Homotopy and Homology for Simplicial Abelian Hopf Algebras by Paul G. Goerss1 and James Turner Abstract Let A be a simplicial bicommutative Hopf algebra over the field F2 with the property that ss0A ~=F2. We show that ss*A is a functor of the Andre-Quillen homology of A, where A is regarded as an F2 algebra. Then we give a method for calculating that Andre-Quillen homology independent of knowledge of ss*A. Let G be an abelian group. Since the work of Serre [19] and Cartan [6], we * *have know that the mod p homology of an Eilenberg-MacLane space K(G; n), n 1, depends on* *ly on Tor s(Z=p; G), s = 0; 1. More is true: the structure of H*K(G; n) = H*(K(G; * *n); Fp) as an unstable coalgebra over the Steenrod algebra depends only on there Tor gr* *oups and the Bockstein fi : Tor 1(Z=p; G) ! Tor 0(Z=p; G) = Z=p G which is the connecting homomorphism of the six term exact sequence obtained by* * ten- soring G with the short exact sequence 0 ! Z=p ! Z=p2 ! Z=p ! 0: The purpose of this paper to expand on this observation; indeed, our principal * *result will be that this is an algebraic fact, not a topological one, and an instance of a * *phenomenon that arises naturally in the study of simplicial bicommutative Hopf algebras. With an appropriate model for an Eilenberg-MacLane space - for example, the* * sim- plicial abelian group model - the mod p homology groups of the Eilenberg-MacLan* *e space K(G; n) are the homotopy groups of the simplicial bicommutative Hopf algebra Fp* *K(G; n); that is, H*K(G; n) = H*(K(G; n); Fp) ~=ss*FpK(G; n): Since K(G; n) is connected (as n > 0), we have that ss0FpK(G; n) ~=Fp; we will * *say that the simplicial Hopf algebra FpK(G; n) is homotopy connected. We will prove, at * *least when _________________________ 1The first author was partially supported by the NSF. 1 p = 2, that if A is a simplicial bicommutative Hopf algebra that is homotopy co* *nnected in the sense that ss0A ~=F2, then ss*A is determined by easily computable data - n* *amely, the Andre-Quillen homology of A. We now give a few details. To shorten notation, we often call a bicommutati* *ve Hopf algebra abelian. Such a Hopf algebra is an abelian group object in the category* * of cocom- mutative coalgebras. We will state our results at the prime 2, although we will* * prove many of them at an arbitrary prime p > 0. If A is a simplicial abelian Hopf algebra, A is a simplicial cocommutative * *coalgebra; thus ss*A is a graded abelian Hopf algebra and it supports an unstable action o* *f the Steenrod squares Sqi, i 0 compatible with the Hopf algebra structure. In gene* *ral, Sq0 6= 1. Also, A is a simplicial commutative algebra, so ss*A supports an act* *ion of the Cartan-Bousfield-Dwyer operations, here labeled ffij as in [10]. (But see * *also [6] and [5].) These operations satisfy axioms similar to the unstable, Cartan, and Adem* * axioms for Steenrod operations, and they also interact with the Sqi through some Nishi* *da-style relations. When all is said and done, we will say that ss*A belongs to the cate* *gory of Hopf D-algebras. All the formulas are spelled out in section 1, and are proved in t* *he second author's paper [22]. The analogous calculations have not been made at odd prime* *s, which is why we give many of our results at the prime 2. If H is a Hopf D-algebra, then the vector space of indecomposables QH suppo* *rts an action of the operations ffij and the Steenrod operations. The operations ffij * *assemble into an algebra B and we can collapse out the action of these operations by forming * *the vector space F2B QH. Because of the form of the Nishida-style relations, this quotient* * does not inherit an action of full Steenrod algebra (see formula 1.4.2 below), but only * *an action of the sub-algebra A(0) = F2[Sq0; Sq1]=(Sq1)2. Not every module over A(0) can be o* *btained by such a quotient: F2 B QH will always be torsion as a F2[Sq0] module. With this in mind, we define two categories. Let HD be the category of Hop* *f D- algebras and HD+ HD the full subcategory of objects that are graded connected* *2 - thus, H 2 HD if H0 ~=F2. For example, if A is a homotopy connected simplicial a* *belian Hopf algebra, then ss*A 2 HD+ . Also, let L+ be the the category of positively* * graded A(0) modules that are torsion as F2[Sq0] modules. Our first result is that the * *functor F2 B Q(.) : HD+ -! L+ is an equivalence of categories. Along the way, we will give a method for reco* *vering H 2 HD+ as a Hopf D-algebra from F2 B QH. This is done in section 3. The princi* *ple technical result is that the relations between the ffij and Sqi spelled out in * *1.4.2 force the Hopf D-algebra H to be free, in some sense, as an algebra over the operations f* *fij. See Lemma 3.1 and compare the results of Andre [1]. _________________________ 2The term connected, which we have somewhat obviously avoided, is reserved for * *its clas- sical geometric use. See section 5, especially 5.2 2 The next result, proved at the end of section 4, is that for any simplicial* * abelian Hopf algebra A, there is a natural isomorphism F2 B Qss*A ~=HQ*A where HQ*A is the Andre-Quillen homology of A regarded as a simplicial commutat* *ive Hopf algebra over F2. This is essentially formal, given the technical Lemma 3.1* * mentioned above. However, it leaves us with a dilemma: to calculate ss*A, at least for A * *homotopy connected, one first needs to know HQ*A; however, at least historically, the be* *st way to calculate HQ*A is to first know ss*A. To get out of this circle, we present ano* *ther way to calculate the Andre-Quillen homology of A. This is the point of sections 5 and 6. Our method is to use Dieudonne theor* *y, which can be summarized as follows: the category of abelian Hopf algebras over a perf* *ect field is an abelian category (a non-trivial result, by the way - see [7] xII.6 and [8] x* *III.3.7) with a set of generators. As such, it is equivalent to a category of modules over a ri* *ng. Dieudonne theory explains which modules over which ring. The details are explained in sec* *tion 5, as is the method, really quite simple, for calculating HQ*A. This program is marre* *d by one difficulty: it takes some work to keep track of the A(0) structure. This is why* * section 6 is long. As an amusing sidebar, we give a completely independent calculation of the * *homology of connected Eilenberg-MacLane spaces. See examples 5.15 and 6.19 below. Intere* *stingly, the method is not inductive. However, our primary interest is different. There * *are many spectral sequences of interest (see [4] or [12], for example) where the E2 term* * is the ho- motopy of some simplicial abelian Hopf algebra A. It is often relatively easy t* *o calculate HQ*A, especially if one avails oneself of Dieudonne theory. Then, by the resul* *ts of this paper, one knows the desired E2 term. This paper develops some of the algebraic theory of simplicial abelian Hopf* * algebras. We alert the reader to a companion paper [14], which concentrates more on the h* *omotopy theory. Incidentally, we see no essential obstacle to carrying out the program * *of this paper over any perfect field. Extending to other fields of characteristic two is str* *aightforward, but work at odd primes awaits the analog of [22] at those primes or, at the ver* *y least, the analog of formula 1.4.2 modulo indecomposables. Finally, the second author would like to thanks Jean Lannes for several fru* *itful con- versations on Hopf algebras and Dieudonne theory. x1. Hopf D-algebras. The section is devoted to the definition of the basic algebraic objects, an* *d to a few easy preliminary lemmas. In the first three sections we will be working at the * *prime 2. 