v. Also pwi = vi = pu on m*
*orphisms and
wi = u on objects and p is faithful; it follows that wi = u on morphisms, as r*
*equired.
Now remove the assumption that p is an equivalence, and suppose instead that*
* i is an equiva
lence. We must again define a functor w: H !K making everything commute. As i*
* is injective
on objects we can choose r :obj(H) !obj(G) with ri = 1. As ss0(i) is a biject*
*ion we find that
ss0(r) = ss0(i)1 so we can choose isomorphisms jb:b !ir(b) for all b 2 H. If*
* b = i(a) for some
(necessarily unique) object a, we have rb = a and irb = b, and we choose jb = *
*1b in this case.
There is a unique way to make r a functor H !G such that j is natural: explic*
*itly, the map rb;b0
is the composite
* i1rb;rb0
H(b; b0) j*j!H(irb; irb0) !G(rb; rb0):
Next, if b 2 image(i) we define wb = urb and ib = 1: wb ! urb. If b 62 image*
*(i) we instead
apply the fibration axiom for p to the map vjb:vb !virb = purb to get an obje*
*ct wb 2 K and
a morphism ib:wb !urb such that pwb = vb and pib = vjb. Note that these last *
*two equations
also hold in the case b 2 image(i). There is a unique way to make w into a fun*
*ctor such that
12 N. P. STRICKLAND
i :w ! ur is natural. Clearly wi = u as functors, and pw = v on objects. Given*
* h: b ! b0in
H we can apply p to the naturality square for i and then use the naturality of *
*j to deduce that
pwh = vh; thus pw = v on morphisms, as required.
MC5: Consider a functor u: G !H. Let K be the category whose objects are tri*
*ples (a; b; k),
with a 2 G and b 2 H and k :u(a) !b. The morphisms from (a; b; k) to (a0; b0; *
*k0) are the pairs
(g; h) where g :a !a0and h: b !b0and the following diagram commutes:
u(g)
u(a)_____wu(a0)
 
  0
k  k
 
u u
b _______wb0:h
We also consider the category L with the same objects as K, but with L(a; b; k;*
* a0; b0; k0) =
H(b; b0), so there is an evident functor v :L ! K. There is also a functor i: *
*G ! K given by
i(a) = (a; ua; 1ua) and a functor q :L !H given by q(a; b; k) = b; we put j = *
*vi and p = qv. It
is clear that u = qvi = qj = pi and that i and j are cofibrations and that i is*
* full and faithful.
If (a; b; k) 2 K then (1a; k): i(a) !(a; b; k) so i is essentially surjective *
*and thus an equivalence.
The functor q is clearly full and faithful, and its image is the repletion of t*
*he image of u.
We next claim that p and q are fibrations. Suppose that (a; b; k) 2 obj(K) a*
*nd h: b =
q(a; b; k) ! b0. Then (a; b0; hk) 2 obj(K) and (1a; h): (a; b; k) ! (a; b0; *
*hk) and q(1a; h) = h.
This shows that q is a fibration, and the same construction also shows that p i*
*s a fibration.
We now have a factorisation u = pi as required by axiom M3. If u is essentia*
*lly surjective
then the same is true of q and thus q is an equivalence and so the factorisatio*
*n u = qj is also as
required. If u is not essentially surjective then we let L0be the full subcateg*
*ory of H consisting of
objects not in the repletion of the image of u and let q0:L0! H be the inclusi*
*on. We then have
an acyclic fibration (q; q0): L q L0! H and a cofibration G j!L !L q L0whose*
*_composite_is u,
as required. *
*__
Proposition 6.8.The above model category structure is right proper (in other wo*
*rds, the pullback
of a weak equivalence along a fibration is a weak equivalence.)
Proof.Consider a pullback square as follows, in which v is a weak equivalence a*
*nd q is a fibration.
G _____wKu
 
p q
 
u u
H _____L:wv
Suppose that a; a02 G and put d = qu(a) = vp(a) and d0= qu(a0) = vp(a0). By the*
* construction
of pullbacks in G, we see that the following square is a pullback square of set*
*s:
ua;a0
G(a; a0)____wK(u(a); u(a0))
 
 
pa;a0 qu(a);u(a0)
 
u u
H(p(a); p(a0))___L(d;wd0):vp(a);p(a0)
As v is a weak equivalence, the map vp(a);p(a0)is a bijection, and it follows t*
*hat the same is true
of ua;a0. This means that u is full and faithful.
Next suppose we have c 2 K, so q(c) 2 L. As v is essentially surjective there*
* exists b 2 H
and l: q(c) !v(b) in L. As q is a fibration there is a map k :c !c0in K with *
*q(c0) = v(b) and
q(k) = l. By the pullback property there is a unique a 2 G with u(a) = c0and p(*
*a) = b. Thus __
u(a) ' c, proving that u is essentially surjective and thus an equivalence. *
* __
K(N)LOCAL DUALITY FOR FINITE GROUPS AND GROUPOIDS *
*13
6.2. Classifying spaces. Let N be the nerve functor from groupoids to simplicia*
*l sets, and put
BG = NG; this is called the classifying space of G. It is easy to see that N *
*converts groupoids
to Kan complexes and fibrations to Kan fibrations, and that it preserves coprod*
*ucts and finite
limits. The geometric realisation functor preserves coproducts (easy) and finit*
*e limits [10, Theorem
4.3.16] and it converts Kan fibrations to fibrations [24] (see also [10, Theore*
*m 4.5.25]). Thus, the
composite functor B :G !B preserves coproducts, finite limits and fibrations.
6.3. Homotopy pullbacks.
Definition 6.9.Suppose we have functors G u!H v K of groupoids. We define a n*
*ew groupoid
L whose objects are triples (a; c; h) with a 2 G and c 2 K and h: u(a) !v(c). *
*The morphisms
from (a; c; h) to (a0; c0; h0) are pairs (r; s) where r :a !a0and s: c !c0and*
* the following diagram
commutes:
u(r)
u(a)_____wu(a0)
 
 
h  h0
 
u u
v(c)_____wv(c0):v(s)
0 v0
We also define functors K u L ! G by u0(a; c; h) = a and v0(a; c; h) = c, an*
*d a natural
transformation_OE: uv0! vu0by OE(a;c;h)= h. This gives a square as follows, wh*
*ich commutes in
G:
0
L ______Gwv
 
u0 u
 
u u
K _____wH:v
*
* __
We call L the homotopy pullback of u and v. We say that an arbitrary commutativ*
*e square in G
is homotopycartesian if it is isomorphic to one of the above form.
Remark 6.10. We can also consider the actual pullback rather than the homotopy *
*pullback,
which can be identified with the full subcategory M L consisting of pairs (a; *
*c; 1) where u(a) =
v(c). One checks that the inclusion M !L is an equivalence if u or v is a fibr*
*ation.
Remark 6.11. Suppose that H is a group and u and v are inclusions of subgroups.*
* Then M is
the group G \ K. Let T H be a set containing one element of each double coset *
*in`G \ H=K;
we may as well assume that 1 2 T . We find that L is equivalent to the groupoid*
* TGt\ K, and
the term indexed by t = 1 is just M. It follows that the map M !L is an equiva*
*lence if and only
if H = GK. Note that this is only predicted by the previous remark when G = H o*
*r K = H.
Remark_6.12. By standard methods of abstract homotopy theory, we see that a squ*
*are S in
Gis homotopycartesian iff there_is a pullback square S0 in G whose maps are fi*
*brations, which
becomes isomorphic to S in G.
Remark 6.13. It is easy to see that if G, H and K are finite then so is their h*
*omotopy pullback.
Definition_6.14.Suppose we have functors u; v; s; t such that the following squ*
*are is commutative
in G.
F _____Gwt
 
s u
 
u u
K _____Hwv
14 N. P. STRICKLAND
Let L be the homotopy pullback of u and v, and let u0; v0be as above. Choose an*
* isomorphism
oe :ut !vs. We can then define a functor ^oe:F ! L by ^oe(d) = (t(d); s(d); o*
*ed); this has u0^oe= t
and v0^oe= s. If i :s !s0and :t0! t and oe0= v(i) O s O u() then it is easy *
*to see that oe ' boe0.
Lemma 6.15. A square as in the above definition is homotopy Cartesian if and on*
*ly if there
exists oe :ut !vs such that ^oe:F ! L is an equivalence.
__
Proof.If there exists such a map oe then the square is visibly equivalent in G *
*to a homotopy
pullback square, and thus is homotopy cartesian. For the converse, suppose that*
*_the square is
homotopy Cartesian. We can then find a diagram as follows which commutes in G, *
*such that the
outer square is a homotopy pullback, and the diagonal functors are equivalences.
v01
L1 ____________________wG1
[^ 
 [ ffi ff aeo
 [ ae 
 ae 
 F _____Gwt 
   
