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 K-theory of the classifying space * *of a finite group G. If n = 1 and G is a p-group 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 K-theory (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 Poincare-Atiyah 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- up-to-homotopy, as outlined in [30, Remark 3.1]. In Section 8 we turn to the sp* *ectra LG. In [17] we used the Greenlees-May 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 Hopkins-Kuhn-Ravenel 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 R-bilinear 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 + 1-dimensional cob* *ordism category, whose objects are closed 1-manifolds 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 A-module. We claim th* *at ft:B -!A is automatically a map of A-module 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 multiplication-by-a 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 well-known 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 Atiyah-Poincare 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 even-dimensional 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 one-point 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 "S-algebra") 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 Pontrjagin-Thom 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 left-exact 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 limm-or(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 non-identity morphisms u: 0 -! 1 and u-1: 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 p-1wb;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 * i-1rb;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 homotopy-cartesian 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 homotopy-cartesian 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 left-hand 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-! fiuv0-fi(OE)--!fivu0-! v1flu0); where the first and third maps come from the commutativity of the right-hand 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 quasi-coverings. 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 quasi-covering 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 quasi-covering 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 k0k-1 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 = q-1{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 well-defined, 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 k-1k0 = 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 well-defined 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 g-1: 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 quasi-covering 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 quasi-coverings 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)g-1: a -!u(d0) satisfies q(u(l)g-1) * *= vp(l)v(k)-1 = 1. As q is a covering, it reflects identity maps, so u(l)g-1 = 1a and a = u(d0). T* *hus u is surjective on objects, as claimed. Now consider the following diagram: L ______Lw1 | oeo|eo 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(g||00) 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 = r-1((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 cartesian-closed. 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 well-known 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 ffl-1Y{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 fibre-homotopy 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 fibre-homotopy 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 well-known 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* * p-1+. The well-known 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 Bv-1+) O T p: To see that this is well-defined, 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 Bw-1+we see that (1 B(wv)-1+) O T (pw-1) 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)v-1 = 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) = u-1T (q). The Mackey property gives uT (p) = T (* *q)v so T (p)T (v) = T (p)v-1 = u-1T (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 homotopy-cartesian 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 K-theory 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 Greenlees-May 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 G-spectra and G2-spectr* *a, indexed over complete universes [21]. We write S0 and S00for the corresponding 0-sphere obj* *ects. Also, we can regard bSas a naive G-spectrum with trivial action and then extend the univ* *erse to obtain a genuine G-spectrum, which we denote by bS0. We define a genuine G2-spectrum 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 G2-spectra. 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 G-spectra. 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 homotopy-cartesian: 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 5-tuples (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; k-1h). 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 homotopy-cartesian, 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 homotopy-cartesian 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 homotopy-cartesian. 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 5-tup* *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) = l-1k. 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|) l-1| |v(1) | | | |u |u |u s(a0)_____v(c) _____v(c) This shows that our square is homotopy-cartesian. 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 homotopy-cartesian, 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 homotopy-equivalent 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 = gug-1. 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 well-known, 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 homotopy-cartesian: 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; v-1u). 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 homotopy-cartesian, 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) = gh-1. Proposition 8.11.We have b = (Ri)(L): LA^LA -!S. We also have ff = |A|j and = * *|A|ffl. Proof.We have a commutative diagram as follows, which is easily seen to be both* * cartesian and homotopy-cartesian: 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 p-local 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 two-periodic 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 two-perio* *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 -Lffl-LG 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 E-good 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 compact-cl* *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 E-good * *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 |A|ffl(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 torsion-free 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 E-good 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 two-periodic version of K(n) (with n > 1), and* * let x be the usual p-typical 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 < pnm-1 and t(xpnm-1) = 1. In the case n = 1 we h* *ave ffl(xpm-pj) = 1 for 0 j m and ffl(xk) = 0 for all other k. P nk Proof.For the integral two-periodic 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(xk-pmndx) and the claim follows easily. In the case n = 1 we have* * ! = k0_xpk-1dx 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 homotopy-cartesian 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))|-1|K(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))|-1|K(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 ss-1{[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))|-1|M((a0; c))|-1|K(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 p-local (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 n-tuples of commuting elements of G of p-power order. More generally, if G i* *s a finite groupoid then a functor : * -!G consists of an object a 2 G together with an n-tuple of * *commuting p-elements 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 E-theory, 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 E-theory) 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 D0-algebras. By juggling various adjunctions we see that it suffices to cons* *truct, for each functor : * -!G, a map o : E0BG -!D0of E0-algebras, 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 quasi-covering. 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 := h-1h maps * into G. The right * *hand side can be rewritten as |G|-1 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) = |G|-1 |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()|-1|ZH ()|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 p-subgroup of G(a). The* * morphisms from (a; A) to (b; B) are maps g :a -!b in G such that B = gAg-1. For any finite Abe* *lian p-group 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 fixed-points 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 p-adic completion of the complex K-theory 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 p-power 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 p-adically 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 p-adically 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 p0-group. Note that [14, 5.3.10] is slightly inaccurate in this regard; the pro* *of given there really shows that OnBG = On-1(B[Zp; G]), rather than On-1(B[Z; G]). References [1]L. Abrams. Two-dimensional topological quantum field theories and Frobenius * *algebras. J. Knot Theory Ram- ifications, 5(5):569-587, 1996. [2]D. W. Anderson. Fibrations and geometric realizations. Bull. Amer. Math. Soc* *., 84(5):765-788, 1978. [3]M. Barr and C. Wells. Toposes, Triples and Theories, volume 278 of Grundlehe* *ren der math. Wiss. Springer- Verlag, Berlin, 1985. [4]H. J. Baues. Algebraic Homotopy, volume 15 of Cambridge Studies in Advanced * *Mathematics. 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