The Homology of Homotopy Inverse Limits by Paul G. Goerss1 Abstract: The homology of a homotopy inverse limit can be studied by a spec* *tral sequence with has E2 term the derived functors of limit in the category of coal* *gebras. These derived functors can be computed using the theory of Dieudonne modules if* * one has a diagram of connected abelian Hopf algebras. One of the standard problems in homotopy theory is to calculate the homolog* *y of a given type of inverse limit. For example, one might want to know the homology * *of the inverse limit of a tower of fibrations, or of the pull-back of a fibration, or * *of the homotopy fixed point set of a group action, or even of an infinite product of spaces. T* *his paper presents a systematic method for dealing with this problem and works out a seri* *es of examples. It simplifies the foundational questions present when dealing with inverse * *limits to work with simplicial sets rather than topological spaces. So this paper is writ* *ten simpli- cially; that is, a space is a simplicial set. As usual this doesn't affect the * *homotopy theory. Homology is with Fp coefficients. Here are some simple examples of the type of result we obtain. An abelian * *Hopf algebra is one for which both diagonal and multiplication are commutative. Theorem A. Let {Xff} be an arbitrary set of connected nilpotent spaces and supp* *ose for all ff, H*Xffis an abelian Hopf algebra. Then there is a natural isomorphism Y Y H*( Xff) ~= H*Xff: ff ff This is a companion to a result of Bousfield's [3, 4.4] where slightly diff* *erent hypotheses Q yield a slightly different result. The product H*Xffis in the category of gra* *ded connected ff coalgebras. If the set is finite it is simply the graded tensor product. If the* * set is infinite something more sophisticated is required. See section 1 for limits of coalgebra* *s. _________________________ 1The author was partially supported by the National Science Foundation. 1 Theorem B. Consider a tower of fibrations over the natural numbers X1 X2 X3 . . . so that for all i, Xi is connected and H*Xi+1 ! H*Xi is a morphism of abelian H* *opf algebras. Suppose each Xiis simple and lim1ss1Xi= 0. Then under either of the f* *ollowing conditions, there is an isomorphism H*(limXi) ~=limiH*Xi: 1) for all i, H*Xi is of finite type; 2) the tower H*X1 H*X2 . .i.s Mittag-Leffler. These are two of the usual conditions for vanishing of lim1for abelian grou* *ps. Again limiH*Xi must be calculated in the category of coalgebras. Under various condi* *tions, however, this will be isomorphic to the inverse limit as vector spaces. One com* *mon example of this is if the tower {H*Xi}i is pro-isomorphic, as a tower of Hopf algebras,* * to a constant tower. This is a stronger condition than Mittag-Leffler. Behind these results is a technique which I will now explain. In general, g* *iven a small category I and an I-diagram X of spaces there is a map of graded coalgebras H*(holimIX) ! limIH*X where holimis the Bousfield-Kan homotopy inverse limit and the limit on the rig* *ht is in the category of coalgebras. There is no reason to think that it is either injective* * or surjective; however, it is the edge homomorphism of a spectral sequence. This spectral sequ* *ence is a variant of one due to Anderson [1] and studied by Bousfield in [3]. The major a* *dvantage of our variant is that the E2 term can be identified and computed by non-abelian h* *omological algebra. Let CA+ be the category of graded connected coalgebras over Fp and CAI+the * *category of I-diagrams in CA+ . The point of this paper is that the functor limI: CAI+! CA+ has right derived functors RsCAlimI, s 0, so that R0CAlimI~=limIand the bigrad* *ed vector space R*CAlimIC is a bigraded, cocommutative coalgebra. Then we have 2 Theorem C. Let X : I ! S be an I-diagram of connected spaces. Then there is a n* *atural second quadrant homology spectral sequence. R*CAlimIH*X ) H* holimIXp: Here Xp : I ! S is the I-diagram obtained by applying the Bousfield-Kan p-c* *omple- tion functor to X. Because this is a second quadrant spectral sequence, conver* *gence is a problem. For this we appeal to [3] or [13]. This accounts for some of the h* *ypotheses of Theorems A and B. Beyond this there is the problem of computing R*CAlimIH*X.* * For example Theorems A and B are based on the assertion that RsCAlimIH*X = 0 for s * *> 0 under the listed hypotheses. It is here that we use the assumption that H*X : I* * ! CA+ is actually a diagram of abelian Hopf algebras. The category of abelian Hopf al* *gebras is equivalent to a category of modules called Dieudonne modules and one can pass f* *rom the ordinary derived functors of limIfor modules to R*CAlimI, at least in some case* *s. We give examples, at least when limIs= 0 for s > 1 as modules. Even in this case RsCAli* *mImay not be zero for any s 0. Using these computations we make some calculations of the* * spectral sequence in cases where R*CAlimIH*X is not concentrated in degree zero. Include* *d is an example of homotopy fixed points. Finally, there is the problem of relating H* * *holimX to H* holimXp. Again there are techniques in [3] that apply. Despite the title of this paper and the general thrust of this introduction* *, this is pri- marily a work in non-abelian homological algebra - five of the six sections are* * devoted to the definitions and calculations of R*CAlimI. Only the sixth and last section c* *ontains homo- topy theoretic applications. Sections 1 and 2 are devoted to the definition and* * homotopical algebra foundations of R*CAlimI- these derived functors are, in fact, the cohom* *otopy groups of a homotopy inverse limit of cosimplicial coalgebras, although this language * *is avoided until section 2. Section 3 is on Dieudonne modules, Section 4 gives some calcul* *ations, and Section 5 contains the proofs of some technical lemmas. The paper began as a meditation on Example 4.4 of [3], spurred on by a conv* *ersation with John Hunton. In fact, the proof of Theorem A given here may be regarded as* * a wildly expanded version of Bousfield's argument, and, in general, this paper owes a gr* *eat debt to [3]. Finally, a conversation with Brooke Shipley on bicosimplicial spaces wa* *s useful for widening my scope. Several results for [13] make this paper go much more smooth* *ly. x1. Derived functors of limits in the category of coalgebras Let CA be the category of graded cocommutative coalgebras over a field k. * *Later, we will specialize to the case where k = Fp. As always in homotopy theory appli* *cations, 3 commutativity is with a sign. Thus if C 2 CA and x 2 C has diagonal x = yi zi then yi zi= (-1)|yi| |zi|zi yi where |w| is the degree of w. Let CA+ CA denote the full sub-category of conn* *ected coalgebras. Thus C 2 CA+ if and only if C0 ~=Fp. A useful fact is: Lemma 1.1. Let C 2 CA. Then C is isomorphic to the filtered colimit of its fi* *nite dimensional sub-coalgebras. Proof: This is clear in C 2 CA+ . The general case follows from [14], p. 46. Now let I be a small category and CAI the category of I-diagrams in CA. By * *definition, limI: CAI ! CA is right adjoint to the constant diagram functor. Such limits exist for all I. * *To see this, first suppose I is filtered. Then if C : I ! CA is an I-diagram we can form th* *e vector space limit limIkC. This is not, in general, a coalgebra; however, it is a comp* *lete coalgebra in the following sense. For i 2 I, let E(i) = ker{limIkC ! C(i)}: Then limIkC is a complete topological vector space with respect to the neighbor* *hood base for zero {E(i)}. Then there is a coproduct limIkC ! limIkC blimIkC where b denotes the completion of the tensor product with respect to {E(i) E(j* *)}. This is because limIkC blimIkC ~=limIk(C C): A sub-coalgebra D limIkC is a sub-vector space equipped with a coproduct D ! D* * D making D a coalgebra and such that the evident diagram commutes: D _____________wD D | | | | |u |u limkC _______wlimkC limkC: 4 Then, for the limit in the category of coalgebras, (1.2.1) limIC = colimffDff where Dffruns over all sub-coalgebras over limIkC. Because of Lemma 1.1, we cou* *ld equally require the Dffto be finite. Next, suppose I is discrete, so that limI= . In * *the object I set of I is finite, then IC = I C: Then, in general, (1.2.2) C = lim C I J J where limis in the category CA and J runs over the filtered diagram of finite s* *ub-categories of I. Proposition 1.3. The category CA has all limits. Proof: We now have products, by 1.2.2, so to get all limits we need only supply* * pull- backs. However, the pull-back of a diagram C1 ! C12 C2 is the cotensor product C1C12C2. Because of the colimit in formula 1.2.1, limIwill not, in general, preserve* * surjections, even in the case of (infinite) products 1.2.2. Thus one will have derived funct* *ors. There are several possible definitions of these. After some preliminaries, we will gi* *ve a definition in (1.8) which, while complicated, arises in homotopy theory applications. Let J : CA+ ! n+ k be the "coaugmentation coideal" functor from connected c* *oal- gebras to positively graded vector spaces. Thus JC is nothing more than the ele* *ments of L positive degree. The functor J has a right adjoint S with SW = SnW where S0W* * = k n0 and SnW = (W__._._.W-z_____")n : n where n acts by permuting factors, up to signs required by the grading. If W is* * of finite type, (SW )* is the free graded commutative algebra on W *and, hence, if char(k* *) 6= 2, is a tensor product of polynomial and exterior algebras. 5 Let S = S O J : CA+ ! CA+ be the composite functor. It is the underlying fu* *nctor of a triple on CA+ . Thus if C 2 CA+ , one has a canonical cosimplicial resolut* *ion (1.4) j : C ! SoC: In particular, ss*So C ~=C with isomorphism induced by j. This can be seen by a* *pplying J to (1.4) and noting that the resulting cosimplicial vector space has a natural * *contraction. This resolution can be used to define derived functors. For example, if P * * is the primitive element functor (1.5) RsP C = sssP SoC: These derived functors were one of the main topics of [4]. We also use this res* *olution to define derived functors of limit. But we need another preliminary. Fix a small category I. The classifying space of the category I is the sim* *plicial set BI with 0-simplices the objects of I and n-simplices, n 1, strings of arrows i* *n I * i= in ! in-1 ! . .!.i1 ! i0 with face and degeneracy operators given by * d0 i = in ! in-1 ! . .!.i1 * dj i = in ! . .!.ij+1 ! ij-1 ! . .!.i0 * dn i = in-1 ! . .!.i1 ! i0; * 1 and sj i = in ! . .!.ij! ij ! . .!.i0: Note that for dj, ij+1 ! ij-1 is the composition. Now let C be a category with * *products and X : I ! C an I-diagram. Then we get a cosimplicial objects o X in C with (1.6) nX = * X(i0): i2BIn with coface and codegeneracy maps induced by those in BI, with the evident twis* *t in d0: the composite 0 ss*j * X(i0) d-!* X(j0) -! X(j0) i2BIn-1 j2BIn 6 fits into a commutative diagram 0 * X(i0)|_______wd*X(j0)| i | j | | | |ss * |ss*: | d0 j | j |u |u X(j1) _________wX(j0) Note that if A : I ! ab is a diagram of abelian groups, then (1.7) limIsA ~=sssoA: See [5, p. 305]. Now let C : I ! CA+ be an I-diagram of connected coalgebras. Then for each * *i 2 I, one can form the augmented cosimplicial coalgebra j : C(i) ! SoC(i) and the naturality of this construction yields an I-diagram of augmented cosimp* *licial coalgebras. Hence one obtains a bicosimplicial coalgebra oSoC which has a diag* *onal cosimplicial coalgebra diagoSoC. Definition 1.8. Let C : I ! CA+ be an I-diagram of coalgebras. Then the deriv* *ed functors of limIare defined by RsCAlimIC = sss diag(oSoC): Remarks 1.9: 1) The augmentation j : C ! SoC induces a map oC ! diagoSoC which may or may not be an equivalence. See Example 1.10 below and Corollary 4.* *9. 2) Since R*CAlimIC is the cohomotopy of a cosimplicial coalgebra, it is a b* *igraded cocommutative coalgebra. The switch sign depends on the total degree. 3) R0CAlimIC ~=limIC since ss0 diagoSoC is an equalizer and limits commute. The remainder of this section is devoted to two examples. Example 1.10: If I has finite object set and C : I ! CA+ is an I-diagram - so C* * is a finite diagram - then j : oC ! diagoSoC 7 is a ss* isomorphism. To see this, filter the bicosimplicial coalgebra oSoC = {nSm C} by degree in n to get a spectral sequence En;m1= ssm (ssnSo C) ) ssn+m diag oSoC: N Since I is finite, BIn is finite, so nSo C = SoC(i0) and the Eilenberg* *-Zilber *iBI n Theorem says ae En;m1= 0nC mm>=00: The claims follows. Thus one has ss*(oC) ~=R*CAlimIC: * * N If I is finite and discrete, one has RnCAlimIC = 0 for n > 0 and R0CAlimIC ~= * *C ~= C. I* * I If I is the category with diagram 1 ! 12 2, so and I-diagram is of the form C1 ! C12 C2 one has R*CAlimIC ~=ss*(oC) ~=Cotor*C12(C1; C2) as we would hope. Example 1.11: Let I be discrete, but not necessarily finite. Then I claim the * *other augmentation ISo C ! oSoC induces a ss* isomorphism ISo C ! diagoSoC so that ss*( SoC) ~=R*CA( )C: I I To see this, filter oSoC = {nSm C} by degree in m, to get a spectral sequence En;m1= ssnoSm C ) ssn+m diag oSoC: 8 For fixed m, Sm C = (S O . .O.S)C, with the composition taken m + 1 times. Then oSm C ~=S(oJ O Sm-1 C) since S is a right adjoint and o on the inside is in the category of vector spa* *ces. Then ssnoJ O Sm-1 C ~=limIn(J O Sm-1 C) ( ~= 0 m-1 n > 0 IJ O S C n = 0 since these derived functors are in vector spaces. Thus oJ O Sm-1 C has a cosi* *mplicial contraction and ( 0 n > 0 En;m1= Sm C n = 0 I where this product is in CA+ . The result follows. x2. The homotopical foundations of R*CAlimI. The point of this section is to demonstrate that the derived functors of li* *mits in the category CA can be given a homotopical foundation flexible enough to be use* *ful in computations. Afterward we supply some examples. Let cCA+ be the category of cosimplicial coalgebras. We will show that R*CA* *limIare the cohomotopy groups of a homotopy inverse limit in cCA+ . To begin with cCA+ is a simplicial model category in the sense of Quillen [* *11, II.2]. This is a result of [7]; see also [2], although the latter reference is not exp* *licit about the simplicial structure. To spell out the details, first note that cCA+ is a simpl* *icial category by the results of Quillen. Indeed if K 2 S is a simplicial set and Co 2 cCA+ * * then hom (K; Co) 2 cCA+ is the cosimplicial coalgebra with (2.1.1) hom (K; Co)n = Cn k2Kn with the induced face and codegeneracy operators. There is also an object CoK u* *niquely determined by the natural isomorphism (2.1.2) Hom eCA+ (Co K; Do) ~=Hom cCA+(Co; hom(K; Do)): 9 Finally, there is a mapping space functor on cCA+ with n-simplices given by (2.1.3) map cCA+(Co; Do) = Hom cCA+(Co; hom( n; Do)): Now define a morphism f : Co ! Do to be 2.1.4) a weak equivalence if ss*f is an isomorphism; 2.1.5) a cofibration if the induced map Nf : NCo ! NDo of normalized cochain c* *om- plexes is an injection in positive degrees; 2.1.6) a fibration if it has the right lifting property with respect to all ma* *ps which are at once cofibrations and weak equivalences. Then one has: Theorem 2.2 [5]. With these definitions, cCA+ becomes a simplicial model catego* *ry. Part of the proof is to describe fibrations. A morphism f : Co ! Do is almo* *st-free if for all n 0 there are vector spaces W nand maps oeo : W n! W n-1, 0 i n - 1,* * so that 2.3.1) there are isomorphisms Cn ~=Dn S(W n) that fit into commutative diagra* *ms ~= n n Cn ________wD S(W ) | f | |ss | | 1 |u |u Dn ____________w=Dn; 2.3.2) the following diagrams commute in-1 Cn ______________________wsC ~ | |~ = | |= |u i i |u Dn S(W n) ____________wsDnsoe-1 S(W n-1): Proposition 2.4. Almost-free maps are fibrations in cCA+ and any fibration is a* * retract of an almost-free map. If the unique map " : Co ! k is a fibration, Co will be called fibrant; if * *" is almost-free, Co will be called almost-free. Since any retract of a coalgebra of the form S(W* * ) is of the form S(W0) (by [4], Proposition 4.2), one has