0   u
u1 s u  1
 u u 
 K _____Hwv[ 
 
 ae [ 
 aefl fi [] 
uaeAE u
K1 ____________________wH1v1
There is a "tautological" natural isomorphism OE1:u1v01!v1u01, and we write ae*
* = OE1ffi_:u1v01ffi !
v1u01ffi so that ffi = ^ae. As the top and lefthand regions of the diagram com*
*mute in G, we have
natural maps fft !v01ffi and u01ffi !fls, which we can use to form a natural *
*map
= (u1fft !u1v01ffi ae!v1u01ffi !v1fls):
Using the remark in the preceeding definition, we see that ^ ' ^ae= ffi :F ! L*
*1. As ffi is an
equivalence, we see that the same is true of ^. Next, we note that the functors*
* u1ffv0; v1flu0:L !L0
are joined by the natural map
o = (u1ffv0! fiuv0fi(OE)!fivu0! v1flu0);
where the first and third maps come from the commutativity of the righthand an*
*d bottom regions
of the diagram. This gives a functor ^o:L !L0; we leave it to the reader to ch*
*eck directly that
this is an equivalence.
Next, consider the composite
fiut !u1fft !v1fls !fivs:
As fi is full and faithful, this composite has the form fi(oe) for a unique nat*
*ural map oe :ut !vs,
which gives rise to ^oe:F ! L. One checks directly that ^o^oe= ^, and both ^oa*
*nd ^are_equivalences,_
so ^oeis an equivalence, as required. *
* __
6.4. Coverings and quasicoverings.
Definition 6.16.A functor u: G !H is a covering if for each a 2 G and each h: *
*u(a) !b in
H there is a unique pair (a0; g) such that a02 G and g :a !a0and u(a0) = b and*
* u(g) = h. More
generally, we say that u is a quasicovering if it can be factored as an equiva*
*lence followed by a
covering.
Remark 6.17. It is easy to check that pullbacks, products and composites of cov*
*erings are cov
erings.
Remark 6.18. A group homomorphism is only a covering if it is an isomorphism. W*
*e will see
later that it is a quasicovering iff it is injective.
K(N)LOCAL DUALITY FOR FINITE GROUPS AND GROUPOIDS *
*15
Definition 6.19.A functor u: G ! H reflects identities if whenever g :a ! a0a*
*nd u(g) = 1b
for some b, we have a = a0and g = 1a. Such a functor is easily seen to be faith*
*ful.
We leave the following easy lemma to the reader.
Lemma 6.20. A functor u: G !H is a covering iff it reflects identities and is *
*a fibration. ___
Proposition 6.21.If u: G ! H is a covering, then Bu: BG ! BH is a covering ma*
*p of
topological spaces.
Proof.Suppose for the moment that H is indiscrete and G is connected. Then for *
*a; a02 G we
have G(a; a0) 6= ; and u: G(a; a0) !H(ua; ua0) is injective but the codomain h*
*as only one element
so the same is true of G(a; a0). Thus u is full and faithful. It is also a fibr*
*ation and H is connected
so it is surjective on objects. If ua = ua0 then the unique map a ! a0 in G m*
*ust become an
identity map in H but u reflects identities so a = a0. We now see that u is an *
*isomorphism so Bu
is a homeomorphism and thus certainly a covering.
If H is indiscrete and G is disconnected, we can still show that Bu is a cove*
*ring by looking at
one component at a time.
Now suppose merely that H is connected. We can then split H as A x K, where A*
* is indiscrete
and K is a group, as in Lemma 6.3. Let K0 be the indiscrete category with objec*
*t set K, and
define q :K0! K by sending the unique morphism k !k0to k0k1 2 mor(K). One ch*
*ecks that
BK0= EK and that Bq :EK ! BK = EK=K is the usual covering map. Thus, H0= A x K0
is indiscrete and r = 1 x q :H0 !H is a covering with the property that Br is *
*also a covering.
Now form a pullback square as follows:
u0
G0 _____H0w
 
