__g*_//; in which P and Q are projective objects in M. Both implications in part (b) fol* *low from the equivalence of the lifting problems represented by the two diagrams IOOeeKK tP K "K t | k || KKK t t |"k | K zzt fflffl| P L _1_f_//P M__f*_// ; in which P and I are projective and injective in M, respectively. * *|___| WHEN PROJECTIVE DOES NOT IMPLY FLAT 7 2.Day's symmetric monoidal closed structures The categories of greatest interest in this article are functor categories wi* *th symmetric monoidal closed structures of the sort described by Day in [6]. Howev* *er, only a rather simple case of Day's general approach is needed for our examples. In this section, that case is reviewed, and our basic examples of functor categ* *ories carrying Day's structures are introduced. Let S be a symmetric monoidal Ab-category whose tensor product ^ is bilin- ear, and O be a skeletally small full subcategory of S. The relevant case of Da* *y's approach applies to any reasonably well-behaved subcategory O, and provides a symmetric monoidal closed structure on the category MO of additive functors fr* *om O into Ab. In fact, Day's machinery applies equally well to functors into the c* *ate- gory R-Mod of modules over a commutative ring R. However, restricting attention to functors into Ab somewhat simplifies our notation. Our entire discussion of MO is plagued by an annoying problem with the vari- ance of our functors. The general observations in the next section about the re* *lation between Day's structures and our axioms are most easily presented in terms of t* *he category of covariant functors out of O. However, some of our examples are drawn from geometric sources, like cohomology theories, which naturally yield contrav* *ari- ant functors. Most of the time, this difference in the preferred variance is no* *thing more than a notational nuisance. The preference for covariant functors in the n* *ext section arises from Proposition 3.3, which necessarily applies only to covarian* *t func- tors. All of the other results in that section apply equally well to covariant* * and contravariant functors since they depend only on self-dual properties of S and * *O. In many of our examples, the question of a preferred variance is moot, either beca* *use O and Oop are isomorphic or because functors of both variances are of interest. Nevertheless, in a few key cases, variance is very significant. The preferred v* *ariance in these cases is perversely split about evenly. Since there is no clearly pref* *erred variance and the choice of variance is often irrelevant, MO is a deliberately * *am- biguous symbol denoting either of the two categories of additive functors from * *O to Ab. In the sections where functors of both variance are considered, this notati* *on is used only in remarks applicable to both categories. When variance really matter* *s, the symbols McovOand McontOare used to denote the categories of covariant and contravariant additive functors from O into Ab, respectively. The case of Day's approach considered here is quite similar to that described* * in section 4 of [21]. However, two differences are worth noting. In [21], it is as* *sumed that O contains the unit for the tensor product on S. Here, that constraint is * *re- placed by a weaker condition on O which fits more naturally into our discussion* * of the connection between the properties of O and our axioms. The second difference is that all the functors in [21] are contravariant. Since the contravariant cas* *e is dis- cussed in detail in [21], the introductory discussion here is focused on the co* *variant case. Readers interested in the contravariant case may either make the necessary notational adjustments for themselves or look them up in [21]. The definition below provides the basic components of Day's symmetric monoidal closed structure on the category McovO. In this definition, and throughout the * *rest of the article, objects of O are denoted by A, B, C, etc., and the objects of S* * that need not be in O are denoted by Z, Y , X, etc. The unit for the tensor product * *of S is denoted o(or oS). 8 L. GAUNCE LEWIS, JR. Definition 2.1.(a)For X in S, let HcovX: O __//_Aband HcontX: Oop __//_Abbe the functors given by HcovX(A) = S(X; A) and HcontX(A) = S(A; X). Note that Hco* *vX is in McovOand HcontXis in McontO. In remarks applicable to the functor categor* *y of either variance and in contexts where the desired variance should be obvious, t* *he notation HX is used for the functor of the appropriate variance associated to X. (b) If M and N are objects of McovO, then the functor M N in McovOis given on an object A of O by Z B;C2O (M N)(A) = M(B) N(C) S(B ^ C; A): The naturality of coends provides the definition of M N on the morphisms of O. (c) If M and N are objects of McovO, then the functor in McovOis given on an object A of O by Z (A) = hom M(B) S(A ^ B; C); N(C) : B;C2O The naturality of ends provides the definition of on the morphisms of O. (d) Let A, C be in O and X, Y be in S. Then the evaluation map S(X; C) S(Y ^ C; A) __//_S(Y ^ X; A) induces a homomorphism Z C aecov(X; Y ;:A) S(X; C) S(Y ^ C; A)___//S(Y ^ X; A) of abelian groups. Note that, by Lemma 4.3(b) of [21], this map is an isomorphi* *sm if X is in O. There is an an analogous map Z C aecont(X; Y ;:A) S(C; X) S(A; Y ^ C)___//S(A; Y ^ X) which must be used instead of aecov(X; Y ; A) in the context of categories of c* *on- travariant functors. This map is denoted in [21] and is discussed there in the* * proof of Proposition 5.2. In remarks applicable to functors of either variance, ae(X;* * Y ; A) is used to denote the appropriate one of these two maps. It is easy to see that the operation is symmetric, and a simple end calcula- tion gives the desired adjunction relating and < ; >. Thus, to show that the* *se constructions provide MO with a symmetric monoidal closed structure, it suffic* *es to exhibit a unit for , construct an associativity isomorphism, and prove that* * the appropriate diagrams commute. The unit for MO should be HoS. The associativity isomorphism for MO should be easily derived from the associativity isomorphism for S, and the commutativity of the required diagrams for MO should follow from the commutativity of the analogous diagrams in S. Nevertheless, these last three pieces of the monoidal structure need not fit properly into place unless the map ae(X; Y ; A) is an isomorphism under the appropriate conditions. Theorem 2.2. Let S be an Ab-category with a symmetric monoidal structure de- rived from a bilinear tensor product ^, and let O be a skeletally small full su* *bcate- gory of S. If the map ae(X; Y ; A) is an isomorphism whenever X = oS and Y 2 O and whenever X and Y are both finite ^-products of objects in O, then MO is a symmetric monoidal closed category. WHEN PROJECTIVE DOES NOT IMPLY FLAT 9 Proof.We prove this result for McovO; the proof for McontOis analogous. In this* * and several other proofs, we make use of a variety of folklore results about ends a* *nd coends. The contravariant analogs of these results are discussed in section 4 o* *f [21]. The proofs of these results are formal, and obviously translate to the covariant context. Note, however, that the results in section 5 of [21] typically apply * *only in the contravariant case since their proofs make use of more geometric argumen* *ts. Let M, N, and P be in McovO. The unit isomorphism M HcovoS~=M of McovOis given at A 2 O by the composite Z B;C2O (M HoS)(A) = M(B) S(oS; C) S(B ^ C; A) RB ae Z B2O ______// M(B) S(B ^ oS; A) Z B2O ~= M(B) S(B; A) ~= M(A); in which the last isomorphism is given by Lemma 4.3(c) of [21]. The associativi* *ty isomorphism for McovOis obtained by using the maps aecov(X; Y ; A) to identify * *both ((M N) P )(A) and (M (N P ))(A) with Z B;C;D2O M(B) N(C) P (D) S(B ^ C ^ D; A) for each A 2 O. The maps aecov(X; Y ; A) are used in much the same fashion to reduce the question of the commutativity of each of the necessary diagrams in_M* *covO to the commutativity of a corresponding diagram in S. |__| Scholium 2.3.The requirement in this theorem that ae(X; Y ; A) be an isomorphism whenever both X and Y are finite ^-products of objects in O was unfortunately overlooked in [21]. However, the map introduced in the proof of Proposition 5.2 of [21] is our map aecont(X; Y ; A). In the proof of that proposition, is show* *n to be an isomorphism under far broader conditions than those needed to ensure that the categories considered in [21] are symmetric monoidal. Remark 2.4.