3 A graded algebra A is a D-algebra if there are functions ffii: An ! An+i; 2 i n satisfying the axioms 1.1.1)ffii is a homomorphism if 2 i < n and ffin(x + y) = ffinx + ffiny + xy; 1.1.2)ffii(xy) = x2ffii(y) if the degree of x is 0, ffii(xy) = ffii(x)y2 if the* * degree of y is 0, and ffii(xy) = 0 otherwise. X j - i + s - 1 1.1.3)if i < 2j, ffiiffij(x) = ffii+j-sffis(x) i+1_2si+j_3 j - s 1.1.4)if x has positive degree, then x2 = 0. Then, using [10] along with [13] to get the exact form of the Adem relation* *s stated in 1.1.3, we have that if B is a simplicial commutative F2-algebra, then ss*B is a* * D-algebra. Let AD be the category of D-algebras. Note that, among other things, a D-algebra A is a divided power algebra. (S* *ee [1], among many references.) If x 2 A has positive degree n, define fl2(x) = ffin(x* *) and the higher divided power operations by fl2k+1(x) = fl2fl2k(x); k 1 and in n = 2i1+ 2i2+ . .+.2itwith t > 1, fln(x) = fl2i1(x) . .f.l2it(x): There is also a category CA of unstable coalgebras over the Steenrod operat* *ions. Thus C 2 CA is a graded cocommutative coalgebra equipped with a right action by Stee* *nrod operations (1.2) (.)Sqi : Cn ! Cn-i; i 0 These are subject to the usual Cartan and Adam relations and are unstable in th* *e sense that Sqi = 0 if 2i > n and if 2i = n, then Sqi is the Verschiebung of the coalg* *ebra C. Note that Sq0 6= 1; however, we do require that for all n, the vector space Cn * *is torsion as a F2[Sq0] module. If C is a simplicial coalgebra, then ss*C 2 CA. In this case, Sq0 : ssnC !* * ssnC is induced by the Verschiebung on the coalgebra Cn. That Cn is torsion as a F2[Sq0* *] module follows from the fact that every coalgebra is the filtered colimit of its finit* *e dimensional sub-coalgebras. (See [7], xI.6.) 4 An unstable Hopf algebra over the Steenrod algebra is an abelian group obje* *ct in CA. Thus H is an unstable Hopf algebra if there are morphisms in CA (1.3) : H H ! H j : F2 ! H making H into a graded, commutative F2 algebra. Notice that since is a morphis* *m in CA, the Steenrod operations satisfy the multiplicative Cartan formula. Let HA * *be the category of unstable Hopf algebras. If A and B are D-algebras, then 1.1.2 shows how to impose a D-algebra struc* *ture on A B. The resulting object in AD is the coproduct of A and D. A Hopf D-algebra H is at once an unstable Hopf algebra and a D-algebra and * *the two structures interact as follows: 1.4.1)the coproduct H ! H H and the counit ffl : H ! F2 are morphisms of D-alge* *bras; 1.4.2)we have the Nishida-style relations for x of simplicial degree n: X 2s - i - 1 (ffijx)Sqi= ffij-i+s(xSqs) if i > j s iX- j = fl2(x) + ffis(xSqs) if i = j 2s>j X j - i - 1 = ffij-i+s(xSqs) if i < j 6* *= n s i - 2s X X j - i - 1 = (xSqs)(xSqi-s) + ffij-i+s(xSqs) if i < j =* * n. 2s j and j = n, both sides of * *the equation are zero by instability. If i = j = n, then (ffinx)Sqn = fl2(x), again by insta* *bility. Finally, if i < j, the two possibilities for (ffijx)Sqi agree modulo indecomposables. Le* *t HD be the resulting category of Hopf D-algebras. If A is a simplicial bicommutative Hopf* * algebra, then ss*A is an object in HD. The purpose of this paper is to analyze the categ* *ory HD. We devote the rest of this section to building various adjoint functors. We* * begin by analyzing indecomposables. Let B be the algebra built from the operations ffii; thus, B is the graded * *tensor algebra on symbols ffii, i 2, of degree i, modulo the two sided ideal created from the* * relations of 1.1.3. If A is an augmented D-algebra, then, by 1.1.1 and 1.1.2, the vector spa* *ce QA of indecomposables of A is B module. Note that the top operations acts linearly on* * QA. We will be interested in the quotient vector space F2 B QA obtained by collapsing * *out the action of ffii, i 2. 5 If H is an unstable Hopf algebra or a Hopf D-algebra, then QH is an unstabl* *e right module over the Steenrod algebra, because of the multiplicative Cartan formula.* * However, if H is a Hopf D-algebra, then F2B QH does not inherit such a structure, becaus* *e of the formula for ffij(x)Sqj given in 1.4.2. Nonetheless, since the formula for ffij(* *x)Sqj requires j 2, the vector space F2 B QH does inherit an action by Sq0 and Sq1. Note that* * QH is torsion as a F2[Sq0] module. Let A(0) A be the sub-algebra of the Steenrod algebra generated by Sq0 and* * Sq1. Then A(0) ~=F2[Sq0; Sq1]=(Sq1)2: Let L be the category of unstable right A(0) modules, which are torsion as F2[S* *q0] modules. Unstable here means that xSq1 = 0 if the degree of x is 1. Lemma 1.5. The functor F2 B Q(.) : HD ! L has a right adjoint . Proof: The functor F2 B Q(.) preserves all colimits. Also, the category HD has * *a set of generators_one can either use a set of representatives for the isomorphism c* *lasses of objects H 2 HD so that F2 B QH is finite dimensional, or one can use the objects SD (C) 2 HD where C 2 CA ranges of a set of representative for the isomorphism * *classes of finite dimensional coalgebras. (The notation SD is explained below in 2.4-2.* *6.) In any case, the result now follows from the special adjoint functor theorem [15, xV.8] We will give a more concrete description of below in Proposition 1.8. If H 2 HA is an unstable Hopf algebra, then QH is an unstable module and, h* *ence, we may regard it as an object in L. An argument similar to that supplied for L* *emma 1.5 will show the functor Q : HA ! L has a right adjoint , but we wish to const* *ruct it concretely. To do this, let U be the category of right unstable modules over the Steenr* *od algebra. Again Sq0 6= 1, but an object in U is torsion when regarded as an F2[Sq0] modul* *e. The forgetful functor CA ! U has a right adjoint V given, essentially, by the dual* * of the Steenrod-Epstein universal enveloping algebra functor U [20, xII.5]. In fact, i* *f M 2 U is of finite type V (M) ~=U(M*)*; where asterisk denotes the F2 dual. In general, colimffV (Mff) ~=V (M) where Mff M ranges over the sub-modules of finite type. If M 2 U, note that V (* *M) is naturally an object in HA. The diagonal V (M) ! V (M) V (M) is obtained by app* *lying V to the diagonal M ! M x M. The resulting functor V : U ! HA is right adjoint * *to Q : U ! HA. 6 The forgetful functor U ! L also has a right adjoint : if M 2 L let Hom A(0* *)(A; M) be the vector space of right A(0)-module maps A ! M with right A-module structu* *re given by ('a)(x) = '(ax); and suitable grading. Then (M) = {x 2 Hom A(0)(A; M)|xSqi = 0 if2i > deg(x)} is the largest sub-module of Hom A(0)(A; M) which is unstable. Since composition of right adjoints yields a right adjoint we have: Lemma 1.6. The functor Q : HA ! L has right adjoint given by the formula (M) ~=V (M): To relate the functor : L ! HD to the functor : L ! HA we will use the fo* *llowing result. Lemma 1.7. The forgetful functor HD ! HA has a left adjoint F with the property* * that there is a natural isomorphism in L F2 B QF (H) ~=QH: This follows from the theory of algebras over triples, and will be proved i* *n the next section. First we have: Proposition 1.8. Let M 2 L. The there is a natural isomorphism in HA (M) ~=(M): Proof: Let H 2 HA. Then the result follows from the following sequence of natu* *ral isomorphisms: Hom HA (H; (M)) ~=Hom HD(F H; (M)) ~=Hom L(F2 B QF H; M) ~=Hom L(QH; M) ~=Hom HA(H; (M)): x2. The existence of the adjoint. 