r0 r
 
u u
G _____wH:u
Note that u0is a covering. As H0is discrete we know that Bu0is a covering by th*
*e first paragraph.
Thus, the pullback of Bu along the surjective covering map Br is a covering, an*
*d it follows easily
that Bu is a covering. *
* __
Finally, if H is disconnected we just look at one component at a time. *
* __
Proposition 6.22.Fix a groupoid H. Then the category of coverings q :G !H is e*
*quivalent to
the category of functors X :H ! Sets, and thus (by [22, Section 1]) to the cat*
*egory of covering
spaces of BH.
Proof.This is a simple translation of Quillen's analysis of coverings of BG.
Suppose we start with a functor X :H !Sets. We then define a category G whos*
*e objects are
pairs (b; x) with b 2 H and x 2 Xb; the morphisms (b; x) !(b0; x0) are the map*
*s h: b !b0in H
such that Xh: Xb !Xb0sends x to x0. There is an evident forgetful functor q :G*
* !H sending
(b; x) to b; one checks that this is a covering.
Conversely, suppose we start with a covering q :G ! H. For each b 2 H, we d*
*efine Xb =
q1{b} obj(G). Given a morphism h: b !b0in H and an element a 2 Xb, the defin*
*ition of a
covering gives a unique morphism g :a !a0in G with q(g) = h; we define a map X*
*h:Xb! Xb0
by Xb(a) = a0. *
* __
We leave it to the reader to check that these constructions give the claimed *
*equivalence. __
We next let C be the class of all coverings, and let E be the class of functo*
*rs that are full and
essentially surjective.
Proposition 6.23.The pair (C; E) is a factorisation system in G; in other words
(a)Both C and E contain all identity functors and are closed under composition*
* by isomorphisms
on either side.
(b)Every functor u: G !H can be factored as u = pr with p 2 C and r 2 E.
16 N. P. STRICKLAND
(c)Every functor in E has the unique left lifting property relative to every f*
*unctor in C. In
other words, given functors u, w, r 2 E and p 2 C making the diagram below *
*commute,
there is a unique functor v such that pv = w and vr = u.
L ______Gwu
 oeo
r  v p
 oe 
uoe u
K _____wH:w
See [3, Exercises 5.5] (for example) for generalities about factorisation sys*
*tems.
Proof.(a): This is clear.
(b): Let u: G ! H be a functor. We define a new groupoid K as follows. The o*
*bjects are
equivalence classes of triples (a; b; h), where a 2 G and b 2 H and h: u(a) !b*
*; the equivalence
relation identifies (a; b; h) with (a0; b0; h0) if and only if b = b0and there *
*is a map g :a !a0such
that h = h0Ou(g). The maps from [a; b; h] to [a0; b0; h0] are the maps k :b !b*
*0in H such that there
exists a map j :a !a0in G with k Oh = h0Ou(j). Equivalently, k gives a map [a;*
* b; h] ![a0; b0; h0]
if and only if [a0; b0; h0] = [a; b0; kh].
There is an evident functor r :G !K defined by r(a) = [a; u(a); 1u(a)]. Give*
*n c = [a; b; h] 2 K
we find that h can be thought of as a map r(a) !b in K, so r is essentially su*
*rjective. Moreover,
we find that K(r(a); r(a0)) is just the image of G(a; a0) in H(u(a); u(a0)), an*
*d thus that r is full.
Thus we have r 2 E.
There is also an evident functor p: K !H defined by p[a; b; h] = b. It is ea*
*sy to check that p
is a covering and u = pr as required. In terms of Proposition 6.22, the coverin*
*g p corresponds to
the functor X :H !Setsdefined by Xb= ss0(u # b).
(c): Suppose we have a square as in the statement of the proposition. We firs*
*t define a map
v :obj(K) !obj(G) as follows. Suppose that c 2 obj(K). As r is essentially sur*
*jective, we can
choose d 2 obj(L) and k :r(d) ! c in K. We apply w to get w(k): pu(d) = wr(d)*
* ! w(c).
As p is a covering, there is a unique pair (a; g) with a 2 obj(G) and g :u(c) *
*! a such that
p(a) = w(c) and p(g) = w(k). We would like to define v(c) = a. To check that th*
*is is welldefined,
consider another d02 obj(L) and another k0:r(d0) !c, giving rise to a unique p*
*air (a0; g0). As
r is full there exists l: d0! d such that k1k0 = r(l) and one checks that (a;*
* g O u(l)) has the
defining property of (a0; g0). Thus a = a0as required. This means that we have *
*a welldefined map
v :obj(K) !obj(G) with pv = w on objects. It is easy to check that vr = u on o*
*bjects as well.
Now suppose we have a map m: c !c0in K. We can choose maps k :r(d) !c and k*
*0:r(d0) !
c0with d; d02 L. By the definition of v on objects we have maps g :u(d) !v(c) *
*and g0:u(d0) !
v(c0) such that p(g) = w(k) and p(g0) = w(k0). As r is full we can choose n: d*
* ! d0 such
that r(n) = (k0)1mk. One then checks that the map g00= g0O u(n) O g1: v(c) *
*! v(c0) has
p(g00) = w(m). As p is faithful, there is at most one map v(c) !v(c0) with thi*
*s property, so g00
is independent of the choices made. We can thus define v on morphisms by v(m) =*
* g00, so that
pv = w. Using the faithfulness of p, we check easily that v is a functor and th*
*at vr = u. Thus v
fills in the diagram as required.
Finally suppose that v0:K ! G is another functor making the diagram commute.*
* We must
check that v0 = v. As p is faithful it is enough to check this on objects. Gi*
*ven c 2 obj(K)
we choose k :r(d) ! c as before and write a = v0(c) and g = v0(k): u(d) ! a. *
*We then have __
p(g) = pv0(k) = w(k), so the definition of v gives v(c) = a = v0(c) as required*
*. __
Corollary 6.24. (i)The factorisation in (b) is unique up to isomorphism.
(ii)C \ E is precisely the class of isomorphisms in G.
(iii)C and E are closed under compositions and retracts.
(iv)C is closed under pullbacks, and E is closed under pushouts.
Proof.See [3, Exercises 5.5]. Of course, in our case, many of these things are *
*immediate_from the
definitions. *
*__
Proposition 6.25.A functor u: G !H is a quasicovering if and only if it is fa*
*ithful.
K(N)LOCAL DUALITY FOR FINITE GROUPS AND GROUPOIDS *
*17
Proof.As equivalences and coverings are faithful, we see that quasicoverings a*
*re faithful.
For the converse, let u: G !H be faithful. We can factor u as pr where p is *
*a covering and
r is full and essentially surjective, as in Proposition 6.23. As u = pr is fait*
*hful we see_that r is
faithful and thus an equivalence, as required. *
* __
Lemma 6.26. Suppose we have functors L p!K v!H such that p is a covering and *
*v is an
equivalence. Then there is a pullback square as follows, in which q is a cover*
*ing and u is an
equivalence.
L ______Gwu
 
p  q
 
u u
K _____wH:v
Proof.We can factor vp as qu with q a covering and u full and essentially surje*
*ctive. Now consider
the following diagram:
L _____wMw[_____Gw"v
[  
p []q" q
u u
K _____wH:v
The square is defined to be the pullback of v and q, and w is the unique functo*
*r such that "qw = p
and "vw = u. By Proposition 6.8 we know that "vis an equivalence. It will thus *
*be enough to show
that w is an isomorphism in G.
As u = "vw is full and essentially surjective, and "vis an equivalence, we se*
*e that w is full and
essentially surjective.
We next show that w is surjective on objects. Suppose e 2 M, and put a = "v(*
*e) 2 G and
c = "q(e) 2 K so that q(a) = v(c) = b say. As u is essentially surjective, we c*
*an choose d 2 L and
g :u(d) !a in G. Thus q(g): vp(d) = qu(d) !q(a) = v(c) in H. As v is an equiv*
*alence, there is a
unique k :p(d) !c such that v(k) = q(g). As p is a covering, there is a unique*
* d02 L and l: d !d0
such that p(d0) = c and p(l) = k. Thus u(l)g1: a !u(d0) satisfies q(u(l)g1) *
*= vp(l)v(k)1 = 1.
As q is a covering, it reflects identity maps, so u(l)g1 = 1a and a = u(d0). T*
*hus u is surjective
on objects, as claimed.
Now consider the following diagram:
L ______Lw1
 oeoeo
w  p
 oez 
uoe u
M _____K:w"q
It follows from Proposition 6.23 that there is a unique map z making everything*
* commute. In
particular, we have zw = 1. It follows that (wz)w = w and w is full and surject*
*ive on_objects_so
wz = 1. Thus w is an isomorphism, as required. *
* __
Lemma 6.27. Suppose that we have a homotopy cartesian square as follows, in whi*
*ch p is a
quasicovering.
L _____wGu
 
q p
 
u u
K _____Hwv
18 N. P. STRICKLAND
Then there is a diagram as follows, in which p0and q0 are coverings, r and s ar*
*e equivalences,
the bottom square is cartesian, the top square commutes up to homotopy, and p =*
* p0r, q = q0s.
L ______Gwu
 