(a) If oS is in O, then the map ae(oS; Y ; A) is an isomorphism by Lemma 4.3(b) of [21]. Thus, the restriction imposed on oS in the theorem above * *is weaker than that imposed in [21]. (b) If O contains oS and is closed under finite ^-products, so that it is a s* *ym- metric monoidal subcategory of S, then both conditions on the maps ae(X; Y ; A) are satisfied, again by Lemma 4.3(b) of [21]. We conclude this section with a list of examples of pairs (S; O) satisfying t* *he hypotheses of Theorem 2.2. These examples serve a three-fold purpose. The next section is devoted to positive results giving conditions on a pair (S; O) which* * ensure that the associated functor category MO satisfies one or more of our axioms. T* *he power of those positive results is illustrated by the fact that they apply to s* *everal of the most important special cases of the examples below, and thus assure us that* * the associated functor categories are well-behaved. The limitation of our positive * *results is that they give sufficient, but not necessary, conditions for the good behavi* *or of MO . Some special cases of the examples below are used to provide a measure of * *this lack of necessity. Section 6, and most of the sections following it, describe f* *unctor 10 L. GAUNCE LEWIS, JR. categories which fail to satisfy our compatibility axioms. These badly behaved categories are also special cases of the examples below. Example 2.5. (a) Let R be a commutative ring, S be the category R-Mod of R-modules, and O be the full subcategory of S containing R as its only object. In this case, S is a symmetric monoidal closed category, and O is a symmetric monoidal closed subcategory. Of course, MO is just R-Mod , and Day's symmetric monoidal closed structure is identical to the standard one. In some sense, this example is frivolous in that we have applied a vast machine to R-Mod with its symmetric monoidal closed structure only to recover that category with the same structure. However, in our discussion of the connections between the structure * *of O and our compatibility axioms, this example nicely illustrates the benefits of* * a very well-behaved category O. (b) Let R be a commutative ring, G be a finite group, and S be the category R[G]-Mod of R[G]-modules. In this context, there are two reasonable choices for O. The smallest, which we denote O1, is the full subcategory of S containing R[* *G] as its only object. The other category, O2, is the full subcategory of S contai* *ning the n-fold direct sum of copies of R[G] for all n 1. Note that neither of these subcategories contains the unit for the tensor product on S, which is R with tr* *ivial G-action. The advantage of O2 over O1 is that O2 is closed under both the tensor product and the internal hom operations on S. Of course, MO1 and MO2 are just R[G]-Mod with its usual symmetric monoidal closed structure, so this example is just as frivolous as the previous one. However, it too serves to illustrate the* * relation between the structure of O and the compatibility axioms satisfied by MO . (c) Let G be a compact Lie group, U be a G-universe, and S be the equivariant stable category hGSU of G-spectra indexed on U (see chapter I of [23]). The obv* *ious choice for the associated category O is the stable orbit category OG (U). This * *is the full subcategory whose objects are suspension spectra 1UG=H+ associated to the orbits G=H derived from the closed subgroups H of G. The category OG (U) contains the unit for S. However, if G is nontrivial, OG (U) is closed under ne* *ither the tensor product nor the internal hom on S. A contravariant functor out of OG* * (U) is called a (G; U)-Mackey functor, and the category of such functors is denoted MG (U). If U is a complete G-universe, then OG (U) and MG (U) are abbreviated to OG and MG , respectively. If G is finite and U is complete, then (G; U)-Mack* *ey functors are the classical Mackey functors introduced by representation theoris* *ts for the study of induction theorems (see [8, 11, 16, 22] and Proposition V.9.9 * *of [23]). The case in which U is incomplete plays a role in the equivariant Hurewi* *cz and suspension theorems [18, 19] and in the study of change of universe functors in equivariant stable homotopy theory [20]. The category MG (U) is discussed in greater detail in sections 4, 6, 7, 8, 9, and 10. (d) If the group G in the previous example is finite, then the Burnside categ* *ory BG (U) is another choice for O. This is the full subcategory of S containing t* *he suspension spectra 1UX+ of the finite G-sets X. Note that OG (U) is a subcatego* *ry of BG (U). The advantage of BG (U) over OG (U) is that it is closed under the t* *ensor product operation on S. If the universe U is complete, then BG (U) is a symmetr* *ic monoidal closed subcategory of S. Its closed structure is even nicer than that * *of the category of finite dimensional vector spaces over a field in that objects in BG* * (U) are canonically self-dual. However, if U is incomplete, then BG (U) is not clo* *sed under the internal hom operation on S. As with OG (U), one is usually intereste* *d in WHEN PROJECTIVE DOES NOT IMPLY FLAT 11 contravariant functors out of BG (U). For either variance, the categories of fu* *nctors out of OG (U) and BG (U) are equivalent via the restriction functor induced by * *the inclusion of OG (U) into BG (U). Thus, we abuse notation and employ MG (U) to denote the category of contravariant functors from BG (U) to Ab. (e) The objects of the global Burnside category B* are the finite groups. If G and H are finite groups, then the set of morphisms from G to H in B* is the Grothendieck group of isomorphism classes of finite (G x H)-sets. The compositi* *on of a (G x H)-set X, regarded as a morphism from G to H, with an (H x K)-set Y , regarded as a morphism from H to a finite group K, is (X x Y )=H, where the passage to orbits is over the diagonal action of H on X x Y . The cartesian pro* *duct of groups makes B* into a symmetric monoidal Ab-category. There is an obvious duality functor D : B*___//Bop*which is the identity on objects and which sends* * the (G x H)-set X to itself regarded as an (H x G)-set. This duality functor provid* *es B* with a closed structure like that for finite dimensional vector spaces; the * *internal hom object associated to groups G and H is D(G)xH. Covariant additive functors from B* to Ab are the most structured kind of globally defined Mackey functors. The category of such functors is denoted M*. Most globally defined Mackey functors carry a much less rich structure than t* *hat carried by the functors in M*. These less structured Mackey functors are additi* *ve functors from some subcategory of B* into Ab. The subcategories of B* of intere* *st to us here can be described in terms of pairs (P; Q) of sets of integer primes.* * A (GxH)-set X is said to be a (P; Q)-set if, for each x 2 X, the G-isotropy subgr* *oup Gx of x has order divisible only by the primes in P and the H-isotropy subgroup Hx of x has order divisible only by the primes in Q. The subcategory B*(P; Q) of B* has the same objects as B*, but the set of morphisms from G to H in B*(P; Q) is the Grothendieck group of isomorphism classes of finite (G x H)-sets which a* *re also (P; Q)-sets. Observe that B*(P; Q)opis just B*(Q; P). The category B*(P; Q) inherits a symmetric monoidal structure from B*, but the internal hom on B* does not restrict to give B*(P; Q) a closed structure. In fact, B*(P; Q) is typicall* *y not a closed category. Covariant additive functors from B*(P; Q) to abelian groups * *are called global (P,Q)-Mackey functors. The category of such is denoted M*(P; Q). The category B*, and each of its subcategories B*(P; Q), can serve as both S * *and O in a pair (S; O) satisfying the hypotheses of Theorem 2.2. Note that this is * *our only example in which S need not be a closed category. The categories B*(P; Q) appear, under various names and in various guises, in [1, 2, 4, 5, 7, 9, 10, 12* *-14, 24- 26, 28, 29], and are discussed in greater detail in sections 5, 6 and 11. Remark 2.6.The pairs (S; O) from Examples 2.5(a), 2.5(d), and 2.5(e) satisfy the hypotheses of Theorem 2.2 by Remark 2.4 since, in each of these cases, O is a symmetric monoidal subcategory of S. Simple direct computations indicate that the pairs (S; O) from Example 2.5(b) satisfy the hypotheses of Theorem 2.2. If the group G is finite in Example 2.5(c), then the pairs introduced in that exam* *ple must satisfy the hypotheses of Theorem 2.2 since the resulting functor categori* *es MG (U) can be identified with the functor categories introduced in Example 2.5(* *d). If G is a nonfinite compact Lie group, then the argument needed to show that the pairs (S; O) of Example 2.5(c) satisfy the hypotheses of Theorem 2.2 is describ* *ed in Scholium 2.3. 