Corollary 2.5. An object 2.5 is fibrant if and only if it is almost-free. 10 Example 2.6: Let C 2 CA+ be regarded as a constant cosimplicial coalgebra. Then* * the canonical map j : C ! SoC is a cofibration and a weak equivalence, and SoC is f* *ibrant. Because cCA+ is a simplicial model category with all products and coproduct* *s, it has homotopy inverse limits in the style of Bousfield and Kan [5, XI, x4ff]. To re* *capitulate, let I be a small category. For i 2 I, let I # i be the category of objects ove* *r I and let B(I # i) be its classifying space. Each of these spaces is contractible since * *I # i has a terminal object. The assignment i 7! B(I # i) is an I-diagram of spaces. Now let Co : I ! cCA+ be an I-diagram of cosimplic* *ial coalgebras. Then holimC* = holimICo is defined by an equalizer diagram in cCA+ 1 (2.7) holim Co ! hom (B(I # i); Co(i)) d=) hom (B(I # i); Co(i0)) i d0 i!i0 where d0 and d1 are induced respectively by hom(B(I # i;Co(i)) ! hom (B(I # i); Co(i0)) and hom (B(I # i0); Co(i0)) ! hom (B(I # i); Co(i0)): However, combining this definition with the description of hom (K; Co) given in* * 2.1.1 one has (2.8) holim Co ~=diag(oCo): where o is the cosimplicial construction of the previous section. One would like holimCo to have the following invariance property: if Co(i) * *! Do(i) is a weak equivalence for all i, then holimCo ! holimDo should be a weak equivalence. This fails in general; however, one has: Lemma 2.9. Let Co ! Do be a morphism of I-diagrams in cCA+ and suppose Co(i) ! Do(i) is a weak equivalence for all i 2 I. Then if Co(i) and Do(i) are fibrant * *for all i, holimCo ! holimDo is a weak equivalence. 11 Proof: Write the bicosimplicial vector space oCo = {pCq}: Filtering by degree in p gives a spectral sequence Ep;q1= ssq(pCo)) ssp+q diag(oCo) ~= ssp+q holimCo: Thus we need only show pCo ! pDo is a weak equivalence for all p. Since every object of cCA+ is cofibrant, any weak equivalence between fibrant objects is a * *homotopy equivalence, which, in cCA+ is the same as a cosimplicial homotopy equivalence * *in the sense of [10]. Thus pCo ! pDo is a cosimplicial homotopy equivalence, hence a w* *eak equivalence. For this reason one demands, before taking a homotopy inverse limit in cCA+* * , that one has a diagram of fibrant objects. One of the results of [7] is that for all* * Co 2 cCA+ there is a natural cofibration and weak equivalence Co ! Do with Do fibrant. T* *hus if Co : I ! cCA+ is an I-diagram Do : I ! cCA+ is an I-diagram of fibrant objects * *and one sets (2.10) R holimCo = holimDo: The boldface R means Quillen's total right derived functor. Corollary 2.11. If Co1! Co2is a morphism of I-diagrams in cCA+ and Co1(i) ! Co2* *(i) is a weak equivalence for all i, then R holimCo1! R holimCo2 is a weak equivalence. Proof: Combine Lemma 2.9 with the definition 2.10. More is true however. Lemma 2.12. Let Co : I ! cCA+ be an I-diagram of cosimplicial coalgebras and l* *et Co ! Co1be a morphism of I diagrams so that Co(i) ! Co1(i) 12 is a weak equivalence for all i and Co1(i) is fibrant for all i. Then there is * *a weak equivalence R holimCo ' holimCo1: Proof: Consider the diagram Co _______wDo | | | | |u |u Co1_______wDo1 where Do and Do1are the natural fibrant replacements indicated above. Then Lemm* *a 2.9 guarantees weak equivalences holimCo1! holimDo1 holimDo: Example 2.13: Let C : I ! cCA+ be an I-diagram of coalgebras. We may regard each C(i) as a constant cosimplicial coalgebra. Then, by Example 2.6, j : C ! * *SoC is a morphism of I-diagrams in cCA+ so that C(i) ! SoC(i) is a weak equivalence wi* *th So C(i) fibrant. Hence R holimC = holimSoC ~=diag(oSoC) and ss*R holim C~=R*CAlimIC: As a first example of the kind of flexibility we've acquired, I supply: Example 2.14: Let C1; C2 : I ! CA+ be two I-diagrams. Then there is a natural isomorphism of bigraded coalgebras R*CAlimI(C1 C2) ~=R*CAlimIC1 R*CAlimIC2: As a result of H : I ! CA+ is actually a diagram of Hopf algebras, R*CAlimIH i* *s a bigraded Hopf algebra. If H is commutative, so is R*CAlimIH. To see the isomorphism, I claim there is a weak equivalence R holim(C1 C2) ' R holimC1 R holimC2 of cosimplicial coalgebras. Here the tensor product is taken level-wise; the r* *esult above follows from the Eilenberg-Zilber Theorem. To see the claim, let C1 : I ! cCA+ and C2 : I ! cCA+ be any two I-diagrams. Choose morphisms of I-diagrams C1 ! D1 C2 ! D2 13 where D1(i) and D2(i) are fibrant for all i, and C1(i) ! D1(i) and C2(i) ! D2(i* *) are weak equivalences. Then C1 C2 ! D1 D2 has the same properties: D1(i) D2(i) is fibrant, since products of fibrant objects are fibrant and C1(i) C2(i) ! D1(i)* * D2(i) is a weak equivalence. Hence, by Lemma 2.12 and the fact that holim commutes with * *finite products, we have weak equivalences R holim(C1 C2) ' holim(D1 D2) ' holimD1 holimD2 ' R holimC1 R holimC2: x3. Recollections on Dieudonnne modules. Let HA + be the category of graded, bicommutative Hopf algebras - hereinaft* *er known as abelian Hopf algebras because an object in HA + is an abelian group object i* *n CA+ . The point of the next two sections is to address the problem of computing R*CAlimIH* * where H : I ! HA + is an I-diagram of abelian Hopf algebras. There are several adva* *ntages to this situation. One is the existence of the classification scheme known as D* *ieudonnne modules. Another is the natural presentation proof of Proposition 4.1 below. Se* *e 4.3 and Remark 4.8. We begin by recalling Schoeller's work on Dieudonnne modules [12]. The poin* *t is that HA + is an abelian category with a set of projective generators. As such it is* * equivalent to a category of modules over a ring. Dieudonnne theory says which modules over* * which ring. Let us assume the ground field k = Fp. Definition 3.1. Suppose p > 2. Let D be the category of positively graded Zp mo* *dules M equipped with homomorphisms V : M2pk ! M2k F : M2k ! M2pk so that 1) pM2k+1 = 0 for all k and if (k; p) = 1, ps+1M2psk= 0; 2) V F (x) = px and F V (y) = py. If p = 2, D is the category of graded Z2 modules with homomorphisms V : M2k ! Mk and F : Mk ! M2k so that V F = F V = 2 and 2s+1M2s(2t+1)= 0: 14 Theorem 3.2 [12]. There is an equivalence of categories D* : HA + ! D: The category D is the category of Dieudonnne modules; if H 2 HA + , then D** *H is the associated Dieudonnne module. The homomorphisms V and F are to be taken as reflecting the Verschiebung and Frobenius (pth power map) respectively. In the sequel, I'll often state results for p > 2 and leave the case p = 2 * *implicit. Example 3.3: The category D has an obvious set of projectives; indeed, for all * *n > 0, there is a module F (n) 2 D so that there is a natural isomorphism Hom D (F (n); M) ~=Mn where Mn denotes the elements of degree n. If n is odd F (n) = Z=pZ concentrat* *ed in degree n and if n = 2psk with (p; k) = 1 8 < Z=ps+1Z t = 2pik; i s F (n)t ~=: Z=pi+1Z t = 2pik; i s 0 otherwise and F i: F (n)n! F (n)2ps+ik V i: F (n)n! F (n)2ps-ik are surjective for all i. Let H(n) be the unique (up to isomorphism) Hopf algebra so that D*H(n) ~=F * *(n). Then if n is odd H(n) ~=(x) where denotes the exterior algebra and the degree * *of x is n. If n = 2psk with (p; k) = 1, H(n) ~=Fp[x0; x1; : :;:xs] where the degree of xiis 2pik and H(n) has the "Witt vector diagonal." The Vers* *chiebung is described by xi = xi-1. See [12]. Of course, the H(n) are the projective gen* *erators of HA + and Schoeller identifies them as such before proving her theorem. Then D** * : HA + ! D is given, in degree n, by DnH = Hom HA +(H(n); H): 15 The operators V and F are described by V = '* and F = * where ' : H(2n) ! H(2p* *n) is the inclusion and : H(2pn) ! H(2n) is the unique map making the diagram co* *mmute H(2n) [ [] |' [ | [ [p] |u H(2pn) _______wH(2pn) where [p] is p times the identity in HA + . In particular xi= xpi-1. Example 3.4: Again assume p > 2. Let n = 2k (with no assumption on (p; k)) and G(n) 2 D the module with ae i+1 i i G(n)t = Z=p0 Z to=t2phke=rpwnise and V is surjective. Then if n = 2psk with (p; k) = 1, there is a short exact s* *equence in D V s+1 s+1 0 ! F (n) ! G(2k) ---! G(2p k) ! 