s r
 
u u0 u
L0 _____G0w
 
q0 p0
 
u u
K _____wHv
Proof.Using Lemma 6.15, it is not hard to reduce to the case in which L is the *
*standard homotopy
pullback of p and v. As p is a quasicovering we can factor it as p = p0r where *
*p0is a covering and
r is an equivalence. We can then define L0, u0and q0so that the bottom square i*
*s cartesian, which
implies that q0is a covering.
Our next task is to define the functor s. An object d 2 L is a triple (a; c; *
*h: p(a) !v(c)). As
p0is a covering and h: p0r(a) ! v(c), we see that there is a unique morphism g*
*0:r(a) ! a0in
G0such that p0(a0) = v(c) and p0(g0) = h. Thus (a0; c) 2 L0and we can define s *
*on objects by
s(d) = (a0; c). Note that u(d) = a and u0s(d) = u0(a0; c) = a0so we can define
ffd := g0:ru(d) !u0s(d):
Next, consider a morphism (g; k): d0 ! d1 in L, where di = (ai; ci; hi:p(ai)*
* ! v(ci)) for
i = 0; 1. We define a0iand g0ias above, and define
_g= g0 0 1 0 0
1O r(g) O (g0) :a0 !a1;
so that the following diagram commutes.
p0r(g)
p0r(a0)_______________________wp0r(a1)'
 '')h0 h1 [ 
 [[^ 
p0(g00) v(c0)_____v(c1)w p0(g01)
 [ v(k) ''' 
 [[ [ ' '' 
u [[ u
p0(a00)________________________p0(a01):wp0(_g)
We now define s on morphisms by putting s(g; k) = (_g; k). It is easy to check *
*that this makes
s into a functor, and that ff: ru ! u0s is a natural map. Thus, the top square*
* in our diagram
commutes up to homotopy. It is also clear that q0s = q.
Thus, all that is left is to check that s is an equivalence. Let d0 and d1 be*
* as above, and suppose
given k :c0 !c1. As p0is faithful and r is an equivalence, we see that there i*
*s at most one map
g making the upper trapezium of the above diagram commute, and at most one map *
*_gmaking
the lower trapezium commute. Moreover, g exists if and only if _gdoes, and they*
* determine each
other by _g= g01O r(g) O (g00)1 and g = r1((g01)1 O g O g00). Note also that*
* L(d0; d1) is the set of
pairs (g; k) such that the top trapezium commutes, and L0(s(d0); s(d1)) is the *
*set of pairs (_g; k)
making the bottom trapezium commute. It follows easily that s is full and faith*
*ful.
Now consider an object d0 = (a0; c) 2 L0, so v(c) = p0(a0). As r is essentia*
*lly surjective we
can choose an object a 2 G and a map g0:r(a) ! a0 in G0. We thus have an obje*
*ct d =
(a; c; p0(g0): p(a) = p0r(a) !v(c)) of L. Clearly s(d) = d0so s is surjective *
*on objects,_and thus
an equivalence as claimed. *
* __
6.5. Cartesian closure. Let G and H be groupoids, and let [G; H] denote the cat*
*egory of func
tors from G to H. It is easy to see that this is a groupoid and that functors K*
* ! [G; H] biject
naturally with functors_K x G ! H. It follows that G is cartesianclosed. One *
*can also check
that this descends to G in the obvious way. __
We next want to check how this works out in the equivalent category B.
K(N)LOCAL DUALITY FOR FINITE GROUPS AND GROUPOIDS *
*19
Lemma 6.28. Suppose that X and Y are objects of B and they have the homotopy ty*
*pe of CW
complexes. Then the space C(X; Y ) of maps from X to Y also lies in B.
Proof.By wellknown results of Milnor, the space C(Sk; Y ) also has the homotop*
*y type of a CW
complex. Evaluation at the basepoint of S2 gives a surjective Hurewicz fibratio*
*n fflY :C(S2; Y ) !
Y whose fibres have the form ffl1Y{y} = 2(Y; y). As Y 2 B we know that these f*
*ibres are acyclic
and so fflY is a weak equivalence, and thus a homotopy equivalence. By a standa*
*rd result (the dual
of [4, Corollary II.1.12], for example) we deduce that fflY is fibrehomotopy e*
*quivalent to 1Y . One
can also see that for any f :X !Y we have
2(C(X; Y ); f) ' {g :X !C(S2; Y )  fflY O g = f}
and our fibrehomotopy equivalence shows that this is contractible. The result *
*follows. ___
__
Proposition 6.29.If G; H 2 G then B[G; H] ' C(BG; BH) in B.
__ __
Proof.It follows from the lemma that C(BG; BH) 2 B. Recall that B :G! B is an *
*equivalence.
Thus, for any K we have
__ __
B(BK; C(BG; BH)) = B(BK x BG; BH)
__
= B(B(K x G); BH)
__
= G(K x G; H)
__
= G(K; [G; H])
__
= B(BK; B[G; H]):
__ __ *
* __
As B is an equivalence we conclude that B(X; C(BG; BH)) = B(X; B[G; H]) for all*
* X 2 B,_and_
it follows by Yoneda's lemma that C(BG; BH) ' B[G; H] as claimed. *
* __
7.Transfers
Let u: G !H be a covering with finite fibres. Then Bu: BG !BH is a finite c*
*overing map of
spaces, so it is wellknown how to define an associated transfer map T u: 1 BH+*
* !1 BG+ of
spectra. This construction is contravariantly functorial and it converts disjoi*
*nt unions to wedges
and cartesian products to smash products. If p is a homeomorphism then T p = 1*
* p1+. The
wellknown Mackey property of transfers says that if we have a pullback square *
*as shown on the
left, in which p is a covering, then q is also a covering and the square on the*
* right commutes up
to homotopy.
1 Bu+
L ___________wGu 1 BL+u _____w1 BG+u
   
q p Tq Tp
   
u u  
K __________Hwv 1 BK+ _____w11BH+Bv+
It will be convenient to extend this to quasicoverings rather than just cover*
*ings. If u: G !H
is a quasicovering then we can factor u as G v!K p!H where v is an equivalenc*
*e and p is a
covering. We then define
T u = (1 Bv1+) O T p:
To see that this is welldefined, note (using Proposition 6.23) that any other *
*such factorisation
1
has the form G wv!L pw!H for some isomorphism w: K ! L. Using this and th*
*e equation
T w = 1 Bw1+we see that (1 B(wv)1+) O T (pw1) as required.
Now suppose we have quasicoverings G u!H v!K; we want to check that T (vu) *
*= T (u)T (v).
It is easy to reduce to the case where we have functors L p!K v!H such that p*
* is a covering and
v is an equivalence; we need to check that T (p)T (v) = T (p)v1 = T (vp), wher*
*e we allow ourselves
20 N. P. STRICKLAND
to write v instead of 1 Bv+. Lemma 6.26 gives us a pullback diagram as follows,*
* in which q is a
covering and u is an equivalence.
L ______Gwu
 
p  q
 
u u
K _____wH:v
By definition we have T (vp) = u1T (q). The Mackey property gives uT (p) = T (*
*q)v so
T (p)T (v) = T (p)v1 = u1T (q) = T (vp)
as required.
It is easy to check that in this greater generality we still have T (p q q) =*
* T (p) _ T (q) and
T (p x q) = T (p) ^ T (q). We also have an extended Mackey property: if the squ*
*are on the left
is homotopycartesian and p is a quasicovering then q is also a quasicovering a*
*nd the right hand
square commutes up to homotopy (this follows easily from Lemma 6.27).
1 Bu+
L ___________wGu 1 BL+u _____w1 BG+u
   