12 L. GAUNCE LEWIS, JR. 3.Positive results on functor categories Throughout this section, (S; O) is assumed to be a pair of categories satisfy* *ing the hypotheses of Theorem 2.2. The focus of most of this article is on functor categories which do not satisfy our compatibility axioms. However, this section* * is devoted to positive results giving conditions on a pair (S; O) which ensure tha* *t the associated functor category MO satisfies our various compatibility axioms. Our compatibility axioms naturally fall into three pairs. Associated to each of th* *ese pairs of axioms there is a fairly natural closure property on the subcategory O which ensures that the associated functor category MO satisfies that pair of a* *xioms. Unfortunately, as various pairs drawn from Example 2.5 illustrate, these suffic* *ient conditions are far from necessary. Proposition 3.1.Let S be an Ab-category with a symmetric monoidal structure derived from a bilinear tensor product ^, and let O be a skeletally small full * *sub- category of S such that the pair (S; O) satisfies the hypotheses of Theorem 2.2* *. If the unit oS of S is in O, then the unit of MO is projective, and MO satisfies* * the axioms IPiP and IIiI. Proof.The unit for MO is the functor HoS. Since oS is in O, this functor is re* *pre- sentable and therefore projective. The rest of the proposition follows_immediat* *ely_ from Lemma 1.2. |__| In all the pairs (S; O) from Example 2.5 except those from Example 2.5(b), O contains the unit of S so that the proposition above applies. However, in the t* *wo pairs introduced in Example 2.5(b), the unit of S is definitely not in O. In t* *his case, both S and MO are the category R[G]-Mod of modules over the group ring R[G] of a finite group G. The unit for this category is the ring R with trivial* * G- action, which is typically neither projective nor injective in R[G]-Mod . Howev* *er, being the unit, R is necessarily internally projective in R[G]-Mod . Moreover,* * if R is a field, then, regarded as a trivial R[G]-module, it is internally injecti* *ve in R[G]-Mod . Thus, R[G]-Mod illustrates how badly behaved the category MO can be when the hypotheses of the proposition don't hold. On the other hand, if R is a field of characteristic prime to the order of the group G, then R with tri* *vial G-action is projective in R[G]-Mod so that R[G]-Mod satisfies the axioms IPiP and IIiI. Thus, the sufficient condition in the proposition is far from necessa* *ry. Proposition 3.2.Let S be an Ab-category with a symmetric monoidal structure derived from a bilinear tensor product ^, and let O be a skeletally small full * *subcat- egory of S such that the pair (S; O) satisfies the hypotheses of Theorem 2.2. I* *f O is closed under the tensor product operation on S, then the category MO satisf* *ies the axioms TPPP and PiIP. Proof.The representable functors form a set of projective generators for MO . T* *hus, to show that MO satisfies TPPP, it suffices to show that, if A and B are in O, then HA HB is projective. Lemma 4.4(b) of [21] gives that HA HB ~=HA^B . Since A ^ B is in O, HA^B is a representable functor in MO and so projective. * *The_ rest of the proposition follows from Proposition 1.3(a). |* *__| The sufficient condition given by this proposition is not strictly necessary * *since the categories R[G]-Mod and MG (U) associated to a finite group G in Examples 2.5(b) and 2.5(c) both satisfy TPPP and PiIP, but are of the form MO for a WHEN PROJECTIVE DOES NOT IMPLY FLAT 13 category O that is not closed under the tensor product operation on S. However, these examples are misleading in the sense that, in both cases, O is a proper s* *ub- category of a larger subcategory O0 of S which is closed under the tensor produ* *ct and whose associated functor category MO0 is equivalent to MO under the obvious restriction functor. Proposition 3.8 below provides a more convincing example of non-necessity. Proposition 3.3.Let S be a symmetric monoidal closed Ab-category, and let O be a skeletally small full subcategory of S such that the pair (S; O) satisfies* * the hypotheses of Theorem 2.2. If O is closed under the internal hom operation on S, then the category McovOsatisfies the axioms PiF and IiII. As with the previous proposition, the sufficient condition given by this prop* *o- sition is not strictly necessary since the category R[G]-Mod of Examples 2.5(b) satisfies PiF and IiII, but is a category of the form McovOfor a category O that is not closed under the internal hom operation on S. Another such example is provided by the category OG (U) of Example 2.5(c) in the special case where G is finite and U is complete. Again, however, these examples are misleading in the sense that, in each of them, O is a proper subcategory of a larger subcategory O0 of S which is closed under the internal hom operation and whose associated functor category MO0 is equivalent to MO under the obvious restriction functo* *r. Proposition 3.8 gives a more convincing example of non-necessity. Remark 3.4.In all of our examples derived from the equivariant stable category hGSU of G-spectra indexed on a G-universe U, the category S, which was assumed to be hGSU , can be replaced with its full subcategory S0 consisting of the G- spectra with the G-homotopy type of finite G-CW spectra. Equivariant Spanier- Whitehead duality provides a functor D : S0___//(S0)opwhich is an equivalence of symmetric monoidal closed categories between S0 and its opposite category. The internal hom in S0 associated to objects X and Y is just D(X) ^ Y . The category B* of Example 2.5(e) has a similar closed structure. Categories with this sort * *of symmetric monoidal closed structure are sometimes called *-autonomous categories (see, for example, [3]). Whenever S, or some full subcategory S0 of S containing O, has a closed structure of this sort, the category O is closed under the inte* *rnal hom operation on S if and only if Oop is analogously closed. Thus, the restrict* *ion of Proposition 3.3 to covariant functors is unnecessary in this special case. To prove the proposition above, we need to introduce an important endofunctor on McovO. Definition 3.5.Let S be a symmetric monoidal closed Ab -category, and let O be a skeletally small full subcategory of S such that the pair (S; O) satisfies* * the hypotheses of Theorem 2.2. Assume also that O is closed under the internal hom operation on S. If X and Y are in S, then denote the internal hom of this pair * *in S by < >For N in McovOand D in O, let ND be the functor in McovOgiven by ND (A) = N(< >). Lemma 3.6. Let D be in O. (a) The assignment of ND to N 2 McovOis an exact additive functor on McovO. (b) There is an isomorphism HcovD N ~=ND which is natural in both N 2 McovOand D 2 O. 14 L. GAUNCE LEWIS, JR. Proof.Clearly the assignment of ND to N and D is functorial and is additive in each of N and D separately. It is exact in N because the exactness of sequences in the functor category McovOis determined pointwise. On an object A of O, the isomorphism between HD N and ND is given by the composite Z B;C2O (HD N)(A) = HD (B) N(C) S(B ^ C; A) Z B;C2O = S(D; B) N(C) S(B ^ C; A) ZC2O ~= N(C) S(D ^ C; A) ZC2O ~= N(C) S(C; < >) ~=N(< >) = ND (A): Here, the first and third isomorphisms are given by Lemmas 4.3(b) and 4.3(c) of [21], respectively. The second isomorphism comes from the adjunction isomorphism making S a closed category. Clearly the composite above is natural in_A,_D, and N. |__| Remark 3.7.The construction ND is very closely related to the construction ND introduced in the context of Example 2.5(d) by Dress [8]. In fact, if the G-uni* *verse U of that example is complete and O is taken to be BG (U), then the very simple nature of the symmetric monoidal closed structure carried by BG (U) implies that the two constructions are isomorphic. Proof of Proposition 3.3.Note first that the adjunction making S a closed categ* *ory forces the tensor product on S to be bilinear. Since the representable functors* * form a set of projective generators for McovO, showing that McovOsatisfies PiF is ea* *sily reduced to showing that the functor HcovD? is exact for any D 2 O. This is established in Lemma 3.6. Since McovOhas enough projectives, it must also satis* *fy_ IiII by Proposition 1.3(b). |__| The following positive result about the category of Mackey functors for the c* *ircle group S1 stands in sharp contrast to the host of negative results contained in section 6. It also provides clear evidence that the rather formal sufficient co* *nditions contained in Propositions 3.2 and 3.3 above are very far from necessary. This r* *esult is proven in section 10. Proposition 3.8.(a) The category MS1 of S1-Mackey functors satisfies the six axioms PiF, PiIP, IiII, IPiP, IIiI, and TPPP. (b) Let O0be a full subcategory of the complete S1-stable category which cont* *ains the stable orbit category OS1. If O0 is closed under either ^-products or funct* *ion objects, then the restriction functor McontO0//_McontOS1= MS1 is not an equivalence of categories. WHEN PROJECTIVE DOES NOT IMPLY FLAT 15 4. An introduction to (G; U)-Mackey functors Let G be a compact Lie group, and U be a possibly incomplete G-universe. This section is intended to provide a basic introduction to the categories OG (* *U) and MG (U). The results presented here provide both some sense of why these categories have something to do with the classical notion of a Mackey functor, * *and some intuition about why, for various choices of G and U, the category MG (U) fails to satisfy our various axioms. These results also form the foundation for* * the proofs, given in later sections, of Theorems 3.8(b), 6.1, 6.5, and 6.9. The structure of the morphism sets of the stable orbit category OG (U) of Exa* *m- ple 2.5(c) is described in Corollary 5.3(b) of [15] and Corollary 3.2 of [21]. * *An object of OG (U) is the suspension spectrum 1UG=H+ of an orbit G=H of G; however, to avoid unnecessary notational complexity, we hereafter denote this object by G=H. The set of morphisms in OG (U) from G=H to G=K is a free abelian group whose generators are certain allowed equivalence classes of diagrams of the form G=H ooff_G=J _fi//_G=K; in which ff : G=J ___//G=H and fi : G=J ___//G=K are space-level G-maps. Two such diagrams are equivalent if there is a G-homeomorphism fl : G=J ___//G=J0 making the space-level diagram ffmG=JmQQfiQ vvmmmm | QQQ(( G=H hhQ fl|| G=K66 QQQ0QQ|fflffl0mmmmmm ff G=J0 fi commute up to G-homotopy. The morphism in OG (U) represented by the diagram G=H ooff_G=J _fi//_G=K is the composite of the map, denoted o(ff), which is represented by the diagram 1G=J G=H ooff_G=J ______//G=J and the map, denoted ae(fi), which is represented by the diagram 1G=J fi G=J oo____ G=J ___//_G=K: If J is a subgroup of H, hereafter denoted J H, then there is a canonical G-map ssJH: G=J ___//G=Hwhich takes the identity coset eJ of G=J to the identity coset eH of G=H. The associated maps o(ssJH): G=H ___//G=Jand ae(ssJH): G=J ___//G=H in OG (U) are denoted oJHand aeJH, respectively. The connection between the category OG (U) and the classical notion of a Mack* *ey functor introduced by representation theorists can be seen from this sketch of the structure of the morphism sets of OG (U). A contravariant functor M from OG (U) to Ab assigns an abelian group M(G=H) to each orbit G=H of G. This abelian group is the value of the Mackey functor M at the subgroup H of G and is often denoted M(H) rather than M(G=H). The map o(ff) associated to a G- map ff : G=J ___//G=Hinduces an induction (or transfer) map from M(G=J) to M(G=H), and the map ae(ff) induces a restriction map from M(G=H) to M(G=J). 