0 If A(n) is the unique (up to isomorphism) Hopf algebra with D*A(n) ~=G(n) then A(n) ~=Fp[x0; x1; x2; : :]: with deg(xi) = pin and the "infinite Witt vector" diagonal. Thus there is a sho* *rt exact sequence of Hopf algebras Fp ! H(n) ! A(2k) ! A(2ps+1k) ! Fp if n = 2psk with (k; p) = 1. The deeper ideas about HA + rely on the following result. Let n+ Ab be the* * category of positively graded abelian groups. Proposition 3.5. The forgetful functor D ! n+ Ab has a right adjoint J. Proof: The argument, a variant on the proof of the special adjoint functor theo* *rem (which, by the way, would suffice in proof) is adequately covered in [9] in the* * context of unstable modules. So I'll give only an outline. Assume p > 2. Let N 2 n+ Ab . To construct J(N) it is sufficient to assume N is concentra* *ted in a single degree, say k. Now if J(N) exists, then J(N)n ~=Hom D (F (n); J(N)) ~=Hom Ab (F (n)k; Nk) 16 so one simply defines J(N)n = Hom Ab (F (n)k; Nk): The operators V and F in D are induced by maps ' : F (2pn) ! F (2n) and : F (* *2n) ! F (2pn) respectively as representing objects. These induce V and F on J(N). Nex* *t notice that if M 2 D one has maps Hom D(M; J(N)) ! Hom Ab (Mk; Hom Ab(F (k)k; Nk)) ~=Hom Ab (Mk F (k)k; Nk) ~=Hom n+ Ab(M; N) since Mk ! Mk F (k)k is an isomorphism. By construction this map is an isomorp* *hism if M = F (n) for some n. Since both source and target send sums to products and* * are left exact as functors from Dop to Ab , the result follows by analyzing projective r* *esolutions. Remark 3.6: Again take p > 2 and let N 2 n+ Ab be concentrated in a single deg* *ree k. Then J(N)n = Hom Ab (F (n)k; Nk) is often zero. If k is odd, F (n)k = 0 unless n = k, thus J(N)n = 0 unless n =* * k and J(N)k = Hom Ab(Z=pZ; Nk). If k = 2pik0, with (k0; p) = 1, then F (n)k = 0 unl* *ess k = 2psk0. Hence J(N)n = 0 unless n = 2psk and J(N)2psk0= Hom Ab (Z=paZ; Nk) where ae a = si++11ss i i: Corollary 3.7. The categories D and HA + have enough injectives. Proof: For the category D this follows from Proposition 3.5 and the fact that n* *+ Ab has enough injectives. The result follows for HA + because this latter category is * *equivalent to D. Example 3.8: The functor D ! Ab given by M 7! Hom Ab (Mk; Q=Z) is representable. Let N(k) 2 n+ Ab be the module isomorphic to Q=Z in degree k* *. Then if J(k) = J(N(k)), Hom D(M; J(k)) ~=Hom Ab (Mk; Q=Z): 17 The object J(k) is evidently injective since Q=Z is an injective abelian group.* * Also J(k) can be described explicitly using Remark 3.6. Assume p > 2. If k is odd J(k) ~=* *Z=pZ in degree k. If k = 2pik0 with (k0; p) = 1, then J(k)n = 0 unless n = 2psk0 and ae s+1 J(k)2psk0~= Z=pZ=pZ;i+s1Zi;s i and V is always surjective, either by calculation or Corollary 3.15 below. We now examine the implications for Hopf algebras. Definition 3.9. A coalgebra D 2 CA+ is injective if any diagram of coalgebras C0 _______wiC1 | i i | i |u ik D with i an inclusion can be completed. Proposition 3.10. For a coalgebra D 2 CA+ the following are equivalent. 1) D ~=S(W ) for some graded vector space W ; 2) D is injective; 3) RsP D = 0 for s 1; and 4) R1P D = 0. Proof: That 1) implies 2) follows from adjointness. For 2) implies 3), notice t* *hat if D is injective, the inclusion D ! S(D) has a coalgebra retraction. That 3) implies 4* *) is clear. Finally 4) implies 1) is Proposition 4.2 of [4]. The connection to Hopf algebras is supplied by: Lemma 3.11. Let H 2 HA + be an injective Hopf algebra. Then H is an injective c* *oalge- bra. Proof: The forgetful functor HA + ! CA+ has a left adjoint that preserves inje* *ctions - namely, the symmetric algebra functor endowed with the evident diagonal. Henc* *e the forgetful functor preserves injectives. We now show how to recover derived functors of primitives on abelian Hopf a* *lgebras from the Dieudonnne module. If M 2 D let M 2 D be its "double". If p = 2, ae (M)t = Mk0 tt==2k2k + 1 18 where V and F induced from M. If p > 2 ae (M)t = M2k0 to=t2pkherwise with again V and F induced from M. The homomorphism V induces a homomorphism in D, V : M ! M: Proposition 3.12. Let H 2 HA + . Then there is a natural exact sequence 0 ! P H ! D*H V!D*H ! R1P H ! 0: Proof: Assume p > 2. If n is odd or n = 2k with (p; k) = 1, then (D*H)n = (R1P H)n = 0, so here we are asserting (P H)n = DnH. If n is odd, one has (P H)n ~=Hom HA +((x); H) ~=DnH where deg(x) = n; if n = 2k, (k; p) = 1, (P H)n ~=Hom HA +(Fp[x0]; H) ~=DnH where the degree of x0 is n. So we may assume n = 2psk, s > 1, with (k; p) = 1.* * Then there is a short exact sequence of Hopf algebras (3.12.1) Fp ! H(n_p) '!H(n) ! Fp[xs] ! Fp where H(n) ~=Fp[x0; : :;:xs] is the projective Example 3.3 and ' is the inclusi* *on. Since Hom HA +(H(n); H) = DnH and ' defines V we get an exact sequence 0 ! Hom HA +(Fp[xs]; H) ! DnH V!Dn_pH ! Ext1HA+(Fp[xs]; H) ! 0: Since Hom HA +(Fp[xs]; H) ~=(P H)n, the result follows from the next lemma. Lemma 3.13. For all H 2 HA + and all m 0, there are natural isomorphisms ~= m ExtmHA+(Fp[xs]; H) ! (R P H)n: 19 Proof: Let Fp ! H0 ! H1 ! H2 ! Fp be a short exact of Hopf algebras. Then one gets a long exact sequence in Ext* and in R*P (.), the latter by Theorem 3.6 of* * [4]. Since HA + has enough injectives and Rm P H = 0 for m 1 for all injectives H, by Pr* *oposition 3.10, and Hom HA +(Fp[xs]; H) ~=(P H)n the result follows from standard facts about derived functors in abelian catego* *ries. See [6], p. 206, for example. Note that Lemma 3.13 and the two term projective resolution of 3.12.1 immed* *iately imply Corollary 3.14. For all H 2 HA + , RsP H = 0 for s 2. Also, Proposition 3.12 immediately implies Corollary 3.15. A Hopf algebra H 2 HA + is injective as a coalgebra if and onl* *y if V : D*H ! D*H is surjective. In the future we may say V is surjective on M if V : M ! M is surjective. x4. Derived functors of limits for abelian Hopf algebras. We apply the technology of the previous section to calculate R*CAlimIH for * *various diagrams of abelian Hopf algebras. Note that just as the forgetful functor fro* *m abelian groups to sets makes all limits of abelian groups, so too does the forgetful fu* *nctor from HA + to CA+ . Thus if H : I ! HA + is a diagram of Hopf algebras, the expressi* *on limIH is unambiguous. Recall that R*CAlimIC = ss* diagoSoC is a bigraded coalgebra. The elements* * of RsCAlimIC will be said to have cosimplicial degree s. An element in [R*CAlimIC]* *t will be said to have internal degree t. The first result concerns products; that is, the case when I is discrete. Proposition 4.1. Let {Hff} be an arbitrary set of abelian Hopf algebr* *as. Then RsCA( )Hff= 0 for s > 0 and ff R0CA( )Hff= Hff: ff ff The proof is below in 4.4 after some preliminaries. Note that the statement* * about R0 is automatic (See 1.9.3.), so the heart of the matter is the vanishing of the h* *igher derived functors. 20 To set the stage for the proof, let D*Hff2 D be the Dieudonne module of Hff* *. In D, choose an inclusion D*Hff! J0ffwhere J0ffis injective. This is possible by Coro* *llary 3.7. Let J1ffbe the cokernel of the inclusion, so there is an exact sequence in D (4.2) 0 ! D*Hff! J0ff! J1ff! 0: Since J0ffis injective, Lemma 3.11 and Corollary 3.15 imply V : J0ff! J0ffis su* *rjective. Since is exact, V : J1ff! J1ffis surjective. Thus (4.2) corresponds to a sho* *rt exact sequence of Hopf algebras (4.3) Fp ! Hff! K0ff! K1ff! Fp with D*K0ff~=J0ffand D*K1ff~=J1ffand, by Corollary 3.15, K0ffand K1ffare inject* *ive as coalgebras. 4.4 Proof of Proposition 4.1: The central observation is that because the seque* *nce of Dieudonne modules 0 ! D*Hff! J0ff! J1ff! 0 ff ff ff is exact, the sequence of Hopf algebras Fp ! Hff! K0ff! K1ff! Fp ff ff ff is exact. (Compare Remark 4.8 below.) We will argue that the " = 0; 1 (4.5) R*CA( K"ff) ~= K"ff ff ff concentrated in cosimplicial degree 0 and then, using an argument of Bousfield'* *s, that the result follows. To begin, note that if we have a set of coalgebras of the form* * S(Vff) then R*CA( S(Vff)) ~= S(Vff) concentrated in cosimplicial degree zero. To see this* *, note that ff ff the canonical resolution S(Vff) ! SoS(Vff) has a cosimplicial contraction, hence S(Vff) ! ffSoS(Vff) has a cosimplicial contraction. Now use Example 1.11. From this observation and Proposition 3.10, equation 4.5 follows. 