q p Tq Tp
   
u u  
K __________Hwv 1 BK+ _____w11BH+Bv+
8.The K(n)local category
Fix a prime p and an integer n > 0, and let K = K(n) denote the associated Mo*
*rava Ktheory
spectrum. Let K denote the category of spectra that are local with respect to K*
*(n) in the sense
of Bousfield [6, 26], and let D be the full subcategory of strongly dualisable *
*objects in K. These
categories are studied in detail in [17]. We write X ^ Y for the K(n)localised*
* smash product,
which makes K into a symmetric monoidal category. The unit object is S := LK 1 *
*S0.
__
Definition 8.1.We define a functor L: G !D by LG := LK 1 BG+. (We know from [*
*17,
Corollary 8.7] that LG is always dualisable, so this lands in D as indicated.) *
* It is clear that
L(G x H) = LG ^ LH and L(G q H) = LG _ LH.
Definition 8.2.Let 1 denote the terminal groupoid (with one object and one morp*
*hism), and
write ffl for the unique functor G !1. Let ffi :G !G x G be the diagonal func*
*tor. Define
bG = (LG ^ LG LKTffi!LG Lffl!S):
It is not hard to see that bGxH = bG ^ bH and bGqH = bG bH .
The following result is the key to the whole paper.
Proposition 8.3.The map bG is an inner product on LG.
Proof.We can easily reduce to the case where G is a group rather than a groupoi*
*d. It was
observed in the proof of of [17, Corollary 8.7] that a certain map cG :LG !DLG*
* (arising from
the GreenleesMay theory of generalised Tate spectra) is an isomorphism. It is *
*thus enough to
show that cG = b#G.
We will need some notation. Firstly, we will need to consider various unlocal*
*ised spectra, so
in this proof only we write S for the ordinary, unlocalised sphere spectrum, an*
*d bSfor LK(n)S.
Similarly, we write X ^ Y for the unlocalised smash product and Xb^Y = LK(n)(X *
*^ Y ). Next,
we will work partially in the equivariant categories of Gspectra and G2spectr*
*a, indexed over
complete universes [21]. We write S0 and S00for the corresponding 0sphere obj*
*ects. Also, we
can regard bSas a naive Gspectrum with trivial action and then extend the univ*
*erse to obtain a
genuine Gspectrum, which we denote by bS0. We define a genuine G2spectrum bS0*
*0in the analogous
way.
K(N)LOCAL DUALITY FOR FINITE GROUPS AND GROUPOIDS *
*21
We next recall the definition of cG. It is obtained from a certain map dG :b*
*S^ BG+ !
F (BG+; bS) ' DLG by observing that DLG is K(n)local and that any map from bS^*
* BG+ to a
K(n)local spectrum factors uniquely through LK(n)(Sb^ BG+) ' LG. It will be en*
*ough to check
that dG is adjoint to 1 ^ b0G:bS^ BG2+!Sb, where b0Gis the composite
! Bffl
1 BG2+! 1 BG+ ! S:
We thus need to show that two elements of the group [Sb^ BG2+; bS] are equal.*
* Theorem II.4.5
of [21] (applied to G2=G2 ' 1) gives a natural isomorphism
[Sb^ BG2+; bS] ' [Sb00^ EG2+; bS00]G2:
Let i :EG+ !S0 and :G2=+ !S0 be the collapse maps. Desuspending Construction*
* II.5.1
of [21] gives a pretransfer map t: S00!1 G2=+ of genuine G2spectra. By smashi*
*ng this with
EG2+and passing to orbits we get the transfer map !:1 BG2+!1 B+. Using this a*
*nd the
proof of [21, Theorem II.4.5] we find that 1 ^ b0Gcorresponds to the composite
Sb00^ EG2+1^i^i^t!bS00^ G2=+ 1^!bS00
in [Sb00^ EG2+; bS00]G2.
We now return to the definition of dG. We have a map
bS0^ EG+ 1^i!bS0i*!F (EG+; bS0)
of Gspectra. We next apply the fixed point functor, noting that F (EG+; bS0)G *
*= F (BG+; bS) and
that [21, Theorem II.7.1] gives an equivalence "o:bS^ BG+ ! (Sb0^ EG+)G. The r*
*esulting map
bS^ BG+ ! F (BG+; bS) is dG (see [12, Section 5]). To understand this better, *
*we need to follow
through the construction of "o. We use the notation of [21, Section II.7], noti*
*ng that in our case
we have N = G. The construction uses the group = G xcN, the semidirect product*
* of G with
N using the action by conjugation. There are two natural maps ffl; : !G given*
* by ffl(g; n) = g
and (g; n) = gn. In our case we find that the resulting map !G2 is an isomorp*
*hism, so we
can replace by G2 everywhere. The subgroup becomes 1 x G, the standard embedd*
*ed copy
G xc1 of G becomes , and the maps ffl and become the projections ss0; ss1:G2 *
*! G. The
relevant spectrum D is bS^ EG+, so i**D = bS0^ ss*1EG+ and j*i**D = bS00^ ss*1E*
*G+. The map
"ois obtained from
1 ^ t: bS00^ ss*1EG+ !Sb00^ ss*1EG+ ^ G2=+
by shrinking the universe, passing to orbits and adjointing as described in [21*
*, Construction II.7.5].
It follows that dG is obtained from the composite
bS00^ ss*1EG+ 1^t!bS00^ ss*1EG+ ^ G2=+ 1^i^1^i*!bS00^ G2=+ ^ F (s*
*s*0EG+; S)
by a similar procedure. We can identify EG2+with ss*0EG+ ^ss*1EG+, and we find *
*that the adjoint
of dG is obtained by applying another similar procedure to the map
bS00^ EG2+1^i^i^t!bS00^ G2=+:
This procedure amounts to just composing with :G2=+ ! S0 and using our isomor*
*phism
[Sb^ BG2+; bS] ' [Sb00^ EG2+; bS00]G2. It follows that the adjoint of dG is b0G*
*, as required. ___
Definition 8.4.For any functor u: G !H we put Ru = (Lu)t:LH !LG.
Proposition 8.5.If u: G !H is faithful then Ru = LK T u.
Proof.We first claim that the following square is homotopycartesian:
G _________wHu
 
(1;u) 
  H
u u
G x H _____wHuxxH:1
22 N. P. STRICKLAND
To see this, let K be the homotopy pullback of the functors H and u x 1. The s*
*quare is clearly
cartesian, which means that G embeds as a full subcategory of K; we need only c*
*heck that the
inclusion is essentially surjective. The objects of K are 5tuples (a; b; c; h*
*; k) where a 2 G and
b; c 2 H and h: u(a) !c and k :b !c. The maps from (a; b; c; h; k) to (a0; b0*
*; c0; h0; k0) are triples
(r; s; t) making the following diagram commute:
u(a)______bwhu_____c_k
  
u(r) s t
  
u u u
u(a0)____wb0h0u___c0:_k0
The canonical functor v :G !K is given by v(a) = (a; u(a); u(a); 1; 1). We def*
*ine w: K !G by
w(a; b; c; h; k) = a. Then wv = 1, and we have a natural map vw(a; b; c; h; k) *
*!(a; b; c; h; k) given
by (1; k; k1h). This proves that v is an equivalence, and if we compose it wit*
*h the projections
K ! G x H and K ! H we get the functors (1; u) and u. This proves that our or*
*iginal square
is homotopycartesian, so the Mackey property tells us that
T ( H ) O (Bu x 1) = Bu O T (1; u): 1 B(G x H)+ !1 BH+:
We now use the fact that (1; u) = (1 x u) O G and compose with the projection *
*1 BH+ ! S0
to get
bH O (Bu ^ 1) = fflH O Bu O (T G) O (1 ^ T u):
We next note that fflH OBu = fflG and K(n)localise to conclude that bH O(Lu^1)*
*_=_bG O(1^LK T u),
as required. *
*__
We can thus think of the maps Ru as generalised transfers. It turns out that *
*we also have a
generalised Mackey property.
Proposition 8.6.If we have a homotopycartesian square as shown on the left, th*
*en the diagram
on the right commutes.
M ______wGu LMu _____wLGLuu
   
   
t s Rt Rs
   
u u  
K ______wHv LK _____wLHLv
Proof.We may assume that the square is actually a pullback square of fibrations*
* (see Remark 6.12),
so in particular it commutes on the nose. As bH is a perfect pairing, it suffi*
*ces to check that
bG O (1 ^ (Rs)(Lv)) = bG O (1 ^ (Lu)(Rt)). By transposition, this is equivalent*
* to
bH O (Ls ^ Lv) = bM O (Ru ^ Rt): LG ^ LK !S:
To verify this, we consider the following diagram:
M _________Hwvt=su
 
(u;t) 
  H
u u
G x K _____wHsxxH:v
We claim that this is homotopycartesian. It is clearly cartesian, so it suffic*
*es (as in the previous
proof) to show that the obvious functor from M to the homotopy pullback is esse*
*ntially surjective.
Suppose we are given an object of the homotopy pullback, in other words a 5tup*
*le d = (a; b; c; k; l)
where a 2 G, b 2 H, c 2 K and s(a) k!b l v(c). As s is a fibration we can cho*
*ose a02 G and
K(N)LOCAL DUALITY FOR FINITE GROUPS AND GROUPOIDS *
*23
g :a !a0such that s(a0) = v(c) and s(g) = l1k. Thus d0:= (a0; s(a0) = v(c); c*
*; 1; 1) is an object
of M and the following diagram gives an isomorphism d !d0:
s(a)______wbku_____v(c)_l
  
  
s(g) l1 v(1)
  
u u u
s(a0)_____v(c) _____v(c)
This shows that our square is homotopycartesian. The vertical functors are fai*
*thful and thus are
quasicoverings, so the Mackey property tells us that
(R H ) O (Ls ^ Lv) = L(su) O R(u; t): LG ^ LK !LH:
We next compose with the map LfflH :LH ! S, noting that fflH su = fflM :M ! 1*
* and that
(u; t) = (u x t) M :M ! G x K. We conclude that bH O (Ls ^ Lv) = bM O (Ru ^ R*
*t),_as
required. *
*__
Theorem 8.7.For any finite groupoid G, the maps (Rffl; R ; Lffl; L ) make LG in*
*to a Frobenius
object.
Proof.This is formally identical to the proof of Proposition 5.1; we need only *
*check that the
following square is homotopycartesian, and that is easy.
G _________Gwx G
 
