The Hopf ring for Bockstein-nil homology of QSn Takuji Kashiwabara Institut Fourier UMR au CNRS 5582 BP 74 38402, St-Martin-d'H`eres CEDEX FRANCE August 15, 2008 1 Introduction Let QSn denote the space 1 1 Sn ~= colimk k kSn. It represents the stable cohomotopy which is a graded ring functor. Thus it is equipped with two pairings, the one coming from simply the loop space structure 1 1 Sn ~= 1 1 Sn+1 , corresponding to the addition in the stable co- homotopy, and the other coming from the so-called composition pairing, which is nothing but the colimit of the composition of maps k kSix k+i k+iSj = Map*(Sk, Sk+i) x Map*(Sk+i, Sk+i+j) ! Map*(Sk, Sk+i+j) = k kSi+j which corresponds to the multiplication in the stable cohomotopy. These pairings induce in mod p homology pairings which we note by ? and O re- spectively. These products together with the coproduct of H*(QS0; Z=p) are related to each other via the distributivity law [24, 18], making it a ring object in the category of coalgebras (which is called a Hopf ring or a coalge- braic ring). It has been known since the results in [16] that as a coalgebraic ring, H*(QS0; Z=p) is generated by elements Qi[1]'s and fiQi[1]'s, but it is only in [25, 6] that the complete set of relations as a coalgebraic ring for H*(QS0; Z=2) and for H*(QSn; Z=2) n2Z+ was given. In [11] an alternative proof for H*(QS0; Z=2) was given. The purpose of this note is to generalize these results to the odd prime case. Of course, the arguments in [11] show that these results are consequences of the fact that the Quillen's homomor- phism for the symmetric group is an isomorphism for mod 2-cohomology, which is not true for odd primes according to [7]. Thus some modifications 1 are necessary. It turnes out that the notion of the "Bockstein-nil cohomol- ogy" introduced in [8] gives us a better understanding of Quillen's homo- morphism. We show also that the modified object is of interest in view of its relationship with the BP -theory. We also obtain an odd prime "Bockstein- nil" analogue of results of [6]. However, our results rely on the knowledge of H*(QSn; Z=p), so unlike in [6], we don't obtain a "new computation" of the Bockstein-nil homology of QSn. The problem of determining the complete set of relations for H*(QSn; Z=p) as a coalgebraic ring remains open. The author acknowledges the influence of the following works which are not directly used in the arguments : the philosophy of [19] that one can study H*(QS0) by studying the Burnside rings of elementary abelian groups; the point of view in [4, 14] that relations among operations can be obtained in a simple way by consideration of GL2(Z=p) coinvariants in H*(B(Z=p x Z=p)) ; and, of course, the relationship between the Dyer-Lashof algebra and Dickson algebra [17, 20], even though only the p = 2 case is treated in theses works. 2 The colimit over a category and the ring object The purpose of this section is to generalize the result of [10, section 2] and show that the colimit over a category with certain structures becomes a (semi-)ring object. Throughout the paper, a semi-ring will mean a commu- tative and associative semi-ring with unit. First we define : Definition 2.1 An external bipermutative (symmetric bimonoidal, respec- tively) category is a category R equipped with two bifunctors : , R x R ! R each of which makes R a permutative (symmetric monoidal, resp.) category and such that there is a natural transformation (not neces- sarily isomorphisms) ffi : A (B C) ! (A B) (A C) A functor o : R ! R is called a conjugation if there is a functor N and there are natural transformations N ! 