16 L. GAUNCE LEWIS, JR. Not every equivalence class of diagrams represents a generator of a morphism group in OG (U). Each equivalence class of diagrams contains at least one diagr* *am of the form ssJH fi G=H oo___G=J ___//_G=K: The allowed equivalence classes are those having a representative of the above special type in which the subgroup J of H satisfies both the condition that H=J embeds in the universe U as an H-space and the condition that the index of J in its H-normalizer NH J is finite. For our purposes, the essential property o* *f a complete G-universe U is that, for such a universe, the embedding condition on H=J is always satisfied. If the universe U is contained in a larger universe U0, then OG (U) can be id* *enti- fied with a subcategory of OG (U0). In particular, since any G-universe is isom* *orphic to a subuniverse of a complete G-universe, OG (U) is always a subcategory of OG* * . Viewing OG (U) as a subcategory of OG reveals a difference between the two re- strictions imposed on the equivalence classes which index the generators of the morphism sets of OG (U). If H=J does not embed in U, but the finiteness conditi* *on holds, then the diagram ssJH fi G=H oo___G=J ___//_G=K; represents a morphism of OG which has been omitted from OG (U) by our choice of U. However, if the index of J in NH J is not finite, then this diagram does * *not represent a generator in OG (U) for any U. If the index of J in NH J is not finite, then for certain geometric reasons i* *t is best to think of the diagram ssJH fi G=H oo___G=J ___//_G=K; as representing a morphism in OG (U) which, in some vague sense, should have be* *en a generator, but has instead been identified with the zero map. This distincti* *on between a morphism that has been omitted from OG (U) and one that has been identified with zero is important for understanding composition of morphisms in OG (U). It is also important for understanding the homological anomalies descri* *bed in this paper. If the group G is finite and the G-universe U is incomplete, the* *n the morphisms missing from OG (U) may cause the axioms PiF and IiII to fail in the functor category MG (U). If the group G is a nonfinite compact Lie group, then * *the "unexpected" zero morphisms in OG may cause the axioms TPPP, PiIP, TPPP, and IiII to fail in the functor category MG . The arguments establishing these * *two types of failures are rather different since the failures happen for quite diff* *erent reasons. WHEN PROJECTIVE DOES NOT IMPLY FLAT 17 5.An introduction to globally defined Mackey functors Here, we examine the structure of the category B*(P; Q) associated to the cat- egory of global (P; Q)-Mackey functors. Thus, throughout this section, all grou* *ps are assumed to be finite, and P, Q, and R are assumed to be sets of primes. The observations presented here should provide both a sense of why the functors out of B*(P; Q) should be regarded as globally defined Mackey functors and an under- standing of the way in which the choice of the sets P and Q controls the level * *of structure carried by those functors. These remarks also lay the foundation for * *the proof of Theorem 6.10 in section 11. Recall from Example 2.5(e) that the set of morphisms from G to H in B*(P; Q) is the Grothendieck group of isomorphism classes of (G x H)-sets which are also (P; Q)-sets. This is a free abelian group whose generators are the isomorphism classes of (G x H)-orbits which are (P; Q)-sets. For any (G x H)-orbit (G x H)=* *J, the inclusion of J into G x H can be composed with the projections from G x H to G and H to produce group homomorphisms ff : J __//_G and fi : J __//_H. If M is a covariant functor out of B*(P; Q), then the appropriate intuitive understandi* *ng of the map from M(G) to M(H) induced by the (G x H)-orbit (G x H)=J is that this map is the composite of a restriction map from M(G) to M(J) associated to the homomorphism ff and an induction map from M(J) to M(H) associated to the homomorphism fi. The imposed constraint that the (G x H)-set (G x H)=J must be a (P; Q)-set translates easily into the restriction that the kernel of ff mu* *st be a Q-group (that is, have order divisible only by the primes in Q) and the kernel * *of fi must be a P-group. Hereafter, we refer to a homomorphism whose kernel is a P-group as a homomorphism with P-kernel. The most common choices for P and Q are the empty set ; of primes and the set of all primes, which we denote by 1. If P = ;, then the trivial group is the only P-group. On the other hand, if P = 1, then all finite groups are P-groups. Thus, a homomorphism with ;-kernel is a monomorphism, and every group homomorphism has 1-kernel. At the level of global Mackey functors, this means, for example, that a global (;; 1)-Mackey functor has induction maps only for injective group homomorphisms, but restriction maps for all homomorphisms. Note that B*(1; 1) = B*. In order to justify the intuitive description of the maps in B*(P; Q) present* *ed above, we must first identify the morphisms in B*(P; Q) which are derived direc* *tly from ordinary group homomorphisms. Definition 5.1.Let ff : H __//_G be a group homomorphism. Then o(ff) is the set G considered as a (H x G)-set with action given by (h; g)x = ff(h)xg-1; for (h; g) 2 H x G and x 2 G. Also, ae(ff) is the set G considered as an (G x H* *)-set with action given by (g; h)x = gxff(h-1); for (g; h) 2 G x H and x 2 G. Note that, if ff has R-kernel, then o(ff) is an (R; ;)-set, and ae(ff) is an (;; R)-set. If R P, then o(ff) is a generator of * *the free abelian group of morphisms from H to G in B*(P; Q). It should be thought of as the induction, or transfer, map associated to ff. Similarly, if R Q, then ae(f* *f) is a generator of the free abelian group of morphisms from G to H in B*(P; Q). It 18 L. GAUNCE LEWIS, JR. should be thought of as the restriction map associated to ff. If ff is the incl* *usion of a subgroup H into the group G, then o(ff) and ae(ff) are denoted oHG and aeH* *G, respectively. If G = H and ff is the identity map, then o(ff) and ae(ff) are eq* *ual and serve as the identity map 1G of G in B*(P; Q). If ff : J __//_G and fi : J __//_H are the group homomorphisms associated in * *our introductory remarks to the (G x H)-orbit (G x H)=J, then it is fairly easy to * *see that, in B*, (GxH)=J is the composite o(fi)Oae(ff). Further, the orbit (GxH)=J * *is a (P; Q)-set precisely when ff has Q-kernel and fi has P-kernel. These observat* *ions allows us to think of the generators of the free abelian group B*(P; Q)(G; H) as equivalence classes of diagrams of the form G ooff_J _fi//_H; in which ff is a group homomorphism with Q-kernel, fi is a group homomorphism with P-kernel, and the induced homomorphism (ff; fi): J __//_G x His a monomor- phism. This description of the morphism sets of B*(P; Q) plays a central role in the proof of Theorem 6.10. Three notes of caution are, however, necessary. First, s* *ince M*(P; Q) is the category of covariant functors out of B*(P; Q), the Mackey func* *tor interpretation of this diagram is the reverse of the interpretation associated * *to the analogous diagrams for OG (U). Here, ff is plays the role of the restriction ma* *p, and fi plays the role of the induction map. Second, the equivalence relation which * *must be imposed on these diagrams cannot be described as neatly as the corresponding relation for OG (U). For our purposes, it suffices to say that the diagram 0 fi0 G ooff_J0____//H; is equivalent to the unprimed diagram above if and only if the two (G x H)-sets o(fi) O ae(ff) and o(fi0) O ae(ff0) are isomorphic. The third note is that comp* *osition in B*(P; Q) cannot be computed using pullback diagrams like those used to compute composition in OG . Remark 5.2.There is an alternative approach to globally defined Mackey functors in which the category B*(P; Q) is replaced by a somewhat larger category B0*(P;* * Q) whose objects are finite groupoids rather than finite groups. In many ways, the relation between B*(P; Q) and B0*(P; Q) is similar to that between OG (U) and BG (U). In particular, the categories B*(P; Q) and B0*(P; Q) are similar enough that the associated categories of covariant functors into Ab are