 1x
 
u u
G x G _____Gwx Gxx1G:
___
Definition 8.8.Given a finite groupoid G, define G = [Z; G]; Proposition 6.29 t*
*ells us that
BG is homotopyequivalent to the free loop space on BG. The objects of G are pa*
*irs (a; u)
where u 2 G(a; a), and the maps from (a; u) to (b; v) are maps g :a !b such th*
*at v = gug1. It
is thus easy to see that ss(a; u) = a gives a functor G !G, and that this is a*
*ctually a covering.
If G is a group then G is equivalent the disjoint union of`the groups ZG(g) as *
*g runs over the
conjugacy classes in G, so the free loop space on BG is BZG(g); this is actua*
*lly wellknown,
and a more elementary account appears in [5, Section 2.12], for example.
Remark 8.9. It is important to distinguish between [Z; G] and [Zp; G]; see Sect*
*ion 11 for more
discussion of this.
We can now identify the maps = ffl :LG ! S and ff = j :S ! LG discussed *
*in
Proposition 4.3.
Proposition 8.10.We have = (LfflG )(Rss) and ff = (Lss)(RfflG ).
Proof.The key point is that the following square is homotopycartesian:
G _____Gwss
 
ss 
 
u u
G _____wG2:
To see this, let H be the homotopy pullback of and . The objects of H are tu*
*ples (a; b; u; v)
where a; b 2 G and u; v :a ! b. The morphisms from (a; b; u; v) to (a0; b0; u0*
*; v0) are pairs (g; h)
where g :a !a0and g :b !b0and hu = u0g and hv = v0g. We can define a functor *
*OE: G !H
by (a; u) 7! (a; a; u; 1) and a functor in the opposite direction by (a; b; u;*
* v) 7! (a; v1u). We find
that these are equivalences and that either projection H !G composed with OE i*
*s just ss; it follows
24 N. P. STRICKLAND
that the square is homotopycartesian, as claimed. We conclude that = (R )(L *
*) = (Lss)(Rss).
We also know from Proposition 4.3 that = ffl = (Lffl)(R )(L ) and ff = j = (*
*R )(L )(Rffl)._
Everything now follows from the evident fact that fflGss = fflG :G !1. *
* __
We conclude this section by discussing the case of a finite abelian group A, *
*considered as
a groupoid with one object. There is then a unique functor i :1 ! A, and also*
* a division
homomorphism :A x A !A given by (g; h) = gh1.
Proposition 8.11.We have b = (Ri)(L): LA^LA !S. We also have ff = Aj and = *
*Affl.
Proof.We have a commutative diagram as follows, which is easily seen to be both*
* cartesian and
homotopycartesian:
A _______w1ffl
 
 
 i
 
u u
A x A _____wA:
The vertical functors are faithful and thus are quasicoverings. The Mackey prop*
*erty now tells us
that b = (Lffl)(R ) = (Ri)(L) as claimed.
Next, consider the groupoid A = [Z; A]. It is easy to see that this is just a*
* disjoint union of
A copies of A, and that the functor ss :A !A just sends each copy isomorphic*
*ally to A._The
remaining claims now follow easily from Proposition 8.10. *
* __
9. Inner products in cohomology
We next study E*BG for suitable cohomology theories E.
If p is an odd prime, let E be a plocal commutative ring spectrum such that
(a)E0 is a complete local Noetherian ring
(b)E1 = 0
(c)E2 contains a unit
(d)The associated formal group over spec(E0=m) has height n.
In the language of [13, Section 2], these are precisely the K(n)local admissib*
*le ring spectra. In
the case p = 2 we would like to allow E to be a twoperiodic version of K(n), b*
*ut this is not
commutative. We therefore relax the requirement that E be commutative and assu*
*me instead
that there is a derivation Q: E !E and an element v 2 ss2E such that 2v = 0 and
ab  ba = vQ(a)Q(b);
so that E is quasicommutative in the sense of [29, Definition 8.1.1]. This is o*
*f course satisfied if
E is commutative, with Q = 0 and v = 0. Other examples, including the twoperio*
*dic version of
K(n), can most easily be produced by the methods of [27], which also contains d*
*etailed references
to previous work in this direction.
We consider LG as a Frobenius object just as in the previous section. As usua*
*l we use the maps
S LfflLG L!LG ^ LG to make E0LG = E0BG into a ring and E*LG = E*BG into a g*
*raded
ring. We also use (Rffl): S ! LG to give a map ffl := (Rffl)*:E0BG !E0, which*
* in turn gives a
bilinear form b(x; y) = ffl(xy) on E0BG.
Remark 9.1. If G is a group then the inclusion of the trivial group gives a map*
* i :1 !G and
thus an augmentation map (Li)*:E0BG !E0. In other contexts this is often denot*
*ed by ffl, but
it is not the same as the map ffl defined above.
We say that G is Egood if E0LG is free of finite rank over E0 and E1LG = 0. *
*If so then we
have a K"unneth isomorphism E0(LG ^ LG) = E0LG E0 E0LG. Using this and Theorem *
*8.7
we find that the above maps make E0LG into a Frobenius object in the compactcl*
*osed category
of finitely generated free modules over E0. In particular, we deduce that our *
*bilinear form is
an inner product. A functor u: G ! H gives a ring map u*:E0BH ! E0BG induced*
* by
K(N)LOCAL DUALITY FOR FINITE GROUPS AND GROUPOIDS *
*25
Lu: LG !LH, and also a map u!:= (Ru)*:E0BG !E0BH that is adjoint to u*. If u *
*is the
inclusion of a subgroup in a group then u!is the corresponding transfer map (by*
* Proposition 8.5).
The adjointness of u!and u* is thus a version of Frobenius reciprocity.
As usual we have a trace map :E0BG ! E0 (which can be computed using only t*
*he ring
structure) and an element ff = (1) 2 E0BG. Proposition 8.10 tells us how to co*
*mpute ff in
terms of ordinary transfers, and Proposition 4.3 tells us that ffl(xff) = (x). *
*We will see later that
ff becomes invertible in Q E0BG, so the previous equation characterises ffl up*
* to torsion terms.
Now let A be a finite Abelian group. It is known that such groups are Egood *
*for all E; see [13,
Proposition 2.9] for a proof in the present generality, although the basic idea*
* of the proof is much
older [20, 14]. We know from Proposition 8.11 that ff = A in this context so *
*that Affl(x) = (x).
We next give another formula for ffl that is more useful when p = 0 in E0. It i*
*s easy to see that
fflAxB = fflA fflB, and if A is coprime to p then E0BA = E0 with fflA = A:*
*1: E0 !E0. It is
thus enough to treat the case where A = Cpm for some m > 0.
Choose a complex orientation x 2 Ee0CP 1, or equivalently a coordinate on the*
* associated
formal group G. This gives a formal group law F with associated pm series [pm *
*](x), and we have
E0B(Cpm) = OG(m)= E0[[x]]=[pm ](x):
There is a unique invariant differential form ! on G that agrees with dx at zer*
*o: if E0 is torsionfree
this is most easily expressed as ! = d(logF(x)) = log0F(x)dx. Given a function *
*f 2 OG = E0[[x]]we
get a meromorphic differential form f!=[pm ](x) 2 M1G, and the residue of this *
*form clearly only
depends on f modulo [pm ](x). (See [29, Sections 5.3 and 5.4] for an exposition*
* of meromorphic
forms and their residues.)
Proposition 9.2.The canonical Frobenius form on E0BCpm is given by ffl(f) = res*
*(f!=[pm ](x)).
Proof.For any Egood group G, we can define
2 0 2 0 0
c := j(1) = trG(1) 2 E (BG ) = E BG E0 E BG:
We see from Scholium 3.12 that ffl: E0BG ! E0 is the unique map such that (ffl*
* 1)(c) = 1 2
E0BG.
Now take G = Cpm, so E0BG = E0[[x]]=[pm ](x) and E0BG2 = E0[[x; y]]=([pm ](x)*
*; [pm ](y)).
Write (t) = [pm ](t)=t 2 E0[[t]]; we know from [25, Section 4] that trG1(1*
*) = (x) (a simpler
proof appears in [32]). Put z = x F y; it follows from Proposition 8.11 that c*
* = (z).
Now consider the form
0[[y]]
fl = c!=[pm ](x) 2 E_____[pmE](y)0M1G;
so that res(fl) 2 E0[[y]]=[pm ](y). In view of the above, it will suffice to ch*
*eck that res(fl) = 1.
For this, we note that [pm ](y) = 0 so zc = [pm ](z) = [pm ](x) so zfl = ! so*
* fl = !=z. Now,
! = g(x)dx for some power series g with g(0) = 1 and this differential is invar*
*iant under translation,_
which implies that ! = g(z)dz also. Thus res(fl) = res(g(z)dz=z) = g(0) = 1 as *
*required. __
Corollary 9.3.Let E be the usual twoperiodic version of K(n) (with n > 1), and*
* let x be the
usual ptypical orientation. Then the Frobenius form on the ring
nm 0 k nm
E0BCpm = E0[[x]]=xp = E {x  0 k < p }
is given by ffl(xk) = 0 for k < pnm1 and t(xpnm1) = 1. In the case n = 1 we h*
*ave ffl(xpmpj) = 1
for 0 j m and ffl(xk) = 0 for all other k.
P nk
Proof.For the integral twoperiodic version of K(n) we have logF(x) = k0 xp *
*=pk. When
n > 1 it follows easily that ! = log0F(x)dx = dx (mod p). We also have [pm ](x*
*) = xpmnPso
ffl(xk) = res(xkpmndx) and the claim follows easily. In the case n = 1 we have*
* ! = k0_xpk1dx
and the stated formula follows in the same way. *
* __
26 N. P. STRICKLAND
10.Character theory
Let G be a finite groupoid. Write C(G) := Q{ss0G} for the rational vector spa*
*ce freely generated
by the set of isomorphism classes of objects of G. Given a 2 G we write [a] for*
* the corresponding
basis element in C(G). We define a bilinear form on C(G) by
([a]; [b]) := G(a; b):
It is convenient to write G(a) := G(a; a) and to introduce the elements [a]0:= *
*[a]=G(a), so that
([a]; [a]0) = 1. We also write C(G)* = Hom Q(C(G); Q) = F (ss0G; Q) for the dua*
*l of C(G). Given
a functor u: G !H we define Lu: C(G) !C(H) by (Lu)[a] = [u(a)], and we let Ru*
*: C(H) !
C(G) be the adjoint of this, so that
X
(Ru)[b]0= [a]0:
[a]  u(a)'b
The sum here is indexed by isomorphism classes of objects a 2 G such that u(a) *
*is isomorphic to
b in H.
We next show that these constructions have the expected Mackey property.
Proposition 10.1.If we have a homotopycartesian square as shown on the left, t*
*hen the diagram
on the right commutes.
M ________wGu C(M) _____wC(G)Lu
  u u
   