21 To complete the argument, we adapt an argument of Bousfield's (pp. 477-479 * *of [4]). The reader who doesn't like the language of closed model categories can use Bou* *sfield's explicit constructions. Consider, for each ff, the canonical resolution K1ff! SoK1ff: Regarding K1ffas a constant simplicial coalgebra, this is a weak equivalence fr* *om K1ffto a fibrant cosimplicial coalgebra. In cCA+ , factor the composite K0ff! K1ff! SoK1* *ffas K0ffj!Wfofqff!SoK1ff where j is a weak equivalence and qffis a fibration. This can be done explicitl* *y using the cobar construction of [4]. Since compositions of fibrations are fibrations, Wfo* *fis fibrant, so j : K0ff! Wfofis a resolution of K0ff. Now let Wfof= Fp SoK1 W o ff ff and (4.6) Hff~=Fp K1ffK0ff! Wfof the induced map. Since pull-backs of fibrations are fibrations, Wfofis fibrant* *. Also, the pull-back diagram of cosimplicial coalgebras Wfof________wWfof | |q | | ff |u |u Fp ________wSoK1ff yields a spectral sequence, because qffis a fibration: Cotorsss*SoK1ff(Fp; ss*Wfof)t ) sss+tWfof: (To see this, use [11, xII.6] where the simplicial case is considered, or adapt* * [4].) The degree t comes from the cosimplicial degree. Since ss*So K1ff~=K1ff, ss*Wfof~=K0ffin d* *egree 0 and K0ff! K1ffis a surjective map of Hopf algebras, Cotors= 0 for s > 0 and Cotor0~* *= Hff, so (4.6) is a resolution of Hff. 22 Now there are isomorphisms (4.7.1) R*CA( )K0ff~=ss*( Wfof) ~= K0ff ff ff ff (4.7.2) R*CA( )Hff~=ss*( Wfof): ff ff To see this, for example, in the second case, note there is a weak equivalence * *SoHff! Wfofin cCA+ . Since every object of cCA+ is cofibrant, this is a cosimplicial homotopy* * equivalence. Hence SoHff! Wfofis a homotopy equivalence. The second isomorphism of (4.7.* *1) is ff (4.5). There is a pull-back diagram in cCA+ ffWfof________wffWfof | | | |qff |u |u Fp _________w SoK1ff: ff Since products of fibrations are fibrations we get, as above, a spectral sequen* *ce Cotor sK1ff(Fp; K0ff)t ) sss+t( Wfof): ff ff ff Here we use (4.5). Since ffK0ff! ffK1ff is a surjective map of Hopf algebras (this is the observation at the beginning * *of the proof) Cotors = 0 for s > 0 and, in degree 0, Fp K1ff K0ff~= (Fp K1ffK0ff) ~= Hff ff ff ff ff as required. Remark 4.8: This argument, although long, is basically formal except for one po* *int. To elucidate this, consider the situation: Bousfield defines a coalgebra to be nic* *e if there is a sequence of coalgebras Fp ! C j!C0 q!C1 ! Fp so that qj is trivial, q is surjective, C0 is an injective C1 comodule, there a* *re vector spaces W 0and W 1and isomorphisms C0 ~=SW 0, C1 ~=SW 1, and C ~=Fp C1 C0. Given a set of nice coalgebras Cffone has such sequences Fp ! Cff! C0ffqff!C1ff! Fp 23 and can form qff Fp ! Cff! C0ff--!C1ff but it isn't clear that qffis surjective, because of the way the product is def* *ined. (See 1.2.1 and 1.2.2.) So the final step of the argument breaks down. What we have d* *one is arrange a situation where the product map is surjective. Proposition 4.1 has the following immediate consequence. Corollary 4.9. Let H : I ! HA + be a diagram of abelian Hopf algebras. Then t* *he augmentation oH ! diag(oSoH) induces an isomorphism ~= * ss*(oH) ! RCA limIH: Proof: The argument is exactly that of Example 1.10, once one applies Propositi* *on 4.1 to prove R*CA(p)H ~=ss*pSo H ~=P H concentrated in cosimplicial degree zero. We use this to make a sequence of calculations. We'll need the following le* *mma, whose proof is postponed to the next section because the argument uses more model cat* *egory technology. Lemma 4.10. Let Ho ! Ko be a morphism of cosimplicial abelian Hopf algebras. If ss*D*Ho ! ss*D*Ko is an isomorphism, then ss*Ho ! ss*Ko is an isomorphism. Corollary 4.11. Let H : I ! HA + be a diagram of abelian Hopf algebras. If limI* *sD*H = 0 for s > 0, then RsCAlimIH = 0 for s > 0 and R0CAlimIH ~=limIH: Note: In this statement, limIsD*H are the usual derived functors of graded abel* *ian groups. 24 Proof: Again the statement about R0 is formal. For the higher derived functors,* * consider the cosimplicial Dieudonne module D*(oH) ~=oD*H: Then sssoD*H ~=limIsD*H = 0 if s > 0. Hence the augmentation limID*H ! oD*H induces a ss* isomorphism from the constant cosimplicial Dieudonne module of li* *mID*H to oD*H. Thus, by Lemma 4.10, the induced map limIH ! oH from the constant cosimplicial object is a ss* isomorphism and limIH ~=ss*(oH) ~=R*CAlimIC (using 4.9) concentrated in degree zero. Example 4.12: Suppose we have a tower over the natural numbers in HA + H1 H2 H3 . .:. Then under any of the following conditions R*CAlimHi~= limiHi concentrated in cosimplicial degree zero, 1) Each of the Hopf algebras is of finite type; 2) The tower {Hi} is pro-isomorphic, in the category of Hopf algebras, to a co* *nstant tower; 3) The tower {Hi} is Mittag-Leffler. Proof: In each case limsD*Hi= 0 for s > 0. For s > 1, this is always true. So o* *ne need only argue lim1D*Hi= 0. Case by case we have: 1) If H 2 HA + is of finite type then D*H is of finite type. This is because* * DnH ~= Hom HA+(H(n); H) Hom algebras(H(n); H) and H(n) is a free algebra on finitely* * many generators. The result now follows because lim1vanishes on graded abelian group* *s of finite type. 25 2) If {Hi} is pro-isomorphic to a constant tower, so is {D*Hi}. Hence lim1D*Hi* *= 0. 3) Mittag-Leffler is the following condition: for fixed i, let H(k)i= Im{Hi+k ! Hi}: Then H(k+1)i H(k)iand one demands that this descending chain stabilize: for eac* *h i, there is an N so that H(N)i= \H(k)i. The claim is that D*(H(k)i) ~=Im{D*Hi+k ! D*Hi} = (D*Hi)(k): If so, the tower of abelian groups {D*Hi} is Mittag-Leffler and has vanishing l* *im1. (See [5, xIX.3].) To see the claim, note that since DnH = Hom HA +(H(n); H), (D*Hi)(k) D*(H(k)i): To get the other inclusion, given a map H(n) ! H(k)ione gets a diagram Hi+k ________Hi+k i ij | | i | | i |u |u H(n) ________wH(k)i_______wHi: The dotted arrow exists since Hi+k ! H(k)iis surjective and H(n) is projective. We next turn to the case where limIsD*H = 0 for s 2; that is sssoD*H = 0 for s 2. We indicate how to calculate R*CAlimIH. First, suppose Do is a cosimplicial Dieudonne module, and Ho a cosimplicial* * abelian Hopf algebra so that D*Ho ~=Do: Suppose sssDo = 0 for s 2. We now state the calculation of ss*Ho. Note that ss0Do and ss1Do are Dieudonne modules and that D*ss0Ho ~=ss0Do. D* *efine graded vector spaces P ss1 and R1P ss1 by the exact sequence 0 ! P ss1 ! ss1Do V!ss1Do ! R1P ss1 ! 0: If k is a field and V a graded vector space over k which is concentrated in eve* *n degrees if the characteristic of k is not 2, let k[V ] be the "polynomial algebra" on V ; * *that is, k[V ] is the symmetric algebra on V , which becomes isomorphic to a polynomial algebra a* *fter one chooses a basis. 26 Lemma 4.13. Suppose p > 2 and ss1Do is trivial in odd degrees. Then there is an* * isomor- phism of bigraded Hopf algebras ss*Ho ~=ss0Ho (P ss1) Fp[R1P ss1] where P ss1 and R1P ss1 are in cosimplicial degree 1 and 2 respectively. Furth* *ermore the elements of P ss1 and R1P ss1 are primitive. This will be proved in the next section. If ss1Do is not trivial in odd deg* *rees one can still write down the answer, but it is more complicated. The techniques of the * *next section suffice for the computation. If p = 2, the conclusion of 4.13 holds coalgebras without restrictions on s* *s1Do, however there is the possibility of an algebra extension. Lemma 4.14. If p = 2, there is a natural homomorphism fi : P ss1 ! R1P ss1 doubling internal degree and an isomorphism of Hopf algebras ss*Ho ~=ss0Ho F2[P ss1] F2[R1P ss1]=(fix + x2) where P ss1 and R1P ss1 are in cosimplicial degree 1 and 2 respectively. Furthe* *rmore P ss1 and R1P ss1 are primitive. This also will be proved in the next section. Example 4.15: Consider a tower of abelian Hopf algebras over the natural numbers H1 H2 . . . where each Hiis injective as a coalgebra. Then we must calculate with lim1D*Hi.