0 and N ! id o, where 0 is the constant functor that sends every object to 0, the unit with respect to . Example 2.1.1 A bipermutative category is an external bipermutative cat- egory. In particular, if R is a semi-ring, the discrete category whose objects 2 are elements of R, which will be noted [R], becomes an external bipermuta- tive category. If, furthermore, R is a ring, then the multiplication by (-1) on R induces a conjugation on [R]. Example 2.1.2 Let C be a category that admits finite limits, G : C ! Semi - Rings be a contravariant functor, where Semi - Rings is the cat- egory of semi-rings. Denote by C=G the category whose objects are pairs (X, s) where X is an object of C, and s 2 G(X), and whose morphisms from (X, s) to (Y, t) are just morphisms ` : X ! Y in C such that G(`)(t) = s. Then C=G becomes an external bipermutative category in the following man- ner : (X1, s1) (X2, s2)= (X1 x X2, ss*1s1 + ss*2s2) (X1, s1) (X2, s2)= (X1 x X2, ss*1s1 x ss*2s2) where ssi is the projection to the i-th factor X1 x X2 ! Xi. The natural transformation ffi is given by the diagonal map X1 x X2 x X3 ! X1 x X2 x X1 x X3. Furthermore, if G takes its values in the category of rings, one can define a conjugation o on C=G by : o(X, s) = (X, -s), N (X, s) = (X, 0). The construction ss*1s1 x ss*2s2 is what is usually called the external prod- uct, which explains the name of the external bipermutative (symmetric bi- monoidal) category. Definition 2.2 Let R be an external symmetric bimonoidal category, C a category admitting any finite product, F a functor from R to C that is strict monoidal with respect to each of the monoidal structures of R and that of C induced by the product. F is called a compatible functor if the following diagramme (where p and m's are the natural isomorphisms for a strict monoidal functor) is commutative. pB,C pA,B C F (A) x F (B) x F (C) ____________F-(A) x F (B C) ______________F-(A (B* * C)) | | | | | | | F(A) F(ff|* *i) | | | | | | |? mA,BxmA,C pA B,A C |? F (A) x F (B) x F (A) x F (C)_____-F (A B) x F (A C) _________F-(A B) * *F (A C) 3 Furthermore, when R is equipped with a conjugation, we also require that there exists a natural transformation t : F ! F O o such that the following triangle commutes. F | Q Q | Q FN |idxt Q | Q |? Qs F x F O o ______F-(id o) Then we get immediately from the definition : Proposition 2.3 Let R, C be as above, D a category admitting any finite product, F a compatible functor from R to C, and H a functor from C to D that preserves the product. Then the composition H O F is a compatible functor. Example 2.3.1 Let C, G be as in Example 2.1.2. Then the functor Source : C=G ! C, Source(X, s) = X is a compatible functor. Furtermore, if D is a category admitting any finite product, and H is a functor from C to D that respectes the product, then H O Source is a compatible functor. Now we can state the main result of the section Theorem 2.4 Let R be an external bipermutative category, C a category admitting any finite product, and F a compatible functor from R to C. Then ColimR F (-) (if it exists) is a semi-ring object in C. Furthermore, if R is equipped with a conjugation, then ColimR F (-) is a ring object. Proof. As F is strict monoidal with respect to two monoidal structures of R, ColimR F (-) has two monoidal structures. The compatibility condition guarantees the distributivity, so that the colimit becomes a semi-ring object in C. When R is equipped with the conjugation, the natural transformatin t induces the conjugation in ColimR F (-) making it a ring object. Example 2.4.1 Let k, R be rings, R = [R], C the category of k-coalgebras, F the constant functor that sends every object of [R] to k. Then ColimR F (-) is nothing but the "ring-ring" k[R]. 