t  s Rt Rs
   
   
u u  
K ________wHv C(K) _____wC(H)Lv
Proof.We may assume that the square is actually a pullback square of fibrations*
* (see Remark 6.12),
so in particular it commutes on the nose. Fix c 2 K, so (Lv)[c] = [vc] = H(vc)*
*[vc]0. We need to
check that (Rs)(Lv)[c] = (Lu)(Rt)[c]. Because s is a fibration, any isomorphism*
* class in G that
maps to [vc] in H has a representative a 2 G such that sa = vc. Using this, we *
*find that
X
(Rs)(Lv)[c] = H(vc)G(a)1[a]:
[a]  sa=vc
We also know that t is a fibration, so every isomorphism class in M that maps t*
*o c contains a
representative d with t(d) = c, in other words d has the form (a; c) for some a*
* 2 G with sa = vc.
It follows that
X
(Lu)(Rt)[c] = M((a; c))1K(c)[a]:
[a;c]  sa=vc
Fix a 2 G with sa = vc. The coefficient of [a] in (Rs)(Lv)[c] is then H(vc)*
*G(a)1. For
(Lu)(Rt)[c] we need to be more careful, because there will typically be objects*
* a0 2 G with
[a0] = [a] 2 ss0G but [a0; c] 6= [a; c] 2 ss0M. Put
X = {a02 G  a0' a andsa0= vc}= ~;
where a0~ a00iff there exist g :a0! a00and k :c !c such that sg = vk :vc !vc*
*. It is easy to
see that a0~ a00iff (a0; c) ' (a00; c) in M, and it follows that the coefficien*
*t of [a] in (Lu)(Rt)[c] is
X
:= M((a0; c))1K(c):
[a0]2X
To analyse this further, we introduce the set
Y = {(a0; g0)  a02 G ; sa0= vc andg0:a0! a}= ~;
where (a0; g0) ~ (a00; g00) iff sg0= sg00:vc ! vc. Using the fact that s is a *
*fibration, one checks
that the map [a0; g0] 7! sg0 gives a bijection Y ' H(vc), so that Y  = H(vc)*
*. On the other
hand, there is an evident projection ss :Y ! X sending [a0; g0] to [a0]. If ss*
*[a00; g00] = [a0] then we
K(N)LOCAL DUALITY FOR FINITE GROUPS AND GROUPOIDS *
*27
can choose f :a0! a00and k :c !c such that sf = vk, and then observe that the*
*re is a unique
g :a0! a0such that f = (g00)1g0g. One checks that the coset (g; k):M((a0; c))*
* depends only on
the equivalence class [a00; g00] and that this construction gives a bijection
ss1{[a0]} ' (G(a0) x K(c))=M((a0; c)):
Note also that G(a0) = G(a) because a0' a. It follows that
X
= G(a0) x K(c)=M((a0; c))1M((a0; c))1K(c)
[a0;g0]2Y
X
= G(a)1
[a0;g0]2Y
= H(vc)G(a)1:
This is the same as the coefficient of [a] in (Rs)(Lv)[c], as required. *
* ___
If we let 1 fflG ! G2 be the obvious functors, then it follows easily that *
*the maps (Lffl)*,
(L )*, (Rffl)*, (R )* make C(G)* into a Frobenius algebra over Q. The Frobenius*
* form is just
X
ffl(f) = (f; 1) = f(a)=G(a):
[a]2ss0G
Next, let denote the group (Qp=Zp)n, whose dual is * = Hom (; Qp=Zp) ' Zpn. *
*We regard
*
* m
* as a groupoid with one object in the usual way. We also consider (m) = ker( *
*p!), so
(m)* = *=pm .
Note that if G is a finite group and : * ! G then the image of is Abelian a*
*nd plocal
(because it is a quotient of *) and finite (because it is a subgroup of G). It *
*follows that (pm *) =
1 for large m, so that is automatically continuous. It follows that Hom(*; G) *
*bijects with the set
of ntuples of commuting elements of G of ppower order. More generally, if G i*
*s a finite groupoid
then a functor : * !G consists of an object a 2 G together with an ntuple of *
*commuting
pelements of the group G(a; a).
The generalised character theory of Hopkins, Kuhn and Ravenel [14, 15] can be*
* repackaged
slightly to relate Q E0BG to C([*; G])* for admissible cohomology theories E, *
*as we now
explain. Given such a spectrum we need to define an associated ring D0. In the *
*special case of
Morava Etheory, this was defined in [14]; the details necessary for the genera*
*l case are given
in [28]. Associated to E we have a formal group G over spf(E0) and thus a level*
*structure scheme
Level((m); G) with coordinate ring Dm say. These form a directed system in an o*
*bvious way and
we define D0= Q lim!Dm . (This was called L in [14] but we have renamed it to*
* avoid clashes
m
of notation.) If G is the universal deformation of its restriction to the speci*
*al fibre (as is the case
with Morava Etheory) then D0is the integral domain obtained from Q E0 by adjo*
*ining a full
set of roots of [pm ](x) for all x. For any E one can show that D0 is a free mo*
*dule of countable
rank over Q E0.
As mentioned earlier, the following theorem is merely a repackaging of result*
*s of Hopkins, Kuhn
and Ravenel [14].
Theorem 10.2. For any admissible ring spectrum E, there is a natural isomorphis*
*m of Frobenius
algebras over L
D0E0 E0BG = D0Q C([*; G])*:
Moreover, this respects the constructions u 7! (Lu)* and u 7! (Ru)* for functor*
*s between groupoids.
Proof.We first construct a map
o :D0E0 E0BG !D0Q C([*; G])*
of D0algebras. By juggling various adjunctions we see that it suffices to cons*
*truct, for each functor
: * !G, a map o : E0BG !D0of E0algebras, such that o = o when is isomorph*
*ic to
. We know from our previous remarks that must factor through (m)* = *=pm for s*
*ome
28 N. P. STRICKLAND
m. We thus get a map E0BG ! E0B(m)*, and we know from [13, Proposition 2.9] t*
*hat
E0B(m)* = OHom((m);G), and Dm is a quotient of this ring, so we get the require*
*d map o as
the composite
*
E0BG B!E0B(m)* !Dm ! D0:
One checks easily that this is independent of the choice of m. Isomorphic func*
*tors ; give
homotopic maps B(m)* !BG and thus o = o as required. The resulting map o is *
*easily seen
to be natural for functors of groupoids and to convert equivalences to isomorph*
*isms. Both source
and target of o convert disjoint unions to products. Any finite groupoid is equ*
*ivalent to a finite
disjoint union of finite groups, so it suffices to check that o is an isomorphi*
*sm when G is a group.
This is just [14, Theorem B].
To say that this isomorphism respects the construction u 7! (Lu)* is just to *
*say that o is a
natural map, which is clear. We also need to check that for any functor u: G !*
*H, the following
diagram commutes:
oG
D0E0 E0BG _____wD0Q C([*; G])*