* * However there is an exact sequence 0 ! limD*Hi! D*Hi! D*Hi! lim1D*Hi! 0: Since V is surjective on D*Hi for all i (by Corollary 3.15) it is surjective on* * lim1D*Hi. Hence R1P ss1 = 0 and R*CAlimHi= limHi (P lim1D*Hi): 27 Furthermore, treating D*Hi! D*Hi as a two term cochain complex, one gets a short exact sequence 0 ! R1P (limHi) ! lim1P Hi! P lim1D*Hi! 0: Hence if R1P (limHi) = 0, or equivalently, if limHi is an injective coalgebra, * *lim1P Hi ~= P lim1D*Hi. To be specific, let Hi = p(xi; xi+1; : :):be the divided algebra on infinit* *ely many generators of degree 2t for some t, and let Hi+1 ! Hi be the inclusion. Then li* *mHi= Fp and 1 1M lim1 P Hi= Fp= Fp: i=1 i=1 Hence 1 1M R*CAlimHi~= ( Fp= Fp) i=1 i=1 with the generators in cosimplicial degree 1 and internal degree 2t. Note that * *RsCAlimHi6= 0 for all s 1. Example 4.16: Let p > 2 and Hi = Fp[xi; xi+1; : :]:=(xpi; xpi+1; : :):be the tr* *uncated polynomial algebra on primitive generators of degree 2t and Hi+1 ! Hi the inclu* *sion. 1 1L Then if W = Fp= Fp in degree 2t, then i=1 i=1 R*CAlimHi= (W ) Fp[W ] with W and W in cosimplicial degrees 1 and 2 respectively. The same example at * *p = 2 yields, with W in cosimplicial degree 1, R*CAlimHi= F2[W ]: Note in this case P R*CAlimHi is non-zero in infinitely many cosimplicial degre* *es. Example 4.17: Let G be a free group on finitely many generators and H 2 HA + a * *G- Hopf algebra; that is G acts on H through morphisms in HA + . Then D*H is a G-m* *odule and limD*H = H0(G; D*H) = (D*H)G and limsD*H = Hs(G; D*H) 28 which is zero for s > 1. Finally H*(G; D*H) can be calculated by an exact seque* *nce n 1 (4.18) 0 ! H0(G; D*H) ! D*H @! D*H ! H (G; D*H) ! 0 i=1 where, if oi2 G, 1 i n, are the generators @(x) = (x - o1x; : :;:x - onx): Let's confuse G with category with one object and G as morphisms. Note that in* * CA+ the inclusion limGH HG is usually strict. Assume p > 2 and that H is concentrated in even degrees. Then R*CAlimGH ~=limGH (P H1(G; D*H)) Fp[R1P H1(G; D*H)]: If H is injective as a coalgebra (e.g., H*BU), then V is surjective on D*H, s* *o on H1(G; D*H) by 4.18 and R*CAlimGH ~=limGH (P H1(G; D*H)): The latter formula holds at p = 2 if H is injective as a coalgebra, but not nec* *essarily in even degrees. We finish this example by showing that R*CAlimGH can be identified as a Cot* *or. Fix generators o1; : :;:on of G and define two maps H ! H__._._.H-z____"= Hn+1 n+1 as follows. The first is n+1 : H ! Hn+1 , the (n + 1)stiterated diagonal. The s* *econd is the composite n+1 1o1...on H ---! Hn+1 ------------! Hn+1 : Both of these maps make H into an Hn+1 comodule. Denote the first comodule stru* *cture by H, the second by Ho. Proposition 4.19. Let H be a connected G-abelian Hopf algebra. Then there is a * *natural isomorphism of bigraded Hopf algebras R*CAlimGH ~=CotorHn+1 (H; Ho): 29 Proof: Let Bo(H) = Bo(H; Hn+1 ; Ho) be the cobar construction. we will show the* *re is a natural map (4.20.1) oD*H = D*oH ! D*Bo(H) giving an isomorphism on cohomotopy. The result will then follow from Corollary* * 4.9 and Lemma 4.10. If M is any G-module, then one defines Hs(G; M) = limGsM = sssoM: However, one can calculate H*(G; M) as the cohomology of the two stage cochain * *complex T *(M) = {M x M @!M_x_._.x.M_-z_____"} n+1 where @(x; y) = (x-y; x-o1y; : :;:x-ony). Indeed, H0T *(M)G ~=MG and HsT *(M) =* * 0 for s > 0 and M an injective G-module, so there must be a natural map of cochain complexes (4.20.2) NoM ! T *(M) inducing an isomorphism on cohomology. (This is a standard fact about derived f* *unctors; see [6] x3.1). Now one easily checks that ND*Bo(H) ~=T *D*H. Hence the normal* *ized cochain equivalence of (4.20.2) gives an (unnormalized) cohomotopy equivalence * *in (4.20.1) and the result follows. x5. From cosimplicial Dieudonne modules to cosimplicial Hopf algebras. In this section we prove those technical results needed in Section 4 on pas* *sing from ss*D*Ho to ss*Ho where Ho is a cosimplicial abelian Hopf algebra. We end with * *some generalities. The first result we used was Lemma 4.10, which we record now as: Proposition 5.1. Let Ho ! Ko be a morphism of cosimplicial abelian Hopf algebra* *s. If ss*D*Ho ! ss*D*Ko is an isomorphism, then ss*Ho ! ss*Ko is an isomorphism. 30 This is proved by combining Lemmas 5.3, 5.4 and 5.6 below with some model c* *ategory techniques. The argument is given in 5.7. Because D has enough injectives there is a simplicial model category struct* *ure on the category cD. Indeed, the simplicial structure is the obvious analog of that giv* *en for cCA+ in (2.1) and a morphism f : Ao ! Bo in cD is 5.2.1) a weak equivalence if it is a ss* isomorphism; 5.2.2) a cofibration if the normalized chain complex Nf : NAo ! NBo is an inje* *ctive in positive degrees; and 5.2.3) a fibration if it is a surjection and Ko = ker{f : Ao ! Bo} is level-wi* *se an injective object. The proof is entirely standard; see [2, Proposition 6.5] or [11, xII.4] and* * use the fact that the opposite category of cD is the category of simplicial objects in Dop a* *nd Dop is an abelian category with enough projectives. Lemma 5.3. Suppose Ho ! Ko is a morphism of cosimplicial abelian Hopf algebras.* * If D*Ho and D*Ko are fibrant in cD and ss*D*Ho ! ss*D*Ko is an isomorphism, then ss*Ho ! ss*Ko is an isomorphism. Proof: Every object of cD is cofibrant so a weak equivalence of fibrant objects* * is a homotopy equivalence. Hence Ho ! Ko is a cosimplicial homotopy equivalence. Lemma 5.4. Let H-1 ! Ho be an augmented cosimplicial abelian Hopf algebra and suppose ss*D*Ho ~=D*H-1 concentrated in cosimplicial degree 0. Then ss*Ho ~=H-1* * . Proof: We will define sub-cosimplicial Hopf algebras H-1 = H(-1) H(0) H(1) . . .Ho so that ss*H(n) ~=ss*H(n + 1) for all n and H(n)s = H(n + 1)s for all s n. The* * result follows. To define H(n) consider the normalized augmented cochain complex of Dieudon* *ne modules 0 ! D*H-1 ! ND*H0 @!ND*H1 @!ND*H2 ! . .:. Let Kn = ker(@ : ND*Hn ! ND*Hn+1 ) ~= Im(@ : ND*Hn-1 ! ND*Hn ) and let H(n) 2 cHA + be the augmented object with augmented normalized Dieudonne module 0 ! DH-1*! ND*H0 ! ND*H1 ! . .!.ND*Hn ! Kn+1 ! 0 ! . . . 31 Then there is a pull-back diagram in cHA +, n 0, H(n - 1) _______wH(n) | | | | |u |u Fp __________wK(n) where K(n) 2 cHA + is the object with normalized Dieudonne module (5.5) . .0.! Kn+1 ! Kn+1 ! 0 . . . in degrees n and n + 1. Thus H-1 = H(-1), and there is a spectral sequence Cotor*ss*K(n)(Fp; ss*H(n)) ) ss*H(n - 1): Now the chain complex of (5.5) has a contraction, so K(n) has a cosimplicial co* *ntraction and the ss*K(n) ~=Fp. Lemma 5.6. Let Ho 2 cHA + be a cosimplicial abelian Hopf algebra. Then there * *is a natural map " : Ho ! Ho1in cHA + so that ss*" and ss*D*" are isomorphisms and * *D*Ho1 is fibrant as an object in cD. Proof: For M 2 D, let (with J(n) as in 3.8) J(M) = J(n) n x2Hom (Mn;Q=Z) and let M ! J(M) be the evident inclusion. Then for M 2 D there is a natural in* *jective resolution M ! J0(M) ! J1(M) ! . . . with J0(M) = J(M) and Jn+1 (M) = J(cokerJn-1 (M) ! Jn (M)). Thus in H 2 HA + one gets a natural cosimplicial resolution in cHA + H ! J(H)o where the normalized Dieudonne module of J(H)o is J*(D*H). By Lemma 5.4, H ~= ss*J(H). If Ho 2 cHA + let J(Ho)o be the obvious bicosimplicial object and se* *t Ho1= diagJ(Ho)o. A bicomplex argument shows that " : Ho ! Ho1is an ss* and ss*D* is* *o- morphism. Since D*Ho1is a product of injectives at each level, D*Ho1is fibrant* * by the definition 5.2.3. 