4 3 The main relations In this section, we prove "the main relations" for colimit model coalgebraic rings. We follow more or less the treatment in [2]. First of all we need to fix notations. Definition 3.1 Let I, C, E be categories, G : Iop ! C, E : I ! E be func- tors. Suppose fi 2 E(X) with X 2 obj(I), x 2 G(X). Then we denote by (x, fi) the image of fi by the canonical map E(X) ! ColimI=GEOSource(-). By the definition of the colimit, in a suitable sense, ColimI=GE O Source(-) is generated by the elements (x, fi), subject only to relations generated by (f*x, fi) = (x, f*(fi)) where f 2 MorI(X, Y ), x 2 G(Y ), fi 2 E(X). In some cases this description is good enough. However in other cases we may need a more concrete description. It turns out that often we get these explicit relations in terms of equality between certain formal power series. For this purpose, we need to be more concrete about our categories and functors. Hypothesis 3.2 I admits arbitrary finite products, C is the category of coalgebras over a ring k, F is a functor from Iop to Semirings. E is a functor from I to C. X 2 obj(I) such that E(Xi) ~= E(X) i, E(X) is a free module over E* = E(X0), and E*(X) = HomE*(E(X), E*) has an element t such that E*(X) is (topologically) free over a set of generators ti, i 2 J. From now on throughout the section we assume that this hypothesis is sat- isfied. Now we can introduce some notations. We denote by the basis of E(X) that is dual to 's. Let xF be an element of F (X) (we don't require any particular property on this element, but it will have to be fixed once and for all). We denote bi= (xF , fii) and b(t) = i2Jbiti. Also, given an element ff of F *= F (X0), we denote [ff] = (1, ff). Let oe be an element of F *[T1, T2, . .,.Tk]. Then we define b[ae](t1, t2, . .,.tk) = ?I[ffI] O b(t1, t2, . .,.tk)OI if ae(T1, T2, . .T.k) = IffIT I 5 where T I= T1I1T2I2. .T.Ikk and b(t1, t2, . .,.tk)OI = b(t1)OI1O . .O.b(tk)OIk. This might sound quite scary, however, it is quite simple, we just re- place the multiplication by the circle multiplication, sum by the star prod- uct, and ff by [ff], then apply the resulting "polynomial" to the "vari- ables" b(t1), . .,.b(tk). For example, if ae(T ) = T 2+ 3T , then bae(t) = b(t) O b(t) ? [3] O b(t). In terms of this notation, the Ravenel-Wilson's main relations [24] can be expressed as : Theorem 3.3 Let E*(-), F *(-) be complex oriented cohomology theories. Then we have, in ColimCP=F*(E* O Source(-))[[xE1, xE2]], we have b[xF1+FxF2](xE1, xE2) = b[xF](xE1+E xE2) where xG (G = E, F ) denotes the orientation class of the cohomology theory G, +G denotes the formal sum in G*(-), and CP is the category whose objects are finite products of CP 1's and whose morphisms are homotopy classes of all continuous maps among them. Note that we followed the treatment in [2] and replaced the "formal indeter- minates" in [24] by the orientation classes. We now will prove the following generalisation of this formula. Theorem 3.4 Let f : Xj ! Xk be a morphism in I such that F (f)(_(xF1, . .,.xFk)) = OE(xF1, . .x.Fj) where (_(xF1, . .,.xFk)) 2 F (Xk). Then we have b[OE](xE1, . .,.xEj) = b[_](E*(f)(xE1), . .E.