equivalent und* *er the restriction functor derived from the inclusion of B*(P; Q) into B0*(P; Q). * *The larger category B0*(P; Q) is somewhat more difficult to define because one has * *to deal with functors from finite groupoids into finite sets rather than sets carr* *ying actions by finite groups. However, it has the advantages of an easily described equivalence relation on the generators of its morphism sets and a simple pullba* *ck formula for its composition. Further advantages of B0*(P; Q) over B*(P; Q) are noted at the end of this section and in Remark 6.11(b). One technical result describing the behavior of composition in B*(P; Q) in the context of cartesian products of groups is needed for the proof of Theorem 6.10. The information this result provides about B*(P; Q) is similar to that provided about OG (U) by Lemma 3.3 of [21]. WHEN PROJECTIVE DOES NOT IMPLY FLAT 19 Lemma 5.3. Let : J __//_P , : J __//_Q and i : Q0 __//_Q be homomorphisms between finite groups such that has Q-kernel and i has P-kernel. Assume that the integers {ni}1im and the diagrams Q0ooffi_Ki_fii//_J; for 1 i m, are chosen so that the composite ae() O o(i) in B*(P; Q) is the sum over i of ni times the generator of B*(P; Q)(Q0; J) represented by the ith diagram. Then the composite ae(( ; )) O o(1 x i) of the restriction map associa* *ted to the homomorphism ( ; ): J __//_P x Qand the transfer map associated to the homomorphism 1 x i: P x Q0___//P x Qis the sum over i of nitimes the generator of B*(P; Q)(P x Q0; J) represented by the diagram ( Ofii;ffi)fii P x Q0oo_______ Ki____//J: This result can be proven by brute force computations with the obvious finite sets carrying the appropriate group actions. A further advantage of the category B0*(P; Q) described in Remark 5.2 is that the simple pullback formula for compo- sition in B0*(P; Q) trivializes the proof of this lemma. 20 L. GAUNCE LEWIS, JR. 6. A bestiary of symmetric monoidal closed abelian categories This section contains a catalog of functor categories which fail to satisfy t* *he hy- potheses of Propositions 3.2 and 3.3 and also fail to satisfy various of our co* *mpati- bility axioms. These badly behaved categories come from the families of categor* *ies MG (U) of Example 2.5(c) and M*(P; Q) of Example 2.5(e). The results stated in this section about the misbehavior of our functor categories are proven in t* *he subsequent sections. Let G be a compact Lie group, and U be a possibly incomplete G-universe. The hypotheses of our theorems about the homological misbehavior of the category MG (U) of (G; U)-Mackey functors are certainly not as weak as they could be. Nevertheless, they are weak enough to produce an almost overwhelming supply of badly behaved categories. Theorem 6.1. Let G be a finite group, U be a G-universe, and C D H be subgroups of G such that (i)C is normal in D and D=C ~=Z=p for some prime p (ii)H=C embeds as an H-space in U (iii)H=D does not embed as an H-space in U. Then HG=D is not flat in MG (U), and MG (U) satisfies neither PiF nor IiII. Remark 6.2.One way of understanding the misbehavior of MG (U) implied by The- orem 6.1 is that, if the subgroups C D H satisfy the hypotheses of the theore* *m, then the induction maps oCDand oCHare contained in the subcategory OG (U) of OG* * , but the induction map oDHis not. Thus, even though the map oCHcan be factored as the composite oCDO oDHin OG , it cannot be so factored in OG (U). This failure * *of oCH to factor properly in OG (U) seems to be the source of the homological misbehav* *ior of the category MG (U). Example 6.3. Let p be a prime, G = H = Z=p2, D = Z=p Z=p2, and C be the trivial subgroup of G. Let U be a G-universe whose only irreducible summands are the trivial irreducible G-representation and a free irreducible G-represent* *ation. Then G=C embeds in U as a G-space, but G=D does not. By the theorem, the projective Mackey functor HG=D is not flat in MG (U). Beyond the context of this paper, the special significance of this universe is that it has natural connect* *ions to the study of semi-free actions of G. Corollary 6.4.Let p be a prime, G be a finite p-group, and U be a G-universe. Then the category MG (U) cannot satisfy either PiF or IiII unless, for each sub- group H of G, the set {K : K H and H=K does not embed as an H-space in}U is closed under passage to subgroups. In particular, if the free orbit G=e emb* *eds in U as a G-space, then MG (U) cannot satisfy either PiF or IiII unless every G-orbit G=K embeds in U. The statement of our second main theorem requires a bit of additional notatio* *n. The Weyl group NG H=H of a subgroup H of G is denoted WG H, and the set of G-conjugates of H G is denoted (H)G . If H and K are subgroups of G, then the notation (K)G (H)G indicates that K is subconjugate to H in G. Theorem 6.5. Let G be a compact Lie group, and U be a complete G-universe. Also, let C D H be subgroups of G such that WHEN PROJECTIVE DOES NOT IMPLY FLAT 21 (i)C is normal in D and D=C is a finite p-group for some prime p (ii)WH D is finite (iii)for every K < D such that (C)G (K)G , WH K is not finite. Then HG=D is not flat in MG , and MG satisfies neither PiF nor IiII. Remark 6.6.One way of understanding the misbehavior of MG implied by Theorem 6.5 is that, if the subgroups C D H satisfy the hypotheses of the theorem, th* *en the induction maps oCDand oDH are generators of the morphism sets of OG . Their composite is the induction map oCH associated to the containment C H. This map oCHshould also be a generator of the appropriate morphism set. However, sin* *ce WH C is not finite, oCHis actually the zero map. This vanishing of the composit* *e of two generators of the morphism sets of OG seems to be the source of the homolog* *ical misbehavior of the category MG (U). Example 6.7. (a) Let G and H both be the orthogonal group O(2), D be the dihedral group of order 2n (for n 3) regarded as a subgroup of G, and C be the cyclic group of order n regarded as a subgroup of D. Then C is normal in G, and D=C ~=Z=2. Moreover, NG D is the dihedral group of order 4n, so WG D is finite. If K is a proper subgroup of D such that (C)G (K)G , then K = C, and WG K is G=C, which is not finite. Thus, the projective Mackey functor HG=D is not flat * *in MG , and MG satisfies neither PiF nor IiII. (b) If C D H are subgroups of G which satisfy the hypotheses of the theorem and ffl : G0___//G is a surjective group homomorphism, then the subgrou* *ps C0 = ffl-1(C), D0 = ffl-1(D), and H0 = ffl-1(H) are subgroups of G0 which also satisfy the hypotheses of the theorem. Therefore, MG0 does not satisfy the axio* *ms PiF and IiII. (c) We would like to argue that, if C D H are subgroups of G which satisfy the hypotheses of the theorem and G G00, then C, D, and H, regarded as subgroups of G00, satisfy the hypotheses of the theorem. However, condition (iii) in the hypotheses of the theorem might fail since there might be a subgro* *up K of D such that C was G00-subconjugate to K, but not G-subconjugate to K. Nevertheless, if D=C ~=Z=p and C is the unique subgroup of D of index p, as in part (a) of this example, then the triple C D H, regarded as subgroups of G00, must still satisfy the hypotheses of the theorem. In fact, it suffices to * *assume that D=C ~=Z=p and that every subgroup C0 of index p in D is H-conjugate to C. Combining this observation with part (a) of this example, we see that, if a com* *pact Lie group G contains the orthogonal group O(2), then MG satisfies neither PiF nor IiII. Combining the various parts of the example above yields the following corolla* *ry of Theorem 6.5. Corollary 6.8.If G is any one of O(m); SU(m); U(m); form 2; SO(n); Spin(n); forn 3; Sp(q); forq 1; then MG satisfies neither PiF nor IiII. 22 L. GAUNCE LEWIS, JR. The family of categories MG , for G a compact Lie group, also provides us with some examples of symmetric monoidal closed categories which fail to satisfy TPPP and PiIP. Theorem 6.9. Let G be the orthogonal group O(m), for m 2, or the special orthogonal group SO(n), for n 3. Then MG satisfies neither TPPP nor PiIP. Now we turn to the context of global Mackey functors. Recall that, in Example 2.5(e), we associated a category M*(P; Q) of global (P; Q)-Mackey functors to e* *ach pair (P; Q) of sets of integer primes. Theorem 6.10. Let P and Q be sets of integer primes. If the prime p is not in P, then the representable functor in M*(P; Q) associated to the cyclic group Z=p is not flat in M*(P; Q). Thus, M*(P; Q) satisfies neither PiF nor IiII. Remark 6.11.