7 The functor F : HA ! HD left adjoint to the forgetful functor exists for es* *sentially formal reasons. We belabor the proof only to introduce some notation and ideas. We begin with a reduction. Let HA0 HA and HD0 HD be the full subcategories of graded connected objects H; thus, the unit map F2 ! H0 is an isomorphism. No* *te that if H 2 HA or H 2 HD, then the sub-Hopf algebra H0 H is a sub-object in HA (or HD). Define H+ ~=F2 H0 H: Then there is a natural isomorphism in HA (or HD) H ~=H0 H+ : Furthermore, if the forgetful functor HD0 ! HA0 has a left adjoint F0 : HA0 ! H* *D0, then this functor can be extended to the left adjoint F : HA ! HD0 by the formu* *la (2.1) F (H) ~=H0 F0H+ : If Q is the indecomposables functor, then F2 B QF (H) ~=QH0 F2 B QF0H+ : Hence, Lemma 1.7 is implied by the following result: Proposition 2.2. The forgetful functor HD0 ! HA0 has a left adjoint F0 with the property that there is a natural isomorphism F2 B QF0H ~=QH: This will be proved below. The key point is that both HA0 and HD0 are categ* *ories of algebras over a triple on a suitable category of coalgebras. In fact, let CA* *0 CA be the full subcategory of the category of unstable coalgebras over the Steenrod algeb* *ra specified by the property that C 2 CA is in CA0 if the counit ffl : C ! F2 induces an iso* *morphism C0 ~=F2. Then HA0 and HD0 are categories of algebras over CA0. In the case of HA0 this is straightforward. The forgetful functor HA0 ! CA0* * has a left adjoint S. As an algebra S(C) is the bigraded symmetric algebra on the vec* *tor space JC = ker{ffl : C ! F2}; this algebra acquires the structure of an algebra over the Steenrod algebra and* * the diagonal is the unique algebra map making the following diagram commute C __________wC C (2.3) ||u ||u S(C) _____wS(C) S(C): 8 One checks that S(C) 2 HA0 and then that S(.) is the required adjoint. If we write S : CA0 ! CA0 for the resulting triple on CA0, then HA0 is equ* *ivalent to the category of S algebras. This follows by direct calculation or by Beck's Th* *eorem [15, xVI.7]. In the case of HD0, we must work a little harder, mostly because Hopf D-al* *gebras are relatively complicated. First note that if V is the category of graded ve* *ctor spaces, the forgetful functor from the category AD of D-algebras to V has a left adjoi* *nt SD . This functor is characterized by the formula (see [9]) (2.4) SD (ss*V ) ~=ss*S(V ) where V is a simplicial vector space and S(.) is the symmetric algebra functor* *. In partic- ular, we have the following formulas. (2.5.1)If W = colimffWffis a filtered colimit of bigraded vector spaces, then t* *he natural map of bigraded vector spaces colimffSD (Wff) ! SD (W ) is an isomorphism; (2.5.2)for all direct sums there is a natural isomorphism of D-algebra SD (W1 W2) ~=SD (W1) SD (W2); (2.5.3)if W ~=Fp on a generator x of degree n, then there are isomorphisms SD (W ) ~=F2[x] ifn = 0 and (2.6.1) SD (W ) ~=E(ffiI(x)) ifn > 0 where E(.) denotes the exterior algebra and ffiI(x) runs over all compositions (2.6.2) ffiI(x) = ffii1ffii2. .f.fiis(x): which are admissible (i.e., it 2it+1), have s 0 and is 2, and with excess (2.6.3) e(I) = i1 - i2 - . .-.is n: Now, if C 2 CA0, we can give SD (JC) the structure of an algebra over the * *Steenrod algebra via the multiplicative Cartan formula and the formulas of 1.4. We can * *also give SD (JC) a diagonal by the obvious D-algebra analog of diagram (2.3). What is n* *ot clear is that this diagonal is a morphism of algebras over the Steenrod algebra. We * *isolate this as a separate lemma. 9 Lemma 2.7. Let C 2 CA0. Then SD (JC), with the structure defined above, is an o* *bject in HD0. Proof: Let V0 be the category of bigraded vector spaces V with V0 = 0. Then the* * functor J : CA0 ! V0 has right adjoint G characterized by the formula G(ss*W ) ~= ss*S** *(W ), where W is a simplicial vector space and S* is right adjoint to the forgetful f* *unctor from cocommutative coalgebras to graded vector spaces. If C 2 CA0, then the counit j* * : C ! G(C) is an injection, so SD (C) ! SD (G(C)) is an injection. Thus it is sufficient to prove the result for coalgebras of th* *e form G(V ). Fix V 2 V0 and choose W so that ss*W ~=V . Then, by 2.4, SD (G(C)) ~=ss*SS*(W ) and the result follows from the main theorem of [22]. Write SD : CA0 ! HD0 for the resulting functor. This name is an abuse of no* *tation, but a mild one, as there is a diagram SD HD0 _____wCA0u_____ | |J |u SD |u AD _______wVu_______ where the unlabeled arrows are forgetful functors. Note that SD : CA0 ! HA0 i* *s left adjoint to the forgetful functor. Let SD : CA0 ! CA0 be the resulting triple. T* *hen, again by Beck's Theorem, HD0 is equivalent to the category of SD -algebras. Thus we have two triples S and SD on CA0. Furthermore, if C 2 CA0, then, * *by forgetting to HA0, we may regard SD (C) as a object in HA0. The unit C ! SD (C)* *, then induces a morphism S(C) ! SD (C) in HA0 and, hence, a natural transformation in CA0 (2.8.1) C : S(C) ! SD (C): Note that C is a morphism of triples; that is, the following diagrams commute: C ______wjS(C) S S(C) _______wfflS(C) (2.8.2) = ||u ||uC ||u ||uC C _____wjSD(C) SD SD(C) _____wSD(C) 10 where the unlabeled vertical map on the right-hand square is the composite SS (C) SC-!SSD(C) SDC-!SDSD (C) which is equal to the composite SS(C) SC-!SDS(C) SDC-!SDSD (C): We use this structure to produce the adjoint F0. 2.9: Proof of Proposition 2.2: If H 2 HA0 is of the form H = S(C) for some C 2 CA0, define (2.10.1) F0H = SD (C): Given any morphism f : H = S(C) ! K in HA0 to an object K 2 HD0, the induced map C ! K in CA0 extends uniquely to a morphism f[ : F0H = SD (C) ! K in HD0 and the following diagram commutes in HA0: H ________S(C) [ [ [[] | | | | K: |u |u aeaeaeo F0H _____wSD (C) That is, Hom HA0 (H; K) ~=Hom HD0 (F0H; K) as required. If H 2 HA0 is general, we can write H as a cokernel in HA0 (2.10.2) S(C1) @!S(C0) ! H: Define F0K by the cokernel diagram in HD0 [ (2.10.2) F0S(C1) @!F0S(C0) ! F0H: Then one easily checks Hom HA0 (H; K) ~=Hom HD0 (F0H; K). Finally, if C 2 CA0, * *then F2 QSD (C) ~=QS(C) ~=JC: Hence (2.10.1)-(2.10.3) imply that F2 B QF0H ~=QH 11 for general H. This proof is obviously very general. Compare [2], x3.7, especially Theorem* * 2. x3. The equivalence of categories This section proves the statement that F2 B Q(.) : HD+ ! L+ is an equivalen* *ce of categories. Recall that an object H is graded connected in HD if H0 ~=F2 in deg* *ree 0 and that HD+ HD is the