*(f)(xEk)). We could symbolically write this formula as b[f*_](x) = b[_](f*(x)). Proof. Consider the following diagram : ~= ~= F (Xk) _________-Hom(E(Xk), R) ______R- E* E*(Xk) _______R[[xE1,-. .,.xEk* *]] | | | | | | F(f)| (E(f))|* E*(f)| E*(f)| | | | | |? |? ~= |? ~= |? F ((Xj) _________Hom(E(Xj),-R) ______R- E* E*(Xk) _______R[[xE1,-. .,.xEj* *]] 6 where R = ColimI=F(E OSource(-)). The vertical maps are defined in such a way that the squares commute, with the exception of the leftmost one. The leftmost square commutes by the definition of the colimit. Now, the leftmost horizontal arrows transform + and x into ? and O, as they are induced by the sum and product maps F x F ! F . Similary, the multiplication by a becomes the circle multiplication by [a]. Furthermore, xFi in F *(Xk) maps to X*iin Hom(E(Xk), R) which then maps to b(xEi) by the definition of the dual basis. Thus the image of _ 2 F (Xk) by the horizontal map is b[_](xE1, . .x.Ek) which maps down to b[_](E*(f)(xE1), . .,.E*(f)(xEk)), where as by the left vertical map, it maps down to OE whose image by the horizontal map is nothing but b[OE](xE1, . .x.Ej). 4 The main results and proof First of all, we recall what is already known about the`object of our study. Denote by CS0 the combinatorial model for QS0, CS0 = n B n. It is a semi-ring object in the homotopy category, the multiplication induced by maps m x n ! mn and the addition induced by maps m x n ! m+n . These structures are compatible with those of QS0, and H*(QS0; Z=p) is ob- tained from H*(CS0; Z=p) by the group completion [1, 21]. In our language, the pairings on CS0 makes H*(CS0; Z=p) a semi-ring object in the category of coalgebras, and H*(QS0; Z=p) is the universal coalgebraic ring containing H*(CS0; Z=p). Both CS0 and QS0 are E1 spaces so that the Dyer-Lashof operations act on their mod p homology. For a sequence (allowable or not) I = (ffl1, I1, . .f.fln, In), ffli= 0, 1, Ii> 0, Ii's are integers, we note QI * *the Dyer- Lashof operation fiffl1QI1. .f.ifflnQIn. We define fi(I) to be the sum ffl1+. .* *f.fln. When fi(I) = 0, we say that the sequence I is allowable if and only if all Ij's are multiple of 2(p-1) and Ij-1 Ij. (This agrees with the usual definition when we only consider QI's acting on even degree elements.) If x has degree d, then Qi(x) has the degree pd + i. We also note BP the p-local Brown-Peterson spectrum [3, 22], 1 nBP its associated 0-th infinite loop space, i. e., the space that represents the functor BP n(-). There is a unit map S ! BP , which gives rise to a map QS0 ! 1 nBP . By composing it with the canonical map CS0 ! QS0, we get a map CS0 ! 1 BP . They induce respectively a map H*(QSn; Z=p) ! H*( 1 nBP ; Z=p) and H*(CS0; Z=p) ! H*( 1 0BP ; Z=p). Further- more there is a Thom map BP ! HZ=p which gives rise to maps BP *(X) ! H*(X; Z=p) for any space X. 7 Theorem 4.1 The following subobjects of H*(CS0; Z=p) from i) to iii), quotients of H*(CS0; Z=p) from iv) to vi) are all isomorphic and dual to the subobjects of H*(CS0; Z=p) from vii) to ix). (i)The subalgebra generated by the elements of the form QI([1]) with fi(I) = 0. (ii)The polynomial subalgebra generated by the elements of the form QI([1]) with fi(I) = 0, I allowable. (iii)The coalgebraic subsemiring generated by the elements of the form Qi[1]. (iv)The quotient by the ideal generated by the elements of the form QI([1]) with fi(I) 1. (v) The quotient by the "coalgebraic