(a) Recall that the empty set of primes and the set of all primes a* *re denoted ; and 1, respectively. The three types of global Mackey functors that appear most often in the literature seem to be (;; ;)-, (;; 1)- and (1; ;)-Mack* *ey functors (see, for example, [1, 2, 4, 5, 7, 9, 10, 12-14, 24-26, 28, 29]). The* *orem 6.10 indicates that the first two of these categories of global Mackey functors are * *badly behaved. (b) As noted in Remark 5.2, the category B*(P; Q) used to define global (P; Q* *)- Mackey functors can be replaced by a somewhat larger category B0*(P; Q) whose objects are finite groupoids. One advantage of this replacement is that, for a* *ny set of primes Q, the category B0*(1; Q) is a symmetric monoidal closed category. Proposition 3.3 therefore implies that M*(1; Q) satisfies PiF and IiII. Thus, t* *he theorem above gives sharp conditions under which M*(P; Q) fails to satisfy PiF and IiII. WHEN PROJECTIVE DOES NOT IMPLY FLAT 23 7.Mackey functors for incomplete universes and the proof of Theorem 6.1 Each of Theorems 6.1, 6.5, and 6.10 asserts that a projective object P in a certain functor category MO is not flat. The proofs of these results follow th* *e same pattern. It suffices to show that, for some object A in O, the functor sending * *M in MO to (P M)(A) does not preserve monomorphisms. To show this, we restrict the domain of this functor to a nice full subcategory M0of MO . We then identify a natural direct summand Z : M0 __//_Abof the restricted functor, and show that the summand fails to preserve monomorphisms. In this section, we prove Theorem 6.1 by carrying out the appropriate special case of this general program. Thus, throughout the section, G is a finite group, U is a G-universe, and C D H are subgroups of G. The appropriate projective object P in this context is the representable functor HG=D in the category MG (U), and the appropriate object A of O = OG (U) is the orbit G=H. We begin the proof of Theorem 6.1 by introducing the appropriate subcategory M0of MG (U), and the appropriate direct summand functor Z. Definition 7.1.(a)A Mackey functor M in MG (U) is concentrated over C if, for K G, M(G=K) = 0 unless (C)G (K)G . The subcategory M0appropriate for the proof of Theorem 6.1 is the full subcategory of MG (U) consisting of Mackey functors which are concentrated over C; this subcategory is denoted MCG(U). (b) The Weyl group WG C is contained in the morphism set OG (U)(G=C; G=C) as the set of maps ae(fi) associated to the endomorphisms fi of the G-set G=C. * *The group WG C therefore acts on the value M(G=C) of a Mackey functor M at G=C. Let ZC : MG (U) __//_Abbe the functor sending M in MG (U) to the quotient group M(G=C)=WD C of M(G=C). The technical foundation of the proof of Theorem 6.1 is the following result: Proposition 7.2.Let G be a finite group, U be a G-universe, and C D H be subgroups of G such that (i)C is normal in D, and D=C ~=Z=p for some prime p (ii)H=C embeds as an H-space in U (iii)H=D does not embed as an H-space in U. Then, for M in MCG(U), ZC (M) splits off from (HG=D M)(G=H) as a natural direct summand. Before proving this result, we show how it can be used to complete the proof * *of our theorem. Proof of Theorem 6.1.Assume that C is normal in D, and that D=C ~=Z=p for some prime p. Also assume that H=C embeds as an H-space in U, but that H=D does not so embed. We must construct a monomorphism : A __//_B which is not preserved by the functor ZC. Define the object B of MG (U) by the exact sequence M "aeKC1 HG=K Z=p ________//HG=C Z=p __//_B __//_0: K : MG (U) __//_MG (U)preserve epimorphisms. Associated to any short exact sequence 0 __//_M0___//M _ffl//_M00__//_0 in MG (U), there is a fibre sequence K(M0; 0) _"//_K(M; 0) _"ffl//_K(M00; 0) of zero-dimensional equivariant Eilenberg-Mac Lane spectra in the stable catego* *ry of G-spectra indexed on U. Proposition 5.2 of [21] allows us to use the long ex* *act homotopy sequence derived from this fibre sequence to investigate the exactness* * of the functor . That proposition gives an isomorphism (G=J) ~=[G=J+ ^ G=H+ ; K(M; 0)]G which is natural in M 2 MG (U) and G=H; G=J 2 OG (U). Here, [ ; ]G is used to denote the morphism sets in the G-stable homotopy category. This isomorphism identifies the map ffl*(G=J) 00 (G=J) ________// (G=J); which must be an epimorphism if MG is to satisfy TPPP and PiIP, with the map [G=J+ ^ G=H+ ; K(M; 0)]G _"ffl*//_[G=J+ ^ G=H+ ; K(M00; 0)]G : From the long exact homotopy sequence, it follows that "ffl*is an epimorphism i* *f and only if the map [G=J+ ^ G=H+ ; K(M0; 0)]G _"G_//[G=J+ ^ G=H+ ; K(M; 0)]G is a monomorphism. The change of group isomorphisms given in section II.4 of [2* *3] identify this map with the map [G=H+ ; K(M0; 0)]J _"J//_[G=H+ ; K(M; 0)]J: We have now reformulated the question of whether MG (U) satisfies TPPP and PiIP into the form needed for our discussion of S1 in section 10. Proposition 9.1.Let G be a compact Lie group, and U be a G-universe. Then the category MG (U) satisfies TPPP and PiIP if and only if, for every monomorphism : M0 __//_M in MG (U) and every pair H, J of subgroups of G, the map [G=H+ ; K(M0; 0)]J _"J//_[G=H+ ; K(M; 0)]J: 34 L. GAUNCE LEWIS, JR. induced by is a monomorphism. For the proof of Theorem 6.9, it suffices to exhibit a monomorphism : M0 __/* */_M in MG and two subgroups H and J of G for which the map "Jis not a monomor- phism. It is possible to select these subgroups H and J so that H J. For such a pair of subgroups, the identity coset eH of G=H is J-invariant and so provide* *s a basepoint for G=H regarded as a J-space. The existence of this basepoint implies that G=H+ is equivalent to G=H _ S0 in the J-stable category. Thus, [G=H+ ; K(M0; 0)]J ~= [G=H _ S0; K(M0; 0)]J ~= [G=H; K(M0; 0)]J [S0; K(M0; 0)]J: Similar observations apply to [G=H+ ; K(M; 0)]J. The dimension axiom implies that [S0; K(M0; 0)]J and [S0; K(M; 0)]J are both zero. The map which we wish to show is not a monomorphism can therefore be identified with the map [G=H; K(M0; 0)]J _^J//_[G=H; K(M; 0)]J: The task of showing that MG fails to satisfy TPPP and PiIP is now reduced to the two problems of analyzing the G-orbit G=H as a J-space so that the map ^Jcan be understood and of selecting a short exact sequence in MG for which this map is not a monomorphism. We address these two problems only for the case in which G is either O(n) or SO(n), and the universe U is complete. If G is either O(n) or SO(n), then the subgroup J in the analysis above can be taken to be H. For either choice of G, our argument employs an appropriately ch* *o- sen proper subgroup K of H. If G = O(n), then H = O(n - 1) and K = O(n - 2). If G = SO(n), then H = SO(n - 1) and K = SO(n - 2). Here, O(1) = Z=2, and O(0) = SO(1) = e. For both O(n) and SO(n), standard geometry gives nonequiv- ariant homeomorphisms G=H ~= Sn-1 and H=K ~= Sn-2. Moreover, for either choice of G, the G-orbit G=H, considered as an H-space, may be identified with the unreduced suspension S(H=K) of the H-orbit H=K. Under this identification, the map ^Jbecomes the map [S(H=K); K(M0; 0)]H _^_//[S(H=K); K(M; 0)]H : The unreduced suspension S(H=K) fits into a cofibre sequence S0 __//_S(H=K) __//_H=K+ ___//S1 of H-spaces. From this cofibre sequence, we obtain the diagram 00 [S1; K(M0; 0)]H__________//[S1; K(M; 0)]H | | | | fflffl| 0 fflffl| [H=K+ ; K(M0; 0)]H _____//[H=K+ ; K(M; 0)]H | | | | fflffl| ^ fflffl| [S(H=K); K(M0; 0)]H ____//_[S(H=K); K(M; 0)]H | | | | fflffl| fflffl| [S0; K(M0; 0)]H [S0; K(M; 0)]H k k 0 0 WHEN PROJECTIVE DOES NOT IMPLY FLAT 35 in which the columns are exact sequences derived from our cofibre sequence. Con- sider the pullback diagram P ________________//[S1; K(M; 0)]H | | | | fflffl| 0 fflffl| [H=K+ ; K(M0; 0)]H ____//_[H=K+ ; K(M; 0)]H associated with the top rectangle of the larger diagram. The commutativity of t* *he top rectangle in the larger diagram implies the existence of a map OE : [S1; K(M0; 0)]H__//_P: Assume that the map in these two diagrams is a monomorphism. A simple diagram chase then gives that the map ^of interest to us fails to be a monomorp* *hism if and only if the map OE fails to be an epimorphism. The change of group isomorphisms from section II.4 of [23] allow us to identi* *fy the top rectangle of our larger diagram with the diagram (G=H) M0(G=H) _______//_M(G=H) M0(aeKH)|| |M(aeKH)| fflffl|(G=K) fflffl| M0(G=K) _______//M(G=K): We can think of P as the pullback associated with this rectangle and of OE as a map from M0(G=H) into P . Our geometric argument has therefore reduced the question of whether the functor : M __//_M preserves epimorphisms to the question of whether there exists a monomorphism M0 ___//M in MG for which the map M(aeKH): M(G=H) ___//M(G=K) is a monomorphism and the map OE : M0(G=H) __//_Pis not an epimorphism. Let M be the Mackey functor Z_ given by Proposition V.9.10 of [23]. For any subgroup J of G, M(G=J) = Z. Further, for L J, then the restriction map M(aeLJ): M(G=J) __//_M(G=L)is the identity map, and the induction map M(oLJ): M(G=L) __//_M(G=J)is multiplication by the nonequivariant Euler char- acteristic O(J=L). Let M0 be the image of the canonical map HG=K M ___//M. By Proposition 5.5 of [21], the map (HG=K M)(G=K) ___//M(G=K) is a split epi- morphism. Therefore, M0(G=K) = M(G=K), and the map (G=K) is the identity map. The image of the map (HG=K M)(G=H) ___//M(G=H) is contained in the image of the transfer map M(oKH) : M(G=K) ___//M(G=H) by Proposition 5.7 of [21]. This transfer map is multiplication by the nonequivariant Euler character* *is- tic of H=K ~=Sn-2, which is either 2 or 0. Thus, M0(G=H) is either Z or 0. If M0(G=H) = Z, then the map (G=H) : M0(G=H) __//_M(G=H) is multiplication by 2. In either case, the map OE : M0(G=H) __//_P is obviously not an epimorph* *ism. Thus, the functor : M __//_M does not preserve epimorphisms, and the category MG satisfies neither TPPP nor PiIP. 36 L. GAUNCE LEWIS, JR. 10. S1-Mackey functors and the proof of Theorem 3.8 To prove Theorem 3.8, we must show that MS1 satisfies all six of the axioms PiF, PiIP, IiII, IPiP, IIiI, and TPPP. It