full-sub-category of graded connected objects. Likewise an * *A(0) module M is graded connected if M0 ~=0 and L+ L is the full sub-category of gr* *aded connected objects. We begin with the following lemma. We call a D-algebra A a free D-algebra i* *f there is a vector space V A so that the resulting map SD (V ) ! A is an isomorphism* * of D-algebra. The following is similar to results of Andre [1]. Lemma 3.1. Let H be a graded connected Hopf D-algebra. Then H is a free D-algeb* *ra. Proof: We use the formulas developed in 2.6. By Borel's structure theorem for H* *opf al- gebras, the formula 1.1.4), and a colimit argument, H is automatically an exter* *ior algebra; thus we need only show that QH is a free unstable B module. Here B is the algeb* *ra of ffii operations and unstable means ffii(x) = 0 if i is greater than the degree of x. Let I B be the augmentation ideal. Filter QH by the powers of I. Thus the * *nth term in the associated graded is EnQH ~=In QH=In+1 QH: In particular, E0QH ~=F2 B QH. The operations ffii define homomorphisms ffii: EnQH ! En+1QH: In fact, B is naturally a bigraded algebra, since the relations among the ffiia* *re homogeneous and E*QH is a bigraded module over B. Now, QH is an unstable right module over the Steenrod algebra; however, EnQH is not an unstable module, due to the formulas of 1.4. In fact, the Sqi, i 2,* * induce homomorphisms Sqi : EnQH ! En-1QH with the property that ae (3.2) ffij(x)Sqi = x;0ifi;=ij;fi 6= j: Here we are using that fl2(x) x modulo indecomposables. See [22], Proposition * *2.22. We will now show E*QH is free unstable B module. 12 For a symbol I = (i1; : :;:is), let ffiI = ffii1. .f.fiin, `(I) = i1 and J(I) = (i2; : :;:in): Hence I = (`(I); J(I)). Let {xff} be a basis for E0QH; then we need to show EnQ* *H has a basis given by all elements ffiI(xff) with I = (i1; : :;:in), I admissible, a* *nd e(I) n. We do this by induction on n. It is tautological if n = 0. Fix a linear relation in EnM X ff(I;ff)ffiI(xff) = 0: (I;ff) We need ff(I;ff)= 0. Write this sum as 2 3 X X 4 ff(I;ff)ffiI(xff)5= 0 i `(I)=i and apply Sqi to get an expression, by 3.2: X ff(I;ff)ffiJ(xff) = 0: e(I)=i The inductive hypothesis now implies ff(I;ff)= 0. Since, by the definition of E* **QH, the monomials ffiI(xff) span EnM, the result follows. Corollary 3.3. Let f : H ! K be a morphism of connected Hopf D-algebras. Then f is an isomorphism if and only if F2 B Qf : F2 B QH ! F2 B QK is an isomorphism. Corollary 3.4. Let F2 ! H0 ! H1 ! H2 ! F2 be an exact sequence of connected Hopf D-algebras. Then 0 ! F2 B QH0 ! F2 B QH1 ! F2 B QH2 ! 0 is exact in L1. Proof: Since we have an exact sequence of exterior Hopf algebras 0 ! QH0 ! QH1 ! QH2 ! 0 is exact. Since each QH2 is a free unstable B-module, this sequence is split as* * B-modules. 13 Proposition 3.5. Let M 2 L+ . Then the counit of the adjunction ffl : F2 B Q(M) ! M is an isomorphism. Proof: We will show the result holds for M one-dimensional. Then it holds for a* *ny finite M by Corollary 3.4 and the fact that if 0 ! M0 ! M1 ! M2 ! 0 is exact in L+ , then F2 ! (M0) ! (M1) ! (M2) ! F2 is exact in HD+ . This follows from Proposition 1.8 and structure of . The gene* *ral case then follows from the fact that both the source and the target of the counit ma* *p ffl commute with filtered colimits and our assumption that the objects L are torsion as F2[* *Sq0] modules. So suppose M ~=F2 concentrated in degree n, n > 0. Let C be the unique unst* *able coalgebra so that JC ~=M. (Hence C ~=H*Sn.) Let SD : CA0 ! HD0 be left adjoint * *to the forgetful functor, and consider SD (C). Then F2 B QSD (C) ~=JC ~=M; let SD (C) ! (M) be the induced map in HD. Then F2 B SD (C) ! F2 B Q(M) ffl!M is an isomorphism, so the counit ffl is onto. Since Q(M) ~=Q(M) as vector space* *s (by Proposition 1.8) a dimension count and the fact that Q(M) must be a free unstab* *le B module imply QSD (C) ~=Q(M): Hence, by Corollary 3.4, SD (C) ~=(M) and the counit ffl is an isomorphism. Theorem 3.6. Let H 2 HD be a connected Hopf D-algebra. Then the unit of the adjunction j : H ! (F2 B QH) is an isomorphism and the functors and F2BQ define an adjoint equivalence of c* *ategories between HD+ and L+ . 14 Proof: The induced map F2 B QH ! F2 B Q(F2 B QH) ffl!F2 B QH is an isomorphism by Proposition 3.5, so Corollary 3.4 implies that the unit ma* *p j is an isomorphism. Since both the unit map j and the counit map ffl are isomorphisms* *, the statement about the equivalence of categories follows. x4. The Andre-Quillen homology of Hopf algebras. The Andre-Quillen homology of a simplicial commutative algebra is a total d* *erived functor of abelianization in the category of commutative algebras, and, as such* *, relies on the existence of free commutative algebras. To compute the Andre-Quillen homol* *ogy of a bicommutative Hopf algebra, therefore, one may be forced to leave the categor* *y - the free commutative algebra on a coalgebra is not necessarily a Hopf algebra. Thi* *s section remedies this potential defect. The remainder of the section is devoted to defi* *nitions and examples. In this and the following sections, we will be working at a general p* *rime p > 0, except where explicitly noted. If A is a fixed Fp algebra, we are concerned with the Andre-Quillen homolog* *y of aug- mented simplicial A algebras, which we will call HQ*(B; A). Often we will take * *A = Fp. This homology is defined with the technology of [17] and [18]. Specifically, th* *e augmenta- tion ideal functor from simplicial A-algebras to graded vector spaces has a lef* *t adjoint SA , with SA (V ) = A S(V ) and S(.) the symmetric algebra functor. Write SA for the resulting cotriple of * *augmented A-algebras. If B is an augmented A-algebra, one thus gets a simplicial resoluti* *on SAoB ! B of B; if B is a simplicial augmented A algebra, this resolution is a bisimplici* *al object and (4:1) HQ*(B; A) ~=ss*Qdiag(Fp A SAoB) ~=ss*Q(Fp A diagSAoB): Note that in this equation we replace diag(S AoB) with any X, where X ! B is a * *weak equivalence and X is cofibrant in the usual model category structure on simplic* *ial aug- mented A-algebras. [17, xII.4]. If A = Fp, we will write HQ*B for HQ*(B; Fp). W* *e will have occasion below to work with algebras over fields other than Fp; if so, we will * *be explicit. Now suppose A ! B is a morphism of abelian Hopf algebras. To compute HQ*(B;* * A), we are required, by (4.1), to pass out of the category of Hopf algebras. We rem* *edy this situation. Let C* be the category of pointed coalgebras; that is, graded cocommutative* * coalgebras C equipped with a coalgebra map Fp ! C. The forgetful functor H ! C* has a left* * adjoint 15 S+ . To construct S+ C one first forms S(JC), where JC = coker{Fp ! C} and equ* *ips S(JC) with the unique diagonal which is a morphism of algebras and makes the fo* *llowing diagram commute C ____________wACC C | | |u |u S(JC) _____wS(JC) S(JC): Then S(JC) is an abelian monoid object in C*, hence not yet necessarily a Hopf * *algebra. Then Hopf algebra S+ (C) is a suitable group completion of S(JC). To describe this group completion, first work over the field Fp_the algebra* *ic closure of Fp. If C is a pointed coalgebra over Fp, let SFp(JC) be the obvious analog o* *f the above constructions. Then (4.2) S+Fp(C) = SFp(JC)[X-1 ] where X SFp(JC) is the multiplicatively closed set of elements x so that x = x x: If C 2 C*, then Fp FpC acquires an action of the Galois groups G = Gal(Fp=Fp) v* *ia the formula oe(a x) = oe-1 (a) x. This actions extends to a G-equivariant map SFp(Fp Fp JC) ! S+Fp(Fp Fp C): Thus we get a morphism over abelian monoid objects, by taking invariants (4.3) ffl : S(JC) ! S+Fp(Fp Fp C)G and the target is a Hopf algebra. Lemma 4.4. The morphism ffl of abelian monoid objects is a group completion in * *C*, hence there is a natural isomorphism S+ (C) ~=S+Fp(Fp Fp C)G : Proof: Let C ! H be a morphism in C* from C to a Hopf algebra. Thus one get a diagram over Fp Fp Fp C _____wS+F(F C) fl p p Fp | flf |u fli FpFp H where f is a morphism of Hopf algebras. Now take invariants and use that if V i* *s any Fp vector space, the natural map V ! (Fp Fp V )G is an isomorphism. This construction allows us to prove a crucial lemma. 