ideal" generated by the elements of the form fiQi[1]. (vi)The image of H*(CS0; Z=p) in H*( 1 BP ; Z=p) (vii)The image of H*(( 1 BP ; Z=p) in H*(CS0; Z=p). (viii)The image of BP *(CS0; Z=p) in H*(CS0; Z=p). (ix)f*(-1)(Im(BP *(Y ) ! H*(Y ))), where Y is a certain disjoint union of copies of (BZ=p)m 's and f : Y ! CS0 has the property such that f* is injective. Theorem 4.2 The following subobjects of H*(QSn; Z=p) from i) to iii), quotients of H*(QSn; Z=p) from iv) to vi) are all isomorphic and dual to the subobjects of H*(QSn; Z=p) from vii) to ix). Here oen notes the fundamental class of Hn(QSn; Z=p) (oe0 = [1]). (i)The subalgebra generated by the elements of the form QI(oen) with fi(I) = 0 (and [-1] for n = 0). (ii)The polynomial subalgebra generated by the elements of the form QI(oen) with fi(I) = 0, I allowable (tensored with Z=p[Z] for n = 0) if n is even. The exterior subalgebra generated by single suspension of these elements if n is odd. (iii)The n-th spacelike degree part of the coalgebraic subring generated by the elements of the form Qi[1] and oen. 8 (iv)The quotient by the ideal generated by the elements of the form QI(oen) with fi(I) 1. (v) The n-th spacelike degree part of the quotient by the "coalgebraic ideal" generated by the elements of the form fiQi[1]. (vi)The image of H*(QSn; Z=p) in H*( 1 nBP ; Z=p) (vii)The image of H*( 1 nBP ; Z=p) in H*(QSn; Z=p). (viii)The image of BP *(QSn; Z=p) in H*(QSn; Z=p). (ix)f*(-1)(Im(BP *(Y ) ! H*(Y ))), where Y is a certain disjoint union of copies of (BZ=p)m x (Sn)xk 's and f : Y ! QSn has the property such that f* is injective. Remark 4.3 By the result of Dyer-Lashof in [5], i),ii) and iv) are isomor- phic one another. It was shown by Wilson [26] that they are isomorphic to vi). The equality between i) and iii) was essentially proved by May [16]. vi) is obviously dual to viii). The equality between vii) and viii) were proved by author [12][Proposition 4.3]. The equality between iv) and v) is nothing but the first step of the proof of Proposition 5.1 in [13] (second property with X = Sn, N = 0). Finally the equality between all of these and ix) is the dual of the second step of the proof of Proposition 5.1 in [13]. See also Theorem 3.1 in [9] Let's denote by M0 the subobject (or quotient) of H*(CS0; Z=p) in Theorem 4.1, by M*0its dual, M1,nthe subobject (or quotient) of H*(QSn; Z=p) in Theorem 4.2, and by M*1,nits dual. We also denote by BV the category whose objects are the classifying groups of elementary abelian p-groups, and whose morphisms are homotopy classes of maps among them. For a space X, we note by BV =X the category whose objects are homotopy classes of maps from an object of BV to X, and whose morphisms are homotopy commutative triangles. Finally, for a left module over a Steenrod algebra M, its Bockstein-nil part as defined in [8] will be noted MfiN, and dually if M is a right module over a mod p Steenrod algebra, then MfiN will denote the "Bockstein-nil quotient" of M, that is M=( iP*ifi(M)). Then our main results are Theorem 4.4 (local version) (i)M0 is a polynomial algebra generated by the elements of the form Ei0O Ep(i0+i1)O Epl(i0+i1+...+il)with i0 > 9 0, ij 0(j 6= 0) and [1]. Here Ei is an element of the image of H2i(p-1)(B p; Z=p) H2i(p-1)(CS0; Z=p). (ii)M1,0is obtained from M0 just by inverting [1]. M1,n(n > 0) is a free commutative algebra generated by the elements of the form oeOn1O Ei0O Ep(i0+i1)O Epl(i0+i1+...