32 5.7 Proof of Proposition 5.1: Use Lemma 5.6 to produce a diagram Ho _______wfKo " | |" H| | K |u |u Ho1_______wf1Ko1 so that the conclusions of Lemma 5.6 hold. Now D*f, D*"H , and D*"K are all ss** * isomor- phisms, so D*f1 is a ss* isomorphism. Hence, by Lemma 5.3 and Lemma 5.6, ss*f1 * *is an isomorphism. Since ss*"H and ss*"K are isomorphisms, so is ss*f. The next project is to calculate ss*Ho where sssD*Ho = 0 for s > 1. By Lemm* *a 5.1 we may assume ND*Hs = 0 for s > 1, simply by letting H(1)o Ho be the sub-cosimplicial Hopf algebra with 8 < ND*H0 s = 0 ND*H(1)s = : ker(ND*H1 ! ND*H2) s = 1 0 s 2: Then ss*D*H(1)o ~=ss*D*Ho, so ss*H(1)o ~=ss*Ho. For short, let us write Ds for * *ND*Hs. Then there is a pull-back diagram of cochain complexes ND*Ho __________w {D1 ! D1 ! 0 ! . .}. | | | | : |u |u {D0 ! 0 . .}.__________w{D1 ! 0 ! . .}. Hence, there is an associated pull-back diagram of cosimplicial Hopf algebras Ho ________wA0 | | | | |u |u H0 _______wA1: Therefore we get a spectral sequence Es;t2= CotorsA1(Fp; H0)t ) sss+tHo 33 where A1 is the Hopf algebra so that D*N1 ~=D1. The grading t arises from the c* *osimpli- cial degree on these cosimplicial Hopf algebra. Since A1 and H0 are both in cos* *implicial degree 0, the spectral sequence collapses. Furthermore the obvious change of ri* *ngs implies (5.8) ss*Ho ~=Cotor*A1(Fp; H0) ~=ss0Ho Cotor*H1(Fp; Fp) where H1 is the Hopf algebra, so that D*H1 ~=ss1D*Ho. We now show how to calculate Cotor*H1(Fp; Fo) as a functor of (5.9) P H1 ~=P ss1D*Ho and R1P H1 ~=R1P ss1D*Ho: Choose an exact sequence of Dieudonne modules 0 ! D*H1 ! J0 ! J1 ! 0 where V is surjective on J0 and J1. This can be done because D has enough injec* *tives and V is surjective on injectives. This corresponds to a short exact sequence of Ho* *pf algebras Fp ! H1 ! K0 ! K1 ! Fp where Ki, i = 0; 1, are injective as coalgebras. Let Bo(.) denote the cobar con* *struction, so that ss*Bo(K) = Cotor*K(Fp; Fp): Then we have a pull-back diagram of cosimplicial Hopf algebras Bo(H1) _______wBo(K0) | | | | |u |u Fp __________wBo(K1) and hence a spectral sequence (5.10) Es;t2= Cotorsss*Bo(K1)(Fp; ss*Bo(K0))t ) Cotors+tH1(Fp; Fp) If K 2 HA + is injective as a coalgebra, K ~=S(W ) for some graded vector space* * W and (where if p > 2, W is concentrated is even degrees) ss*Bo(K) ~=Cotor*K(Fp; Fp) ~=(P K): 34 This isomorphism is natural in maps of injective coalgebras. Thus we can write E2 ~=Cotor*(PK1)(Fp; (P K0)): By a change of rings since is isomorphic to (5.11) (P H1) Cotor*(R1PH1)(Fp; Fp) ~=(P H1) Fp[R1P H1] since there is an exact sequence 0 ! P H1 ! P K0 ! P K1 ! R1P H1 ! 0: Since (5.9) is a spectral sequence of Hopf algebras it must collapse, and there* * can be no coalgebra extensions. Thus we can give 5.12: The proof of 4.13: Combine 5.8, 5.9, and 5.11 with the observations that * *if p > 2 there can be no algebra extensions. If p = 2 there can be an algebra extension in the spectral sequence of 5.10* *. For example if H1 is an exterior algebra on a single generator, then CotorH1(F2; F2* *) ~=F2[x]. This example is generic, and is detected by an operation fi : P H1 ! R1P H2. To define fi consider a short exact sequence of Dieudonne modules 0 ! D*H1 "!J0 q!J1 ! 0 where V is surjective on J0. Let x 2 P H1 = ker{V : D*H1 ! D*H1}. Then there is* * a y 2 J0 so that V y = x. The element q(y) 2 P H1 = ker{V : J1 ! J1}: Then fi(x) is the image of q(y) in R1P H1 in the exact sequence 0 ! P H1 ! P H0 ! P J1 ! R1P H1 ! 0: It is an easy exercise that this is independent of the choices and fi(x) = 0 if* * and only if x is in the image of V in D*H1. 5.13: The proof of 4.14: Because the spectral sequence 5.10 collapses and there* * can be no coalgebra extensions, the result follows from 5.8, 5.9, and 5.11 once we * *determine 35 the algebra extensions. Since CotorH1(Fp; Fp) commutes with filtered colimits,* * we may assume H1 is of finite type. Thus CotorH1 (F2; F2) ~=ExtH*1(F2; F2): Use Borel's structure theorem to write an algebra decomposition O ni H*1~= F2[xi]=(x2i ); i where 1 ni 1. Since an algebra decomposition of this sort yields a tensor prod* *uct on Ext , we may assume n H*1~=F2[x]=(x2 ) with 1 n 1. But, if deg(x) = t, we have ae i (D*H)k ~= Z=2Z0 ko=t2ht;er0wiisen - 1 and V is surjective. But since ae CotorH1(F2; F2) ~= F2[P(H1];P H 1 n = 1 1) F2[R PnH1];> 1; the result follows. x6. The homology spectral sequence. This section joins algebra to homotopy theory and writes down a spectral se* *quence passing from R*CAlimIH*X to H* holimIX. Convergence is discussed and examples * *are given. Recall that Bousfield and Kan [5, Chap XI] define homotopy inverse limits i* *n the following manner. Let X : I ! S be an I-diagram of spaces. Then one has the ass* *ociated cosimplicial space oX and (6.1) holimX = holimIX = Tot(oX): If each X(i) is fibrant, then oX is a fibrant cosimplicial space, and the assoc* *iated homo- topy spectral sequence (for a pointed diagram) (6.2) ssssstoX = limssstX ) sst-sholim X 36 is the expected one. If some X(i) is not fibrant, one probably doesn't want to* * take its homotopy inverse limit. Rather one should take a morphism of diagrams X ! Y whe* *re each X(i) ! Y (i) is a weak equivalence with Y (i) fibrant and simply define ho* *limX = holim Y . This is analogous to what occured in section 2 or, in this case, in [* *5,xXI.8]. Then (6.2) holds regardless. Associated to the cosimplicial space oX is a homology spectral sequence sssHt(oX) ) Ht-s(oX): This spectral sequence, due to Anderson [1], is discussed in [3, 4.3]; however,* * since nX is, in general, an infinite product, this spectral sequence does not have an ac* *cessible E1 term. Hence we propose a modification. For a space X, let j : X ! FopX be the Bousfield-Kan resolution of X. It i* *s the cosimplicial resolution of X derived from the triple on space sending a simplic* *ial set X to the simplicial vector space FpX generated by X. (Actually this is Lannes's vari* *ant [8,x1.5] of the Bousfield-Kan resolution.) Then the Bousfield-Kan p-completion on X is d* *efined by Xp = Tot(FopX): Now if X : I ! S is an I-diagram, FopX is an I-diagram of cosimplicial spaces. * *Thus we can form the bicosimplicial space oFopX and its diagonal diag(oFopX). That this is a good thing to do is indicated by t* *he following two results. Lemma 6.3. The cosimplicial space diag(oFopX) is fibrant and Tot(diag oFopX) ~=holimXp; the homotopy inverse limit of the p-completions. Proof: The first claim is proved exactly as in [13, Lemma 7.1]. For the second* *, write oFopX = {nFmpX} and call n and m the horizontal and vertical directions, respec* *tively. Then by [13, Proposition 8.1] Tot(diag oFopX) ~=TothTot v(oFopX) where h and v mean take Tot in the horizontal or vertical directions. Since Tot* * commutes with products, Totv(nFopX) ~=nXp and the result follows. 37 Lemma 6.4. Suppose X : I ! S is an I-diagram of connected spaces. There is a na* *tural isomorphism of bigraded coalgebras ss*H* diagFopXo ~=R*CAlimH*X: Proof: This is a direct application of the results of Section 2. Regard H*X : I* * ! cCA+ as a constant I-diagram of cosimplicial coalgebras. Then H*X ! H*FopX is a morphism of I-diagrams in cCA+ and H*X(i) ~=ss*H*FopX(i) for all i. Furthe* *rmore, for any space connected space Y , there is a functorial isomorphism, with W (Y * *) a vector space functorial in Y , H*FpY ~=S(W (Y )) because FpY is a mod -p generalized Eilenberg-MacLane space. Hence H*FpoX(i) is* * almost- free, hence fibrant and, by 2.12, R*CAlimH*X ~= ss* diagoH*FopX ~= ss*H*(diag oFopX): Thus we have Proposition 6.5. There is a second-quadrant homology spectral sequence [RsCAlimIH*X]t ) Ht-sholimIXp: Proof: This is the homology spectral sequence sssHtdiag oFopX ) Ht-sTot(diag oFopX) using Lemmas 6.3 and 6.4. 