16 Lemma 4.5. Let C 2 C* and ffl : S(JC) ! S+ (C) the group completion map. Then HQ*(S+ (C); S(JC)) = 0: Proof: First, using 4.1, we see that if A ! B is any morphism of Fp algebras, t* *hen Fp Fp HQ*(B; A) ~=HQ*(Fp Fp B; FpFp A): Since Fp Fp S(JC) ~=SFp(J(Fp Fp C)) and, by Lemma 4.4, Fp Fp S+ (C) ~=S+F(F C) p p Fp we need only show that for any pointed coalgebra over Fp HQ*(S+Fp(C); SFp(JC)) = 0: But this follows from the fact that S+Fp(C) is the localization of SFp(JC) at a* * multiplica- tively closed subset. See [18]. Corollary 4.6. Let C 2 C*. Then ae + ~ HQqS+ C ~= QS0;C = JC; ifqi=f0q > 0: Proof: For any algebra B, HQ0B ~= QB. On the other hand, there is a long exact sequence - the transitivity sequence of [18]: ! HQqS(JC) ! HQqS+ (C) ! HQq(S+ (C); S(JC)) ! HQq-1S(JC) . .!.HQ0(S+ (C); S(JC)) ! 0: Fix a Hopf algebra A. Then the category of Hopf algebras under A is Hopf al* *gebras B equipped with algebra map A ! B. Write A # HA for this category. Note Fp # HA =* * HA. Proposition 4.7. Let B 2 s(A # HA) be a simplicial Hopf algebra under A. Suppo* *se X ! B is a weak equivalence in s(A # HA) so that for each s 0, there is a poin* *ted coalgebra Cs and an isomorphism in A # HA Xs ~=A S+ (Cs): Then HQ*(B; A) ~=ss*Q(Fp A X). 17 Proof: First note that HQ*(X; A) ~=HQ*(B; A) since X ! B is a weak equivalence.* * Now form the bicomplex Q(Fp A SAoX) = {Q(Fp A (SA )t+1Xs}: Filtering by degree in t gives a spectral sequence E1s;t= HQt(Xs; A) ) HQs+t(X; A): But, since Xs ~=A S+ (Cs) HQt(Xs; A) ~=HQt(S+ (Cs)): Thus, by the previous result, HQt(Xs; A) = 0 for t > 0 and HQ0(Xs; A) ~=QS+ (Cs) ~=Q(Fp A Xs): Thus E2s;t= 0 if t > 0 and Es;02= sssQ(Fp A X): Resolutions X ! B such as encountered in the hypotheses of Proposition 4.7 * *exist: the forgetful functor A # H ! C* has left adjoint T AC = A S+ C; if TA : A # H ! A # H is the resulting cotriple, we may take (4.8) X = diagToAB: Note that if A = Fp, then the Frobenius (i.e., the pth power map) is an injecti* *on of Xn for all n. We now provide the result which connects Andre-Quillen homology with the id* *eas of the previous sections. For this we are required to work at the prime 2. Let A b* *e a simplicial abelian Hopf algebra over F2 and X ! A a resolution satisfying the hypotheses o* *f Propo- sition 4.7. Then the quotient map from the augmentation ideal to the indecompo* *sables IX ! QX defines a map Iss*A ! HQ*A which factors (by the results of [11]) as a map " : F2 B Qss*A ~=HQ*A: 18 Proposition 4.9. Let A 2 sH be a simplicial Hopf algebra and suppose ss0A ~=F2.* * Then the natural map " defines an isomorphism of vector spaces F2 B Qss*A ~=HQ*A: Proof: We use the "reverse Adams spectral sequence" of [16], explored in detail* * in [11]. It can be obtained by applying the indecomposables functor Q to the bisimplicia* *l algebra ToA = ToF2A of 4.8. Since ss*A is automatically an exterior algebra, this spect* *ral sequence reads Tor pUB(F2; Qss*A)q ) HQp+qA: Here UB is the category of unstable modules over B. The edge homomorphism is ex* *actly the map ". The result now follows from Lemma 3.1. 5. Hopf Algebras, Dieudonne modules, and derived functors of indecompos- ables The category of abelian Hopf algebras over Fp has a set of generators and, * *as such, is equivalent to a category of modules. We describe the details and, in this secti* *on and the next, show how to use this theory to compute the Andre-Quillen homology of a si* *mplicial abelian Hopf algebra. To begin, we give some terminology from coalgebra theory [21]. A coalgebra * *is simple if it has no non-trivial sub-coalgebras. Every coalgebra is a colimit of its fi* *nite dimensional coalgebras; therefore, if C is simple, the augmentation ffl : C ! Fp dualizes t* *o a finite field extension ffl* : Fp ! C*. A coalgebra is connected if it has a unique non-triv* *ial simple sub-coalgebra. If C is any coalgebra and D C is a simple sub-coalgebra, there* * is a unique connected sub-coalgebra C(D) C so that D C(D). The coalgebra C(D) is t* *he connected component of D. Furthermore there is an isomorphism of coalgebras (5:1) C(D) ~=C D where D runs over the simple sub-coalgebras of C. An abelian Hopf algebra H is connected if it is connected as a coalgebra; t* *hus H is the connected component of the unit j : Fp ! H. For a Hopf algebra H, the follo* *wing are equivalent: 5.2.1)H is connected; 5.2.2)if : H ! H is the Verschiebung and K(n) H is the Hopf algebra kernel of n : H ! H, then colimK(n) ~=H. 19 Another way to phrase 5.2.2 is to say that for all x in the augmentation id* *eal of H, there is a positive integer n so that nx = 0. The term "connected" means that t* *he affine group scheme represented by H is connected. See [7]. It is possible for a Hopf algebra H to be the sum of its_simple sub-coalgeb* *ras - for example, if H is a group ring. More generally, let G = Gal(Fp=Fp) be the Galois* * group of the algebraic closure of Fp over Fp. The Frobenius OE 2 G defines an isomor* *phism G ~=Z^ = limZ=nZ. A G-module M is discrete if the orbit of any x 2 M is finite;* * in other words,_M ~=colimMGn where Gn = ker{G_! Z=nZ} = OEnG._If_M is a discrete G modul* *e, let Fp[M]_be the group ring over Fp. Then G acts on Fp[M] by OE(aiyi) = OE-1(ai* *)OE(xi) and Fp[M]G is a Hopf algebra over Fp. For a Hopf algebra H over Fp, the followi* *ng are equivalent 5.3.1)the Verschiebung : H ! H is an isomorphism; __ 5.3.2)there is a discrete G-module M and an isomorphism Fp[M]G ~=H. Such a Hopf algebra is called separable. Given a Hopf algebra H, there are canonical Hopf algebras H0 H and H+ H which are respectively separable and connected. The connected sub-Hopf algebra * *H+ H is the connected component of the unit j : Fp ! H. To describe H0, let __ __ __ M(H) = {x 2 FpFp H|x = x x} ~=Hom CA (Fp; FpFp H) The elements of M(H) are called group-like._Note_that M(H) is an abelian group * *under multiplication and that the action of G on FpFpH that sends ax to OE-1(a)x_rest* *ricts to an action of G on M(H). Thus M(H) is a discrete G module. Let H0 = Fp[M(H)]G* * . The natural map __ H0 = Fp[M(H)]G ! H identifies H0 with the sum of all simple sub-coalgebras of H. The following fac* *t is plausible, given (5.1). See [7, xII.9] or [8, xIV.3] for a proof. Proposition 5.4. The natural inclusions H0 H and H+ H extend to a natural isomorphism ~ H0 H+ -=! H: In other words, every Hopf algebra over H is naturally a tensor product (i.