+il)with n < 2i0. Theorem 4.5 (global version) We have following isomorphisms (i)M0 ~=H*(CS0; Z=p)fiN, M1,n~=H*(QSn; Z=p)fiN. (ii)M0 ~=ColimBV =CS0H*(-; Z=p)fiN, M1,0~=ColimBV =QS0H*(-; Z=p)fiN. (iii)M0 ~=HT*(CS0; Z=p), M1,n~= HT*(QSn; Z=p) where HT*(CS0; Z=p) is the coalgebraic semiring generated over Z=p[Z+ ] by elements Ei's of degree 2i(p - 1), with the diagonal (Ei) = j+k=iEj Ek, subject to the relations (a) E0 = [p]. (b) E(sp-1) O E(tp-1) = E(sp-1) O E((s + t)p-1) in HT*(CS0; Z=p)[s, t] ~=Hom(HfiN(BZ=pxBZ=p), HT*(CS0; Z=p)[s, t] where s, t are standard polynomial generators of HfiN(BZ=p x BZ=p) abd E(X) denotes the formal sum iEiXi. and {HT*(QSn; Z=p)}n2Z+ is the coalgebraic ring generated over Z=p[Z] by same elements together with oe1 (with bidegree (1,1)) subject to the relations (a), (b) and the following : (c) oeO2n1O En = (oeO2n1)?p Proof of Theorem 4.3. This uses the knowledge of H*(QSn; Z=p). However, it is also possible to derive i) and ii) for n = 0 from Theorem 3.4, which would give a "new" computation of H*(CS0; Z=p)fiNand H*(QS0; Z=p)fiN. It is easy to see that it is suffices to prove the results for n even. First we show that these elements are linealy independant in the module of inde- composable. We proceed by induction on l for all n at once, assuming the 10 identity (c) (which will be proved later without using Theorem 3.3). As a matter of fact, by (c) and the distributivity law, we get : oeO2i01OEi0OEp(i0+i1)O. .O.Epl(i0+i1+...+il)= (oeO2i01OEi1O. .O.Ep(l-1)(i1+...+* *il))?p. Since M0 and M1,2n's are known to be polynomial algebras, the p-th power map is injective. Thus by induction on l, we get the linear independence of elements of this form. To prove that they span, it suffices to show that they span the vector space with the correct dimension. For that purpose, we establish an isomor- phism of graded sets between our proposed basis and a known basis. Now note that OE : (i0, i1, . .i.l) ! (2(p-1)(i0-n), 0, 2(p-1)(i0+i1-n), 0, . .2.(p-1)(i0+i1 * *. .+.il-n), 0) gives a bijection between the set of sequences of non-negative numbers with i0 > n and the set of allowable sequences. Thus it only remains to show the equality deg(oeO2n1O Ei0O Ep(i0+i1)O . .O.Epl(i0+i1+...+il)) = deg(Q2(p-1)(i0-n)Q2(p-1)(i0+i1-n).Q.2.(p-1)(i0+i1...+il-n)(oeO2n1)). However, the latter is equal to 2(p - 1)(i0 - n) + pdeg(Q2(p-1)(i0+i1-n).Q.2.(p-1)(i0+i1...+il-n)(oeO2n1)), and by induction on l, we see that this is equal to 2(p - 1)(i0 - n) + pdeg(oeO2n1O E(i0+i1)O . .O.Epl-1(i0+i1+...+il)) = 2(p - 1)(i0 - n) + 2np + pdeg(E(i0+i1)O . .O.Epl(i0+i1+...+il)) = 2n + deg(Ei0O Ep(i0+i1)O . .O.Epl(i0+i1+...+il)) Proof of Theorem 4.4. i). We use the notation of Theorems 4.1, 4.2 ix). Note that for the space Y we have Im(BP *(Y ) ! H*(Y ; Z=p)) = H*(Y ; Z=p)fiN. Since f* is injective, an element of H*(CS0; Z=p) or H*(QSn; Z=p) respectively is in H*(CS0; Z=p)fiN or H*(QSn; Z=p)fiN respectively if and only if its image by f* is in H*(Y ; Z=p)fiN. Thus we get the desired result. 11 ii) By the group completion theorem, it is enough to prove the isomor- phisms just for M0. Since the Quillen's category of a finite group G [23] is cofinal in BV =BG, by dualizing the result of Lesh and Ha [8], one sees that ColimBV =B mH*(B(-); Z=p)fiN~= H*(B m ; Z=p)fiN. By assembling together these isomorphisms for different m, one gets ColimBV =CS0H*(B(-); Z=p)fiN~= H*(CS0; Z=p)fiN. This together with i) prove ii). iii). First we prove it for M0. Again the group completion theorem implies that it holds for M1,0. Now we modify