follows immediately from Proposition 3.1 that it satisfies IPiP and IIiI. Tools from equivariant stable homotopy the* *ory are used to show that it satisfies the other four axioms. Proposition 9.1 provi* *des the means of applying these tools to the question of whether MS1 satisfies TPPP and PiIP. This section begins with an analogous proposition providing a method for applying these tools to the question of whether the category MG (U) associa* *ted to a compact Lie group G and a G-universe U satisfies PiF and IiII. After provi* *ng this proposition, we specialize to the case in which G = S1 and U is a complete G-universe, and prove Theorem 3.8. As in section 9, we assume here that : M0 ___//M is a monomorphism in MG (U), and ": K(M0; 0)___//K(M; 0)is the induced map between the equivariant Eilenberg-Mac Lane spectra associated to M0 and M. Proposition 10.1.Let G be a compact Lie group, and U be a G-universe. Then the category MG (U) satisfies PiF and IiII if and only if, for every monomorphi* *sm : M0 __//_M in MG (U) and every pair H, J of subgroups of G, the map [S0; G=H+ ^ K(M0; 0)]J _J_//_[S0; G=H+ ^ K(M; 0)]J induced by is a monomorphism. Proof.To prove that MG (U) satisfies PiF and IiII, it suffices to show that, for every monomorphism : M0 __//_M in MG (U) and every pair H and J of subgroups of G, the map (1 )(G=J): (HG=H M0)(G=J) ___//(HG=H M)(G=J) is a monomorphism. Proposition 5.2 of [21] provides an isomorphism (HG=H M)(G=J) ~=[G=J+ ; G=H+ ^ K(M; 0)]G which is natural in M 2 MG (U) and G=H; G=J 2 OG (U). Under this isomorphism, the map (1 )(G=J) is identified with the map [G=J+ ; G=H+ ^ K(M0; 0)]G _G__//[G=J+ ; G=H+ ^ K(M; 0)]G induced by the map ": K(M0; 0)___//K(M; 0). The change of group isomorphisms given in section II.4 of [23] allow us to identify the map G with the map_J of* * the proposition. |__| Henceforth, we assume that G = S1 and that the universe U is complete. The remainder of this section is devoted to the proofs of the two parts of Theorem * *3.8. Proof of Theorem 3.8(a).Propositions 3.1, 9.1 and 10.1 reduce the proof of this part of the theorem to showing that, for every monomorphism : M0 __//_M in MG and every pair H and J of subgroups of G = S1, the two maps [G=H+ ; K(M0; 0)]J _"J//_[G=H+ ; K(M; 0)]J: and [S0; G=H+ ^ K(M0; 0)]J _J_//_[S0; G=H+ ^ K(M; 0)]J WHEN PROJECTIVE DOES NOT IMPLY FLAT 37 are monomorphisms. For H = G, the dimension axiom indicates that the morphism set [G=G+ ; K(M0; 0)]J, and the analogous set for M, are both zero. Thus, "Jis trivially a monomorphism. Further, the isomorphism [S0; G=G+ ^ K(M0; 0)]J ~=M(G=J); and the analogous isomorphism for M, identify the map J with the map (G=J) : M0(G=J) ___//M(G=J); which is assumed to be a monomorphism. Thus, we may assume that H 6= G. If J = G and H 6= G, then both "Jand J have vanishing range and domain, and so are monomorphisms. For "J, this follows directly from the dimension axiom. To see this for J , note that in the equivariant stable category G=H+ ^ K(M; 0) ca* *n be identified with a G-CW spectrum whose zero skeleton is a wedge _iG=Ki+of orbits G=Ki associated to certain subgroups Ki of H. A simple connectivity argument gives that the map M [S0; G=Ki+]G ~=[S0; _iG=Ki+]G ___//[S0; G=H+ ^ K(M; 0)]G i induced by the inclusion of this zero skeleton is an epimorphism. Thus, to show that [S0; G=H+ ^ K(M; 0)]G is zero, it suffices to argue that [S0; G=Ki+]G is z* *ero. Corollary 3.2 of [21] gives that [S0; G=Ki+]G = OG (G=G; G=Ki) is a free abelian group whose generators are equivalence classes of diagrams of the form G=G ooff_G=L _fi//_G=Ki in which WG L is finite. However, in any such diagram, L Ki H. Since H 6= G, L must be a finite cyclic group, and so WG L cannot be finite. Thus, there are * *no generators, and OG (G=G; G=Ki) = 0. Analogously, the domain of the map J is zero. We can now assume that neither H nor J is equal to G. If J H, then the identity coset eH of G=H is J-invariant, and so provides G=H, considered as a J-space, with a basepoint. Thus, in the J-stable category, G=H+ ~=G=H _ S0 = S1_S0, where S1 is assumed to have trivial J-action. This decomposition of G=H+ and the dimension axiom provide isomorphisms [G=H+ ; K(M; 0)]J ~=[S1; K(M; 0)]J = M(G=J) and [S0; G=H+ ^ K(M; 0)]J ~=[S0; S0 ^ K(M; 0)]J = M(G=J): These isomorphisms, and the analogous ones for M0, identify the maps "Jand J with the map (G=J) : M0(G=J) ___//M(G=J), which is assumed to be a monomor- phism. We can now assume that H; J 6= G and J 6 H. Thus, J \H is a proper subgroup of J. In this case, G=H, considered as a J-space, can be identified with the un* *it sphere S of an irreducible complex representation of J whose kernel is J \ H. The inclusion of any J-orbit into G=H = S yields a cofibre sequence J=(J \ H)+ ___//G=H+ ___//J=(J \ H)+ : The next map J=(J \ H)+ __//_J=(J \ H)+in this sequence is the difference of the identity map of J=(J \ H)+ and the map j: J=(J \ H)+ __//_J=(J \ H)+ given by multiplication by an appropriate generator j of J (see, for example, t* *he 38 L. GAUNCE LEWIS, JR. proof of Lemma A.1 in the appendix of [17]). To understand the map "J in this context, consider the exact sequence (1-j)* [J=(J \ H)+ ; K(M; 0)]J _______// [J=(J \ H)+ ; K(M; 0)]J ___// [G=H+ ; K(M; 0)]J ___// [J=(J \ H)+ ; K(M; 0)]J = 0 derived from our cofibre sequence. The last group in this sequence is zero by t* *he dimension axiom. The first map in this sequence can be identified with the map "j M(G=(J \ H)) _1-__//M(G=(J \ H)); in which the map "jis given by the action of j, considered as an element of the* * Weyl group WG (J \ H), on M(G=(J \ H)). Since WG (J \ H) = S1 is connected, it acts trivially on M(G=(J \ H)). Thus, the map (1 - j)* in the exact sequence above is trivial, and there is an isomorphism [G=H+ ; K(M; 0)]J ~=[J=(J \ H)+ ; K(M; 0)]J = M(G=(J \ H)): This isomorphism, and the analogous isomorphism for M0, identify the map "Jwith the map (G=(J \ H)) : M0(G=(J \ H)) __//_M(G=(J \ H)), which is assumed to be a monomorphism. The cofibre sequence of the inclusion J=(J \ H)+___//G=H+ can also be used to analyze the map J whenever H; J 6= G and J 6 H. For this analysis, consider the exact sequence ((1-j)1)* 0 [S0; J=(J \ H)+ ^ K(M; 0)]J_________// [S ; J=(J \ H)+ ^ K(M; 0)]J ___// [S0; G=H+ ^ K(M; 0)]J ___// [S0; J=(J \ H)+ ^ K(M; 0)]J = 0; in which the last term is zero by the dimension axiom. Since the group J is fi- nite, equivariant Spanier-Whitehead duality provides an identification of the g* *roup [S0; J=(J \ H)+ ^ K(M; 0)]J with the group [J=(J \ H)+ ; K(M; 0)]J. Using this identification, the first map in our exact sequence can be identified, as in th* *e pre- vious exact sequence, with the map "j M(G=(J \ H)) _1-__//M(G=(J \ H)); which is known to be zero. Thus, there is an isomorphism [S0; G=H+ ^ K(M; 0)]J ~=[J=(J \ H)+ ; K(M; 0)]J = M(G=(J \ H)): This isomorphism, and the analogous isomorphism for M0, identify the map J with the map (G=(J \ H)) : M0(G=(J \ H)) __//_M(G=(J \ H)), which is assumed_to be a monomorphism. |__| Proof of Theorem 3.8(b).Recall that G = S1. In this proof, we make use of the ideas presented in Remark 3.4. The stable orbit category OG is a full subcatego* *ry of the category S0 of G-spectra which have the G-homotopy type of finite G-CW complexes. The category S0is a symmetric monoidal closed category whose internal hom functor <>is derived from a duality functor. Thus, for objects X and * *Y in S0, X ^ Y ~=D(D(X)) ^ Y = < >~=<<< >Y:>> WHEN PROJECTIVE DOES NOT IMPLY FLAT 39 It follows that, if O is a subcategory of S0 which is closed under the internal* * hom operation on S0, then it must also be closed under the ^-product operation on S* *0. To prove Theorem 3.8(b), it therefore suffices to show that, if O0is a full s* *ubcat- egory of the G-stable category which contains OG and which is closed under smash products, then the restriction functor McontO0_//McontOG= MG is not an equivalence of categories. We can regard the functors HG=e and HG=exG* *=e as contravariant functors out of O0. The diagonal map : G=e __//_G=e x G=ein- duces a map " : HG=e __//_HG=exG=e. Since G=e and G=exG=e are path connected, Proposition 3.1 of [21] implies that the map " (G=H) : HG=e(G=H) ___//HG=exG=e(G=H) is an isomorphism for H G. However, by the change of group isomorphisms in section II.4 of [23], HG=e(G=e x G=e) = [G=e+ ^ G=e+ ; G=e+ ]G ~= [G=e+ ; G=e+ ]e ~= [S1 _ S0; S1 _ S0]e ~= Z Z Z=2 and HG=exG=e(G=e x G=e) = [G=e+ ^ G=e+ ; G=e+ ^ G=e+ ]G ~= [G=e+ ; G=e+ ^ G=e+ ]e ~= [S1 _ S0; (S1 _ S0) ^ (S1 _ S0)]e ~= Z Z Z Z=2: Thus, " is not an isomorphism in McontO0, but its restriction to McontOG=_MG is* * an isomorphism. |__| 40 L. GAUNCE LEWIS, JR. 11. Globally defined Mackey functors and the proof of Theorem 6.10 In this section, all groups are assumed to be finite, and P and Q are sets of primes. In proving Theorem 6.10, we follow the general pattern described in sec* *tion 7. First, we identify a natural direct summand Zp(M) of the functor sending a (P; Q)-Mackey functor M to the abelian group (HZ=p M)(Z=p). Then we show that the functor Zp does not preserve monomorphisms. The argument here is somewhat simpler than the previous ones in that no restriction on M is needed to split off Zp(M). Definition 11.1.Assume that the prime p is not in the set P of primes, and