* *e., coprod- uct) of a separable Hopf algebra and a connected Hopf algebra. It follows that * *there are no non-trivial Hopf algebra maps between a separable Hopf algebra and a connect* *ed Hopf algebra. We next discuss the generators of the category HA; that is, we give a set o* *f Hopf algebras HA so that any Hopf algebra is a quotient of a coproduct of elements o* *f that set. The category of discrete G-modules has generators Z = ZZ=n = Hom sets(Z=n; Z) 20 __ where G = Gal(Fp=Fp) acts on the source. Then if M is a discrete G module, we * *have MGn ~= Hom G(Z; M). More is true: if n divides m, there is a surjection Z* * ! Z and for any discrete G-module M, M ~=colimnHom G(Z; M): __ The Hopf algebras Fp[Z]G generate the category of separable Hopf algebras; i* *ndeed, if H 2 HA is any Hopf algebra, then __ G (5:5) M(H) ~=colimnHomHA (Fp[Z] ; H): Note that in this colimit there is a morphism n ! m if n|m. The connected Hopf algebras also have generators, given by the coordinate a* *lgebras of the affine group schemes that assign to each Fp-algebra its Witt vectors of * *finite height. (See [7], xIII.3.) Concretely, the polynomial ring Z[x0; x1; : :;:xn] has a un* *ique abelian Hopf algebra structure so that the Witt polynomials i pi-1 i wi= xp0+ px1 + . .+.p xi are primitive. Let H(n) = Fp[x0; : :;:xn] be the resulting Hopf algebra over F* *p. If : H(n) ! H(n) is the Verschiebung, we have xi = xi-1, i 1, and x0 = 0. Hence H(n) is connected. The Hopf algebras H(n) are related by a series of maps. Let i :* * H(n) ! H(n + 1) be the inclusion and q : H(n + 1) ! H(n) the quotient with qxi = xi-1,* * i 1, and qx0 = 0. There is also a map j : H(n + 1) ! H(n) given by jxi = xpi-1, i 1* *, and jx0 = 0. The morphism j is characterized by the fact that j O i = [p] and i O j* * = [p] where [p] : H(n) ! H(n) is p times the identity in Hom HA (H(n); H(n)). The Hopf algebras H(n) are not projective in HA (which doesn't have enough * *pro- jectives); however, if f : H(n) ! H is any map and K ! H a surjection, then the* *re is an m > n so that there is a diagram aeo K aeaeaeae | aeaeae |u H(m) _____wH(n) _____wH; where H(m) ! H(n) is a composite of the morphisms q. This suggests forming the * *abelian group (5:6) D0H = colimnHomHA (H(n); H): The morphisms i and j induce, respectively, homomorphisms V and F on D0H so th* *at V F = F V = p and for all x 2 D0H, there is an n so that V nx = 0. The operato* *rs V and F reflect the Verschiebung and Frobenius on H respectively. See Lemma 5.8 b* *elow. We say an R = Z[V; F ]=(V F - p) module M is a V -torsion Dieudonne module if f* *or all x 2 M, V nx = 0 for some n. The following is proved in [8], xVI.1. 21 Proposition 5.7. The functor D0 defines an equivalence of categories between co* *nnected abelian Hopf algebras over Fp and V -torsion Dieudonne modules. Note that any discrete G module is also an R-module via the formulas V x = * *OE-1x and F x = pOEx. Propositions 5.4 and 5.7 imply that the functor on HA H 7! D(H) = M(H) x D0H identifies HA with a full sub-category of the category R-modules. Without makin* *g that sub-category explicit, we call it the category of Dieudonne modules and call an* *y of the objects M(H), D0H, or D(H) a Dieudonne module. In particular, D(H) is the Dieud* *onne module of H. The following result justifies the use of F (for Frobenius) and V (for Vers* *chiebung) in the notation. Let (.)p : H ! H denote the pth power map. Lemma 5.8. Let H be a Hopf algebra. Then F = D(.)p : DH ! DH and V = D : DH ! DH: __ __ Proof: First suppose H is separable; that is H ~=Fp[N]G where N = M(H) FpFpH is the discrete G module of group-like elements. (The module M(H) and the acti* *on of G on M(H) was defined_before_Proposition_5.4.) Then there is a commutative diag* *ram, where we assume H = Fp[N]G Fp[N] __ j __ FpFp H _____w Fp[N] OE-1 | | __|u __|u FpFp H _____wjFp[N] __ where j(a x) = ax is an isomorphism. Since Fp[N] is a group ring induces the * *identity on N. Since OE-1 = (OE-1 1) O (1 ) one has, on N, that OED = 1 or D = OE-1 = V: Since [p] = ((.))p = H ! H, one has p = D O D(.)p and the result about D(.)p fo* *llows. Next suppose H is connected and x 2 DH = colimnHomHA (H(n); H). If f : H(n)* * ! H represents x, then F x is represented by H(n + 1) -j!H(n) -f!H 22 where jxi= xpi-1, i 1, and jx0 = 0. The result that D(.)p = F follows from the* * following diagram and the fact that fq also represents x: p H(n + 1) _____w(.)H(n + 1) = | |q |u |u H(n + 1) _______wjH(n) fq | |f |u |up H ___________w(.)H: The argument that V = D is similar. We next show how to calculate Andre-Quillen homology of a simplicial Hopf a* *lgebra B from its simplicial Dieudonne module D*B. If M is a Dieudonne module we define (5.9) QM = M=F M ~=Fp[V ] R M: Since F and V reflect, respectively, the Frobenius ad the Verschiebung in a Hop* *f algebra, it is not surprising that QD*H ~=QH and the Fp[V ] module structure on QD*H is determined by the action of the Vers* *chiebung of H on QH. This is proved in the next section. The functor Q on D is right exact and has left derived functors LsQM ~=Tor Rs(Fp[V ]; M): Since 0 ! R F!R ! Fp[V ] ! 0 is a projective resolution of Fp[V ] as an R-module, we have (5.10.1) LsQ(M) = 0 fors > 1 and there is an sequence of Dieudonne modules (5.10.2) 0 ! L1Q(M) ! M F!M ! QM ! 0: We call a Dieudonne module F -projective if the operator F acts in a one-to* *-one man- ner. Lemma 5.11. Let M 2 sD be a simplicial Dieudonne module. Then there is a natur* *al weak equivalence in sD fflM : Y (M) ! M with the property that Y (M)s is F -projective for all s 0. 23 Proof: Suppose B is the unique (up to isomorphism) simplicial Hopf algebra so t* *hat D*B ~=M. Then we define (5.12) Y (M) = D*diagToB; where T = TFp is as in 4.8. The fact that Y (M) is level-wise F -projective fo* *llows from Lemma 5.8. We define the total left derived functor of Q on sD by the formula (5.13) L QM = QY (M): We will see in the next section that for a simplicial Hopf algebra A ss*L QDA ~=HQ*A: This makes computations accessible. Proposition 5.14. 1) Let M be a simplicial Dieudonne module and suppose N ! M is any weak equivalence of simplicial Dieudonne modules with N level-wise F -pr* *ojective. Then there is a canonical isomorphism of Fp[V ] modules ss*QN ~=ss*L Q(M): 2) Let M be a simplicial Dieudonne module, then there is a short exact sequ* *ence of Fp[V ] modules 0 ! QssqM ! ssqL Q(M) ! L1Qssq-1M ! 0: Proof: If N is any simplicial Dieudonne module which is level-wise F -projectiv* *e, then 0 ! N F!N ! QM ! 0 is exact, so one obtains, by 5.10.2, short exact sequences 0 ! QssqN ! ssqQN ! L1Qssq-1N ! 0: Part 2 follows by setting N = Y (M) and noting that ss*Y (M) ~=ss*M. Part 1 fol* *lows from the 5-lemma and the naturality of Y (M). Examples 5.15: 1) The short exact sequences of Proposition 5.15.2 need not be s* *plit as Fp[V ] modules. For example, consider the simplicial Dieudonne module M with no* *rmal- ization . .!.0 ! 