6.6: Note on convergence: Because each of the spaces nFnpX is simple and p- nilpotent any of Bousfield's Theorems 3.2, 3.4, or 3.6 will apply to give stron* *g convergence. For example, we will have strong convergence if X : I ! CA+ is a pointed I-diag* *ram and either 1) ssssstdiag FopX = 0 for t - s 0 and for each s there are only finitely man* *y n so that ssssss+n diagFopX 6= 0, or 38 2) If [R0CAlimH*X]0 ~=Fp, [RsCAlimH*X]t = 0 for t - s 1 and for each s there * *are only finitely many n so that [RsCAlimH*]s+n 6= 0: There are other convergence results in [3] and [13]. We devote the rest of this section to examples. Example 6.7: We prove here that if {Xff} is a set of connected nilpotent spaces* * with, for all ff, H*Xffthe coalgebra underlying an abelian Hopf algebra, then H*( Xff) ~= H*Xff~=H*( (Xff)p) ff ff where the product on the right is in CA+ . Indeed if ffFopXff! diag(oFopX) is the augmentation we get a diagram of sp* *ectral sequences ss*H*( FopXff)) H*(Tot( FopXff)) ff ff (6.7.1) || || |u |u R*CA( )H*Xff ~=ss*H*(diag oFopX) ) H*( (Xff)p) ff ff As in example 1.11, the map on E2 terms is an isomorphism. Since Tot commutes w* *ith products, Tot( ffFopXff) ~= ff(Xff)p, so we have isomorphic spectral sequence* *s. But in [3], 4.14 it is proved that this spectral sequence converges completely to H*( * * ffXff). By Proposition 4.1, R*CA( )H*Xff~= H*Xff ff ff concentrated in cosimplicial degree zero. This yields one of the isomorphisims* *. On the other hand, since (Xff)p is also nilpotent H*( (Xff)p) ~= H*(Xff)p ~= H*Xff: ff ff ff Note that we have proved that the isomorphic spectral sequences of (6.7.1) conv* *erge strongly. Example 6.8: Let X : I ! S be an I-diagram of pointed, connected simple spaces * *and suppose 1) for all i 2 I, H*X(i) is the coalgebra underlying an abelian Hopf algebra a* *nd 39 2) limssstX = 0 for t-s 0 and for s there are only finitely many n so that li* *mssss+nX 6= 0. Then there is a strongly convergent homology spectral sequence R*CAlimIH*X ) H* holimIX: Note that 2) will happen if each X(i) is simply connected and I diagram is a to* *wer over the natural numbers or a space with an action by a free group. To see this consider the augmented bicosimplicial space oX ! oFopX: Then one gets a diagram of homology spectral sequences ss*H*(oX) ) H* holimX | | (6.8.3) | | |u |u R*CAlimIH*X ~=ss*H* diag(oFopX) ) H* holimXp: A bicomplex argument and Example 6.7 shows the map on E2 terms is an isomorphis* *m. Strong convergence for the top spectral sequence follows from 6.6.1. Note if h* *ypothesis 6.8.2 also holds for the I-diagram Xp : I ! S then the lower spectral sequence of (6.8.3) also converges. This follows from t* *he fact that H*X ~=H*Xp and the diagram ss*H*(oXp) ) H* holimXp |~ |~ |= |= |u |u R*CAlimIH*X ~= ss*H* diag(oFopXp)) H* holimXp obtained from 6.8.3 is an isomorphism of spectral sequences. Example 6.9: Let X1 X2 X3 . . . be a tower of connected spaces so that the resulting diagram H*X : I ! CA+ fact* *ors as through HA + ; in short, one has a tower of abelian Hopf algebras {H*Xn}. Suppo* *se this 40 tower of coalgebras is either of finite type, Mittag-Leffler, or pro-isomorphic* * to a constant tower. Then by Example 4.12, R*CAlimiH*Xi~= limiH*Xiconcentrated in simplicial * *degree 0. As throughout this paper the limit at the right is in the category of coalge* *bras. Thus if each of the spaces is simple and lim1ss1Xi= 0 we have (6.9.1) H* holimiXi~= limiH*Xi: Or if [limiH*Xi]1 = 0 we have (6.9.2) H* holimi(Xi)p ~=limiH*Xi: Note that some hypothesis on degree 1 is necessary. Consider the tower K(Z; 1) xpK(Z; 1) xpK(Z; 1) . . . where xp means the map multiplies by p on homotopy. Then holimK(Z; 1) ' K(Zp=Z; 0): However limiH*Xi~= Fp in degree 0. Note that (6.9.1) doesn't apply, and holimK(* *Zp; 1) = *, as predicted by (6.9.2). Example 6.10: Let X : I ! S be an I-diagram of connected simple spaces, again w* *ith the property that H*X is an I-diagram of abelian Hopf algebras. Suppose (for si* *mplicity) that p > 2 and that for all i 2 I, H*X(i) is concentrated in even degrees. Fina* *lly suppose limsD*H*X = 0 for s > 1, where D*H*X is the associated diagram of Dieudonne mod* *ules. Write lim1for lim1D*H*X. Then the spectral sequence becomes by Lemma 4.13, R*CAlimIH*X ~=limIH*X (P lim1) Fp[R1P lim1] ) H* holimIXp: The exact sequence 0 ! P lim1! lim1D*H V! lim1D*H ! R1P lim1! 0 shows (R1P lim1)t = 0 for t < 2p. So if [lim1D*H]2 = 0 the criterion of 6.6.2 i* *mplies the spectral sequence converges. Also, if the limssstX = 0 for s > 1 and lim1ss1X =* * 0, then the companion spectral sequence R*CAlimIH*X ) H* holimIX 41 will converge. In either case, if R1P lim1 = 0, the spectral sequence will collapse becaus* *e it is a spectral sequence of coalgebras. If I is the natural numbers (so X : I ! S is a tower) or I is the category * *of a finitely generated free group G (so X : I ! S is a G-space) then R1P lim1= 0 if V is sur* *jective on D*H*X(i) for all i 2 I. This is because there are exact sequences 0 ! limiD*H*X ! D*H*X(i) ! D*H*X(i) ! lim1D*H*X(i) ! 0 i i i in the first case and 0 ! H0(G; D*H*X) ! D*H*X ! D*H*X ! H1(G; D*H*X) ! 0 n in the second. We close with the following observation. Our homology diagrams are diagrams* * of Hopf algebras. In practice such may arise from diagrams of loop spaces. Under this a* *ssumption, we'd like to know we have a spectral sequence of Hopf algebras. Note that by [5* *, Corollary I.7.4] the p-completion of a loop space is an homotopy associative H-space. Lemma 6.11. Let X : I ! S be a diagram of loop spaces. Then the spectral sequen* *ce R*CAlimIH*X ) H* holim(X)p is a spectral sequence of Hopf algebras. Note: The Hopf algebra structure on R*CAlimIH*X is defined in Example 2.14. If * *each X(i) is simply connected (X(i))p = (X(i)p) and holim(X)p = holimXp since holim commutes with loops. The proof will show that the companion spectr* *al se- quence of 6.8 is also a spectral sequence of Hopf algebras. Proof: One has a map of I diagrams X x X ! X induced by H-space multiplication hence a map of spectral sequences R*CAlim(H*X H*X) ) H* holim(X x X)p | | | | |u |u R*CAlimH*X ) H* holim(X)p: 42 We show the top spectral sequence is the necessary tensor product spectral sequ* *ence. Consider the maps of cosimplicial spaces diagoFopX x diagFopX oX x oX ~=o(X x X) ! diagoFop(X x X): Then we get a diagram of spectral sequences R*CAlimIH*X R*CAlimIH*X u___~= ss*H*(o(X x X)) ___w~= R*CAlimI(H*X H*X) || || || || || || || |u|u |u|u |u|u H* holim(X)p H* holim(X)pu___f* H*(holim(X X) ____wg* H* holim(X x X)p): Since p-completion and holim commute with finite products, there is a commutati* *ve dia- gram holim(X x X) ________wf holim(X)p x holim(X)p g | hhj | h ' |u h holim(X x X)p with the diagonal map a homotopy equivalence. The result follows. References 1. D. Anderson, "A generalization of the Eilenbger-Moore spectral sequence, " B* *ull. A.M.S 78 (1972), pp. 784-786. 2. D. Blanc, "New model categories from old," manuscript, Haifa University, 199* *3. 3. A.K. Bousfield, "On the homology spectral sequence of a cosimplicial space",* * Amer. J. of Math. 109 (1987), pp. 361-394. 4. A.K. Bousfield, "Nice homology coalgebras", Trans. A.M.S. 148 (1970), pp. 47* *3-489. 5. A.K. Bousfield and D.M. Kan, Homotopy Limits, Completions and Localizations,* * Lec- ture Notes in Mathematics 304, Springer-Verlag 1972. 6. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Spring* *er- Verlag, New York, 1977. 7. F. Kuczmarski, "Cosimplicial Hopf algebras and Andre-Quillen cohomology", Th* *esis, University of Washington, 1995. 8. J. Lannes, "Sur les espaces functionnels dont la source est le classifiant d* *'un p-groupe abelienelementaire", I.H.E.S. Publications Mathematiques 75 (1992), pp. 135-244. 43 9. J. Lannes and S. Zarati, "Sur les U-injectifs", Ann. Sci. Ecole Norm. Sup. 1* *9 (1986), pp. 303-333. 10. J.-P. Meyer, "Cosimplicial homotopies", Proc. A.M.S. 108 (1990), pp. 9-17. 11. D.G. Quillen, Homotopical Algebra, Lecture Notes in Mathematics 43, Springe* *r-Verlag, Berlin 1967. 12. C. Schoeller, "Etude de la categorie des algebras de Hopf commutative conne* *xe sur un corps," Manusc. Math. 3 (1970), pp. 133-155. 13. B. Shipley, "Convergence of the homology spectral sequence of a cosimplicia* *l space," Thesis, MIT, 1994. 14. M. Sweedler, Hopf Algebras, W.A. Benjamin, New York, 1969. November 1994 Department of Mathematics University of Washington Seattle, WA 98195 e-mail: pgoerss@math.washington.edu 44