that M is in M*(P; Q). Denote the trivial group by e, and the direct sum of p copies of an abelian group A by Ap. Then Zp(M) is the pushout in the diagram M(oeZ=p)p M(e)p _______//_M(Z=p)p r || M|| fflffl|je fflffl| M(e) _________//Zp(M): Here, r is the folding map. Regard M(Z=p)p as being the direct sum of a collect* *ion of copies of M(Z=p) indexed on the set of group endomorphisms of Z=p. Given such an endomorphism fl : Z=p___//Z=p, let jfl: M(Z=p) ___//Zp(M)be the composite M(Z=p) M(Z=p)p _M__//_Zp(M); in which the first map is the inclusion of M(Z=p) into the direct sum as the co* *py indexed on fl. Analogously, the map je in the pushout diagram above should be thought of as the map from M(e) into Zp(M) associated to the unique group homomorphism e __//_Z=p. Proposition 11.2.Let p be a prime and P and Q be sets of primes such that p 62 P. Then, for all M in M*(P; Q), Zp(M) splits off from (HZ=p M)(Z=p) as a natural direct summand. As in the previous sections, we postpone the proof of this result until after* * we have shown how it can be used to complete the proof of our theorem Proof of Theorem 6.10.Let p be a prime, and let P and Q be sets of primes such that p 62 P. We must construct a monomorphism : A __//_B which is not preserved by the functor Zp. Let B be the representable Mackey functor He, and let A be the image of the map o"eZ=p: HZ=p___//He = B: induced by the map oeZ=pin B*(P; Q). Note that there are an obvious monomor- phism : A __//_B and an obvious epimorphism : HZ=p___//A. WHEN PROJECTIVE DOES NOT IMPLY FLAT 41 Consider the diagram A(e)LWWW (e) 33 WWWW 33LL WWWWWWWWW 33 %%P___0____++//_B(e) A(oeZ=p)333| | e 33|o0 |B(oZ=p) ssss3fflffl|(|fflfflZ=p) A(Z=p) _____//B(Z=p); in which the rectangle is a pullback, and is the induced map into the pullback. We show first that the map is not surjective, and then use this to show that t* *he map Zp() : Zp(A)___//Zp(B)is not a monomorphism. To show that is not surjective, it is necessary to compute several values of* * the representable functors used to define A and B. It is easy to see that the morph* *ism sets [Z=p; e] = HZ=p(e) and [e; e] = He(e) of B*(P; Q) are infinite cyclic grou* *ps generated by the morphisms aeeZ=pand 1e, respectively. The map "oeZ=p(e) takes * *aeeZ=p to p . 1e. The morphism set [e; Z=p] = He(Z=p) has either one or two generators, de- pending on whether or not p is an element of Q. The morphism oeZ=pis always a generator. If p 2 Q, then there is a second generator represented by the diagram 1Z=p e oo_Z=p _____//_Z=p: The morphism set [Z=p; Z=p] = HZ=p(Z=p) is generated by the composite oeZ=pOaee* *Z=p and by the morphisms in B*(P; Q) of the form ae(ff), where ff is an endomorphism of Z=p. If p 62 Q, then ff must be a nonzero endomorphism; otherwise, it can be any endomorphism. The map "oeZ=p(Z=p) takes oeZ=pO aeeZ=pto p . oeZ=pand ae(ff)* * to oeZ=p. From these computations, it follows immediately that the diagram above has the form Z/HXXXXXXpX //HH XXXXXXXX /p/##/Z______XX++//_Z p// | 1Z | //|1Z i1| /fflffl|i1 fflffl| Z________//Z C: Here, C is Z if p 2 Q; otherwise, C = 0. Also, the map i1 is the inclusion of t* *he first summand of a direct sum, and p denotes the map given by multiplication by p. Clearly, is not surjective. Pick x 2 A(Z=p) and y 2 B(e) illustrating the fact that isn't onto; that is, such that (Z=p)(x) = B(oeZ=p)(y) = z 2 B(Z=p) but there is no element of A(e) hitting both x and y under the obvious maps. The choices x = (1Z=p) and y = 1e are, for example, acceptable. Consider the elements (x; -x; 0; : :;:0) 2 A(Z=p)p (y; -y; 0; : :;:0) 2 B(e)p (z; -z; 0; : :;:0) 2 B(Z=p)p 42 L. GAUNCE LEWIS, JR. in the context of the pushout diagrams defining Zp(A) and Zp(B). From the equa- tions r(y; -y; 0; : :;:0) = 0 and B(oeZ=p)p(y; -y; 0; : :;:0) = (z; -z; 0; : :;:0); it follows that B (z; -z; 0; : :;:0) = 0 2 Zp(B): However, A (x; -x; 0; : :;:0) must be a nonzero element of Zp(A) since x and y illustrate the fact that isn't surjective. Since Zp()(A (x; -x; 0; : :;:0)) = B (z; -z; 0; : :;:0) = 0; it follows that Zp does not preserve monomorphisms. |___| The rest of this section is devoted to the postponed proof of our splitting r* *esult. Proof of Proposition 11.2.Let M be in M*(P; Q). As in the proof of Proposition 7.2, we first construct a natural map : (HZ=p M)(Z=p) ___//Zp(M); and then show that it is a split epimorphism. The map is, as before, derived f* *rom an appropriately behaved map ": M M(Q) [Z=p x Q; Z=p]___//Zp(M): Q Again it suffices to specify the restriction "Q of " to the summand indexed on * *each group Q. That summand is itself a direct sum of copies of M(Q) indexed on the generators of [Z=p x Q; Z=p]. These generators correspond to equivalence classes (ff; fi; ffi) of diagrams of the form (ff;fi) ffi Z=p x Q oo____ J ___//Z=p; in which the induced map (ff; fi;:ffi)J __//_Z=p x Q x Z=pis a monomorphism. De- note the restriction of "Q to the copy of M(Q) indexed on the generator associa* *ted to this diagram by "(ff;fi;ffi). Some manipulation of the diagram above must be done to define "(ff;fi;ffi). L* *et L = ffi(J) so that L is either e or Z=p, and regard ffi as a map into L. Since * *the order of the kernel of ffi is not divisible by p, there is a unique map fl : L * *__//_Z=p making the diagram "J > "" >>>ffi> """ff"""" >> OEOE> Z=p oo___fl_____L commute. This map is perhaps best understood by looking at a p-Sylow subgroup P of J. The map ffi must restrict to an isomorphism from P to L. The map fl is the composite of the inverse of this isomorphism and the restriction of ff to P* * . WHEN PROJECTIVE DOES NOT IMPLY FLAT 43 If the kernel of fi is not a Q-group, then define "(ff;fi;ffi)to be zero. Oth* *erwise, define it to be the composite M(ae(fi)) M(o(ffi)) jfl M(Q) ________//_M(J) ________//M(L) ____//Zp(M); where jflis the map from Definition 11.1 associated to a homomorphism fl from either e or Z=p into Z=p. To show that can be derived from ", it suffices to show that, for each morph* *ism f : Q0___//Q in B*(P; Q), the diagram M(f)1 M(Q0) [Z=p x Q; Z=p]_______//M(Q) [Z=p x Q; Z=p] 1(1xf)* || Q"|| fflffl| "Q0 fflffl| M(Q0) [Z=p x Q0; Z=p]_______________//Zp(M) commutes. Further, we need only check the cases in which f is a generator of the form ae() associated to a homomorphism : Q __//_Q0with Q-kernel, or of the form o(i) associated to a homomorphism i : Q0 __//_Q with P-kernel. As in the previous proof, the commutativity of the diagram is purely formal if f = ae() f* *or some homomorphism . Thus, we assume that f = o(i) for some homomorphism i : Q0___//Q with P-kernel. We can verify the commutativity of the diagram by checking it on each summand M(Q0) of M(Q0) [Z=p x Q; Z=p]. On the summand indexed by a diagram (ff;fi) ffi Z=p x Q oo____ J ___//Z=p; in which the map fi : J __//_Q does not have Q-kernel, it is easy to see that t* *he two composites in the diagram are both zero. Thus, we restrict our attention to the summands indexed on diagrams in which fi does have Q-kernel. For this case, we must use Lemma 5.3 in much the same way that Lemma 3.3 of [21] is used in the proofs of Propositions 7.2 and 8.1. We also need to know th* *at, for any group endomorphism fl : Z=p___//Z=p, the diagram M(oEZ=p) M(e) ______//M(Z=p) KK KKK |jfl jeKKKK%%Kfflffl|| Zp(M) commutes. Ensuring the commutativity of this diagram is, however, the whole point of the pushout diagram used to define Zp(M). The commutativity of our naturality diagram on the appropriate summand M(Q0) of M(Q0) [Z=p x Q; Z=p] follows from these observations by an easy diagram chase. Thus, the desired map can be obtained from the map ". Clearly, is natural in M. We still must show that the map splits naturally. Let D be the summand of [Z=p x Z=p; Z=p] whose generators are represented by the diagrams (fl;1Z=p) 1Z=p Z=p x Z=p oo______ Z=p _____//_Z=p; 44 L. GAUNCE LEWIS, JR. in which fl is an endomorphism of Z=p. Then D ~=Zp, and M(Z=p)D ~=M(Z=p)p. Let "oe: M(Z=p)p __//_(HZ=p M)(Z=p)be the composite M(Z=p)p ~= M(Z=p) D M(Z=p) [Z=p x Z=p; Z=p] M M(Q) [Z=p x Q; Z=p] Q __//_(HZ=p M)(Z=p); in which the last map is the standard projection. It is easy to see that the co* *mposite O "oe: M(Z=p)p ___//Zp(M)is just the projection M : M(Z=p)p ___//Zp(M)of Definition 11.1. Thus, to show that is a split epimorphism, it suffices to sh* *ow that the map "oefactors through the projection M . For each group Q, let Q : M(Q) [Z=p x Q; Z=p]___//(HZ=p M)(Z=p)be the mapLobtained from our description of (HZ=p M)(Z=p) as a quotient group of Q M(Q) [Z=p x Q; Z=p]. The diagram M(oeZ=p)1 M(e) [Z=p x Z=p; Z=p]_______//M(Z=p) [Z=p x Z=p; Z=p] 1(1xoeZ=p)*|| Z=p|| fflffl| e fflffl| M(e) [Z=p x e; Z=p]____________//(HZ=p M)(Z=p) commutes by the definition of (HZ=p M)(Z=p). For any endomorphism fl of Z=p, the image under (1 x oeZ=p)* of the generator of [Z=p x Z=p; Z=p] represented b* *y the diagram (fl;1Z=p) 1Z=p Z=p x Z=p oo______ Z=p _____//_Z=p is just the morphism represented by the diagram Z=p x e oo_e __//_Z=p in which all the maps are the obvious ones. This observation, together with the commutativity of the above naturality diagram for the maps e and Z=p, easily implies that "oefactors through the projection M . It follows that "oeinduces_a* *_map oe which splits . 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