0 ! R=(V 2) F!R=(V ) 24 with the non-zero groups in degrees 0 and 1. Then the exact sequence 0 ! Qss1M ! ss1L QM ! L1Qss0M ! 0 becomes 0 ! Fp ! Fp[V ]=(V 2) ! Fp ! 0: 2) Let A be a simplicial abelian group and Fp[A] its simplicial group ring;* * we com- pute HQ*Fp[A], assuming the results of the next section. Any abelian group bec* *omes a Dieudonne module by setting V = 1 and F = p; furthermore, D*Fp[A] ~=A. Thus, if* * we choose a weak equivalence of simplicial abelian groups Y ! A with Y level-wise * *free, we have, by Proposition 5.14 and 5.15.1, HQ*Fp[A] ~=ss*L QA ~=ss*(Z=pZ Y ): The short exact sequence of Proposition 5.15.2 becomes 0 ! Z=pZ ssqA ! HQqFp[A] ! Tor (Z=pZ; ssq-1A) ! 0: Since V = 1 throughout, the sequence is splittable. The rest of the section is written at the prime 2. We give the graded vect* *or space ss*L QM the structure of a module over A(0) ~=F2[Sq0; Sq1]=(Sq1)2. Take M 2 sD. First, the action of Sq0 is defined by the equation (5.16) Sq0 = V : ssqL QM ! ssqL QM: To get an action of Sq1, choose a weak equivalence N ! M with N level-wise F -p* *rojective. Then there is a long exact sequence . .!.ssqM F!ssqM ! ssqL QM ! ssq-1M ! . . . coming from the short exact sequence of simplicial Dieudonne modules 0 ! N F!N ! QN ! 0: Define Sq1 : ssqL QM ! ssq-1L QM to be the composite (5.17) ssqL QM ! ssq-1M ! ssq-1L QM: Since each of the maps in 5.17 is a morphism of Dieudonne modules, Sq0Sq1 = Sq1* *Sq0, and since Sq1Sq1 composes two adjacent maps in a long exact sequence, we have Sq1Sq* *1 = 0. Thus we have 25 Lemma 5.18. For all M 2 sD, ss*L QM has a natural structure as a right module o* *ver F2[Sq1; Sq0]=(Sq1)2 = A(0). Furthermore, this module is torsion as an F2[Sq0] m* *odule. Note that this action need not be unstable as Sq1 : ss1L QM ! ss0L QM need not be 0; for example choose M so that NM ~=F2 concentrated in degree 0 an* *d with trivial F and V action. 6. Computing Andre-Quillen homology via Dieudonne Theory In this section we close the circle of ideas in this paper by showing that,* * for a simplicial Hopf algebra A, there is a natural isomorphism of A(0) modules ss*L QD(A) ~=HQ** *A. We will then relate the two possible A(0) structures on HQ*A, when p = 2. We begin by describing a natural map : QD(H) ! QH for H 2 HA. If H is any Hopf algebra, then the Verschiebung : H ! H induces a structure of an Fp[V ] m* *odule on QH. Let OE 2 Gal(Fp; Fp) be the Frobenius. __ Lemma 6.1. For the generating Hopf algebras Fp[Z]G and H(n) ~=Fp[x0; : :;:xn* *] there are isomorphisms of Fp[V ] modules __ 1)QFp[Z]G ~=Fpn with V (a) = OE-1(a) 2)QH(n) ~=Fp[V ]=(V n+1) with [xn] 2 QH(n) the generator as an Fp[V ] modul* *e. Proof: The second statement is clear from the fact that xi = xi-1, i 1, and x0* * = 0 in H(n). For the first statement use that __ G __ F pFp Fp[Z] ~=F p[Z] so that __ __ Q(Fp[Z]G )~=QFp[Z]G ~=[__FZ=np]G ~=Fpn: __ The last isomorphism sends a 2 Fpn to the function f : Z=n ! Fp with f(i) = OEi* *(a). The calculation of the action of V follows. The normal_basis theorem ([3] xV.10.9) supplies an isomorphism of discrete * *G modules between Fp and the "coinduced" module Hom cont(G; Fp) ~=colimnFZ=2np: This, in turn, supplies isomorphisms of Fp[V ] modules __ G Z=n (6:2) QFp[Z] ~=Fpn ~=Fp : 26 __ __ If n|m, the injection Z=n ! Z=m induces a surjection Fp[Z]G ! Fp[Z]G , wh* *ich, under the isomorphism of (6.2) yields, as indecomposables, the induced surjecti* *on (6:3:1) FZ=mp! FZ=np: Note that as an Fp[V ] module, FZ=npis generated by an : Z=n ! Fp, where an(0) * *= 1 and an(i) = 0 if 1 i < n. This choice of generator yields an isomorphism Fp[V ]=(V* * n- 1) ! FZ=np, and, if n|m, the surjection (6:3:2) Fp[V ]=(V m - 1) ! Fp[V ]=(V n- 1) is isomorphic to the surjection of (6.3.1). We now define our map : QD(H) ! QH, for H 2 HA._If x 2 D(H) = M(H) x D0(H), then x is represented either by a map f : Fp[Z]G ! H or by a map g : * *H(n) ! H. Define 0 : D(H) ! QH by 0(x) = Qf(an) orQg(bn) __ where an is the generator of QFp[Z]G and bn = [xn] 2 QH(n). Lemma 6.4. The homomorphism 0 commutes with the operator V and, for all x 2 D(H* *), 0F x = 0. __ Proof: If x is represented by a map f : Fp[Z]G ! H, the fact that 0(V x) = V* * 0(x) follows from 6.1.1 and 6.2. If x is represented by a map g : H(n) ! H, then 0(* *V x) = V 0(x) because the following diagram commutes H(n) _____wgH q A A A A | | AD |u |u H(n - 1) _____wiH(n) _____wH: __ We now show 0(F x) = 0. If x is represented by a map f : Fp[Z]G ! H, then x * *2 M(H) and F x = pOEx, so 0(F x) = p0(OEx) = 0. If x is represented by a map g : H(n) * *! H, then F x is represented by the composition H(n + 1) -j!H(n) -g!H with jxi= xpi-* *1. Because of Lemma 6.4, 0 : D(H) ! QH induces a map of Fp[V ] modules (6:5) : QD(H) = D(H)=F D(H) ! QH: We now show Proposition 6.6. The natural map of Fp[V ] modules : QD(H) ! QH is an isomorphism. 27 Proof: Since both QD(.) and Q(.) are right exact_and_take coproducts to coprodu* *cts, we need only show the result for the generators Fp[Z]G and H(n). In the form* *er case, is defined to be exactly the inverse of the isomorphism 6.2. The latter case f* *ollows from the following lemma. Lemma 6.7. The identity map 1H(n) : H(n) ! H(n) generates DH(n) ~=D0H(n) as a Z[V; F ]=(V F - p) = R module and defines an isomorphism of R-modules R=(V n+1) ~=DH(n): Proof: Certainly V n+11H(n) = 0 and we get a map of R-modules R=(V n+1) ! DH(n). This is an isomorphism by induction on n. If n = 0, H(0) ~=Fp[x0], and DH(0) = * *P H(0). For n > 0, use the short exact sequence of R-modules 0 ! DH(n - 1) Di-!DH(n) ! DH(0) ! 0: The isomorphism of 6.6, combined with the computations of HQ*A given by (4.* *7) and (4.8), and the computations of ss*L QD(A) given in (5.13) now imply that there * *is a natural isomorphism ~= Q (6:8) * : ss*L QD(A) -! H* A for simplicial Hopf algebras A 2 sHA. The rest of this section is written at the prime 2. The final task here is to rationalize the various A(0) structures on HQ*A. * * There are two. The isomorphism of of (6.8) and Lemma 5.18 supply the first; however,* * the isomorphism of (6.8) is not canonical, but depends on the choices inherent in t* *he normal basis theorem, which itself supplies a non-canonical isomorphism. (See Remark * *6.19.1 below.) The second A(0) structure on HQ*A is valid for homotopy connected simp* *licial Hopf algebras A, and is supplied by Proposition 4.9. We give preference to the * *latter, even if it is defined less often, because it is canonical. Thus we need to show that* *, for homotopy connected A and regardless of the choices, the isomorphism * of (6.8) is a morp* *hism of A(0) modules. The fact that * commutes with Sq0 follows from the fact that : QD(H) ! QH * *is an isomorphism of F2[V ] modules, the action of V on D(H) induces the action of* * Sq0 on ss*L QD(A) and the action of V = Q, where is Verschiebung, on QH induces the a* *ction of Sq0 on HQ*A. To prove Sq1 commutes with * will be done by examining a universal example.* * Let (.)* denote F2-duality. 