 
(Ru)* (Ru)*
 
u u
D0E0 E0BH _____wD0QoC([*;HH])*
We first make this more explicit. The functor u induces Ru: LH ! LG. By applyi*
*ng E0()
and noting that E0LK = E0BK we get a map (Ru)*:E0BG ! E0BH. After tensoring wi*
*th
D0we obtain the left hand vertical map in the above diagram. On the other hand,*
* u also induces
a functor u*:[*; G] ![*; H] and thus a map R(u*): C[*; H] !C[*; G]. By dualis*
*ing and
tensoring with D0we obtain the right hand vertical map.
We first prove that the diagram commutes when u is a quasicovering. This red*
*uces easily to the
case where H is a group and G is connected. It is not hard to see that in this *
*case u is equivalent to
the inclusion of a subgroup G H and Ru: LH !LG is just the K(n)localisationP*
*of the transfer
map 1 BH+ ! 1 BG+. It follows from [14, Proposition 3.6.1] that o ((Ru)*x) = *
* oh (x),
where the sum runs over cosetsPhG such that h := h1h maps * into G. The right *
*hand side
can be rewritten as G1 h oh (x), where the sum now runs over elements rathe*
*r than conjugacy
classes. Fix a homomorphism : * !G that becomes conjugate to in H. Then the n*
*umber
of h's for which h = is the order of the group ZH () = {h 2 H  h = }, so
X
o ((Ru)*x) = G1 ZH ()o (x):
If we want to index this sum using conjugacy classes of 's rather than the 's t*
*hemselves, we
need an extra factor of G=ZG(), the number of conjugates of . This gives
X
o ((Ru)*x) = ZG()1ZH ()o (x):
[]
On the other hand, ZG() is just the automorphism group of in the category [*; *
*G], so the
map R(u*): C[*; H] !C[*; G] is given by
X
R(u*)[]=ZH () = []=ZG():
[]  u'
The claim follows easily by comparing these formulae.
We have an inner product on D0E0 E0BG obtained from the inner product bG on L*
*G, and
an inner product on D0Q C([*; G])* obtained from the standard inner product on *
*C(K)* for
any K. By taking u to be the diagonal functor G !G x G in the above discussion*
*, we see that
our isomorphism o converts the former inner product to the latter one. Thus o i*
*s compatible with
taking adjoints and with the construction u 7! (Lu)*, so it is compatible with *
*the_construction
u 7! (Ru)* as well. *
* __
K(N)LOCAL DUALITY FOR FINITE GROUPS AND GROUPOIDS *
*29
We next reformulate Theorem 10.2 in the spirit of [13, Theorem 3.7].
Definition 10.3.Given a finite groupoid G, we define a new groupoid AG as follo*
*ws. The objects
are pairs (a; A), where a 2 G and A is a finite Abelian psubgroup of G(a). The*
* morphisms from
(a; A) to (b; B) are maps g :a !b in G such that B = gAg1. For any finite Abe*
*lian pgroup A we
can define a ring D0A= QOLevel(A*;G)as in [28, Proposition 22]. There is an evi*
*dent way to make
the assignment (a; A) 7! D0Ainto a functor AGop! Rings, and we define T G = li*
*m  D0A.
*
*(a;A)2AG
If we write ff0(a;A)= ZG(a)(A) then ff02 T G.
Theorem 10.4. There is a natural isomorphism QE0BG = T G, and this is a finitel*
*y generated
free module over E0. The element ff = (1) 2 E0BG becomes ff02 T G, so the resu*
*lting Frobenius
form on T G is just ffl(x) = (x=ff0), where is the trace form.
Proof.The isomorphism QE0BG = T G can be proved either by reducing to the case *
*of a group
and quoting [13, Theorem 3.7], or by taking the fixedpoints of both sides in T*
*heorem 10.2 under
the action of Aut(). From the latter point of view, the term in T G indexed by *
*(a; A) corresponds
to the terms in C[*; G]* coming from homomorphisms * !G(a) with image A, so ff*
*0becomes
the function ss0[*; G] !L that sends [] to [*; G](). Proposition 8.10 identi*
*fies_this with ff,
as required. *
*__
11.Warnings
We started this paper by considering the representation ring R(G), but unfort*
*unately the
analogy between our rings E0LG and R(G) fails in a number of respects, even in *
*the height one
case. In this section we point out some possible pitfalls.
Let E be the padic completion of the complex Ktheory spectrum, so E is an a*
*dmissible
ring spectrum of height one. Then E0BG is the completion of R(G) at I + (p), w*
*here I is
the augmentation ideal. The ring R(G) is a free Abelian group of rank equal to *
*the number of
conjugacy classes, generated by the irreducible characters. These are orthonorm*
*al, so the inner
product on R(G) is equivalent to the standard diagonal, positive definite inner*
* product on Zh. It
also follows that R(G) is a permutation module for the outer automorphism group*
* of G.
The ring E0BG is a free module over Zpof rank equal to the number of conjugac*
*y elements of
elements of ppower order. The canonical map R(G) !E0BG does not preserve inne*
*r products.
There is no canonical set of generators for E0BG, so there is no reason for it *
*to be a permutation
module for Out(G). In fact, Igor Kriz has constructed examples of extensions G*
* ! G0 !Cp
where G is good but H1(Cp; E0BG) 6= 0 and one can deduce that E0BG is not a per*
*mutation
module in this case [19]. There is also no reason to expect that E0BG has an or*
*thonormal basis.
A related set of issues involves the comparison between the free loop space o*
*f BG (which is
B[Z; G]) and the space of maps from the padically completed circle to BG (whic*
*h is B[Zp; G]).
The former enters into Proposition 8.10, and the latter into Theorem 10.2. The *
*two spaces are
not even padically equivalent: if G is a group and T is a set of representativ*
*es`for the conjugacy
classes of elements whose order is not a power of p then B[Z; G] ' B[Zp; G] q *
*g2TBZG(g), and
each term in the coproduct contributes at least a factor of E0 in E0B[Z; G] eve*
*n if ZG(g) is a
p0group. Note that [14, 5.3.10] is slightly inaccurate in this regard; the pro*
*of given there really
shows that OnBG = On1(B[Zp; G]), rather than On1(B[Z; G]).
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