28 Proposition 6.9. The functor from Dieudonne modules to vector spaces M ! (QM)* is representable: there is a Dieudonne module J(F2) and a natural isomorphism Hom D(M; J(F2)) ~=(QM)*: Furthermore 0 = F on J(F2) and J(F2) ~=J0(F2) x J+ (F2) where J0(F2) ~=colimn(F2)Z=n J+ (F2)~=colimnF2[V ]=(V n+1): Proof: This is a direct calculation. Let K(n) 2 sHA be the unique (up to isomorphism) simplicial Hopf algebra so* * that the normalization NDK(n) is isomorphic to J(F2) concentrated in degree n. Proposition 6.10. Let A 2 sHA. Then there are natural isomorphisms [A; K(n)]sHA ~= (HQnA)* and [DA; DK(n)]sD ~=[ssnL QD(A)]*: Proof: Let X ! A be a resolution as in 4.8 and let ~ denote simplicial homotopy equivalence. Then [A; K(n)]sHA~= Hom sHA(X; K(n))= ~ ~= Hom sD(DX; DK(n))= ~ ~= [ssnLQD(A)]* ~= [HQnA]*: Also Hom sD(DX; DK(n))= ~ ~=[DA; DK(n)]sD. The usual arguments with universal examples now implies that if Sq1 commute* *s with * : ss*L QDK(n) ! HQ*K(n); n 1, then * commutes with Sq1 for all homotopy con- nected A 2 sHA. The following result cuts down the universal examples further. Lemma 6.11. If Sq1 commutes with * : ss*L QDK(1) ! HQ*K(1), then Sq1 commutes with * for all homotopy connected objects A 2 sHA. 29 Proof: We will need two notions of suspension. The first is the suspension of a* * simplicial abelian group X. This is the simplicial abelian group X whose normalization is * *the chain complex with (NX)n ~=NXn-1 where NX-1 = 0 and boundary induced by NX. There is a short exact sequence 0 ! X ! CX ! X ! 0 of simplicial abelian groups so that CX has a natural contraction. The second suspension_applies_to simplicial commutative algebras. This is t* *he gener- alized bar construction W . If A is a simplicial augmented commutative algebra,* * there is a sequence of simplicial algebras ___ (6:11:1) ssp ! A -j!W A -q!W A ! Fp ___ which is exact_in the sense that qj is the composite A !_Fp_! W A, j is an inje* *ction, and Fp A W A ~=W A. In fact, in each degree, (W A)n ~=An (W A)n. The simplicial al* *gebra W A has a natural contraction. If one takes indecomposables of (6.11.1) then on* *e obtains an exact sequence of simplicial vector spaces which is naturally isomorphic to 0 ! QA ! C(QA) ! (QA) ! 0: It follows - once one shows that_the functor W preserves cofibrant simplicial * *commutative augmented algebras - that HQnW A ~=HQn-1A. If A is a simplicial abelian Hopf al* *gebra, define + A to be the simplicial coalgebra with ker{+ A ! Fp} ~= ker{A ! Fp}: ___ We can define C+ A similarly. In addition, W A and W A are simplicial Hopf alg* *ebras. There is a natural diagram of simplicial coalgebras 0 _____wA|_____w|||||C+ A+_____wA _____w0 |||||||| (6:12:1) |||||||||||||||||||||||| |u__ |u 0 _____wA _____wW A ______wWA _____w0 given by the inclusion of the coproduct in the product. Finally, there is a co* *mmutative diagram (where JA = ker{A ! Fp}) ___ J+ A ~=JA _____w JW A (6:12:2) ||u ||u ___ QA _________w~=QW A: 30 Since the isomorphism ssn+ A ~=ssn-1A, n 1, commutes with Sq1, the diagrams 6.* *12.1 and 6.12.2 imply, the isomorphism ___ Q HQnW A ~=Hn-1A commutes with Sq1. Next, we turn to the Dieudonne modules. Applying the functor D(.) to 6.11.1* * yields an exact sequence of Dieudonne modules isomorphic to 0 ! DA ! CDA ! DA ! 0: It follows that there is an isomorphism ___ ssnL QW A ~=ssn-1L QA and (see 5.17), this isomorphism commutes with Sq1. In the end we get a commuta* *tive square, in which the horizontal maps commute with Sq1. ___ ~= ssnL QDW A _____wssn-1L QDA | | |u_ ~ |u HQnW A _________w=HQn-1A: ___ The result_now follows by noticing that W K(n - 1) ~=K(n). This follows from th* *e fact that DW K(n - 1) ~=DK(n - 1) ~=DK(n). This leaves the calculation of the behavior of Sq1_in the case A = K(1). No* *te that K(0) is a constant simplicial Hopf algebra and that W K(0) = K(1). Hence ss*K(1)~= Tor K(0)*(F2; F2) ~= (QK(0)): Here (.) denotes the divided power algebra and the last isomorphism follows* * because the squaring map (.)2 : K(0) ! K(0) is trivial. This, in turn, follows from Lem* *ma 5.8. Lemma 6.13. The Hopf algebra K(0) can be presented in an exact sequence of Hopf algebras 2 Fp ! H (.)-!H ! K(0) ! Fp: Proof: This follows from Lemma 5.8 and the description of DK(0) ~= J(F2) given * *at the end of the proof of Proposition 6.9. In fact, there are compatible exact se* *quences of Dieudonne modules 0 ! Z -F!Z ! (Z=2)Z=n ! 0 31 and 0 ! DH(n) -F!DH(n) ! F2[V ]=(V n+1) ! 0: One immediately has the following facts about the Hopf algebra H: there is * *an iso- morphism QH ~=QK(0) and DH is F -projective. Now let A be the simplicial Hopf algebra so that ae (NDA)n = DH0 nn=61;=21; 2 and @ = F : (NDA)2 ! (NDA)1. Then there is a weak equivalence ___ A ! W K(0) ~=K(1) and one has HQ*K(1) ~=ss*QA and ss*L DK(1) ~=ss*QDA: It follows from the first of these isomorphisms that ae (6:15:1) HQnK(1) ~=HQnA ~= QH0 nn=61;=21; 2: In fact, the normalized chain complex of indecomposables NQA is given by ae (6:15:2) (NQA)n = QH0 nn=61;=21; 2: This defines the isomorphism 6.15.1. We now construct isomorphisms n : QH ! ssnA; n = 1; 2 so that the composites, n = 1; 2 QH -n!ssnA ~=(Qss*A)n ~=HQnA ~=(NQA)n ~=QH are the identity. For n = 1 and x 2 QH, 2(x) is the class of y 2 IH A1, where* * y is any lift of x. For n = 2, y is the class of z = y 1 1 + 1 y y 2 H H H = A2: We've arranged the factors of H in A2 so that @(a b c) = a2bffl(c) + ffl(a)bc + ffl(ab)c 2 A1 = H 32 where ffl : H ! F2 is the augmentation. The reduced diagonal of 2(x) in the Ho* *pf algebra ss*A is 1(x) 1(x); hence 2(x)Sq1 = 1(x)Sq1: Thus, identifying HQ*K(1) via the isomorphism of 6.15 we have a commutative dia* *gram HQ2K(1) _____w~=QH (6:16) ||uSq1 ||u= HQ1K(1) _____w~=QH: Next consider ss*L QDK(1) ~=ss*QDA. The normalized complex NDA is defined by 6.* *14; it follows from the definition of Sq1 that there is a commutative diagram ss2L QDK(1) _____w~=QDH (6:17) ||uSq1 ||u= ss1L QDK(1) _____w~=QDH: Now the isomorphism : QDH ! QH defines the isomorphism * : ss*L QDK(1) ! HQ*K(1) so (6.16) and (6.17) show * commutes with Sq1 in this case. Combined with Lemma* * 6.11 we have Proposition 6.18. For homotopy connected A 2 sHA, the natural isomorphism * : ss*L QDA ! HQ*A is an isomorphism of A(0) = F2[Sq0; Sq1]=(Sq1)2 modules. Remarks 6.19: 1) This isomorphism is natural but not canonical. It can be adju* *sted by any element of AutD (J(F2)), which is isomorphic to the group of units in F2* *[V ]^, the pro-finite completion of the polynomial ring on V . 2) This example refers back to Example 5.15.2. Suppose G is an abelian gro* *up. Consider the simplicial abelian group K(G; n). Then ss*F2K(G; n) ~= H*(K(G; n)* *; F2). The simplicial Dieudonne module DF2K(G; n) is isomorphic to K(G; n) with V = 1 * *and F = 2. Thus 8 < Z=2 G m = n ssm LQDF2K=(G; n) ~=: Tor (Z=2; G) m = n + 1 0 otherwise; and Sq1 : ssm+1 LQDF2K(G; n) ! ssm LQDF2K(G; m) is the Bockstein obtained from * *the six term exact sequence is Tor obtained by tensoring G with 0 ! Z=2 ! Z=4 ! Z=2 ! 0: 33 Thus this paper recovers the results of Serre and Cartan. References 1. M. Andre, "Puissances divisees des algebres simpliciales en characteristiqu* *e deux et series Poincare de certains anneaux locale", Man. Math. 18 (1976), 83-108. 2. M. Barr and C. 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