the arguments in [11] to fit to odd prime case. First we need some notations. Definition 4.6 We denote o C the category of Z=p-coalgebras (graded, cocommutative and couni- tary). o V the category of finite dimensional Z=p-vector spaces. o E the category of sets, o F the category of contravariant functors from V to E (which we con- sider as covariant functors from Vop to E. Let G be an object of F. Then V=G will denote the category whose objects are the pairs (W, g) where W is an object of V, g an element of G(W ), and whose morphisms frome (W1, g1, ) to (W2, g2) are linear maps OE : W1 ! W2 such that G(OE)(g1) = g2. Then as in [11], we have Proposition 4.7 Let c denote the functor which sends G to the colimit of the composite V=G ! V ! * C where *(-) denotes the (graded) divided power coalgebra functor (with the elements of the vector space having degree 2). If G is a semi-ring object in F, then c(G) is a semi-ring object in C. Furthermore, if G is a ring-object, then so is c(G). Proof. One sees from Example 2.3.1 that the composite V=G ! V !* C is a compatible functor. The result follows from Theorem 2.4. 12 Definition 4.8 For a Z=p-vector space W , let a(W ) be the quotient of the free unitary commutative semiring generated by symbols au, u 2 W *where *denotes the Z=p-dual, by the following relations : (i)a0 = p (ii)auav = au+vav8u, v (iii)aqu = au if q 2 (Z=p)x If f is a linear map from W1 to W2, define the map a(f) simply by a(f)(au) = auOf. A(W ) will mean the Burnside semi-ring (and not the Burnside ring), i.e., the semi-ring of all isomorphism classes of W -sets. With these preparations, the proof of Theorem 4.5 iii) will be done in three steps. First we show that c(a) is isomorphic to the algebric model HT*(CS0) (counterpart of Proposition 2.7 in [11]), then we show that c(A) is isomorphic to the real object H*(CS0; Z=p)fiN (counterpart of Proposition 2.9 in [11]) and finally we show that the functors A and a are isomorphic (counterpart of Proposition 2.10 in [11]), thus c(A) isomorphic to c(a). The first step : we show that c(a) is isomorphic to HT*(CS0). For this we introduce several more definitions. Definition 4.9 An object X of V=a is a standard decomposable if it is of the form X ~=(X1,1 . . .X1,r1) (X2,1 . . .X2,r2) . . .(Xn,1 . . .Xn,rn), where each of the factor Xi,j's is isomorphic to the pair (Z=p, aid). For an object Z of V=a, a decomposable morphism of Z is a morphism in V=a whose target is a standard decomposable object. Now note that if Y = (V, v), there exists a decomposable morphism ` : Y ! X where X is as above if and only if v = ni=1 rij=1a`i,jwith `i,jthe map V ! Z=p corresponding to the component Xi,j. Since au is a pull-back by u of aid:Z=p!Z=p, one sees that any object of V=a admits at least one decomposable morphism. If such decomposable morphism were unique (up to permutation of factors), then c(a) would be just the direct sum of i j *(Z=p)i,jwhere *(Z=p)i,jcoming from Xi,jas above, which is nothing but the free (commutative) coalgebraic ring generated by *(Z=p). Thus c(a) is a coalgebraic ring generated by *(Z=p), subject only to relations coming 13 from the failure of the uniqueness of decomposable morphisms. Now, this failure, in turn, arises from the relations in a(W )'s. Note ei a generator of 2i(Z=p). Denote by E"ithe element (aid:Z=p!Z=p; ei). The relation (iii) aqu = au is a pull-back by u of the relation aq:Z=p!Z=p= aid:Z=p!Z=p, which in turn can be written as q*au = au. Therefore, Theorem 3.4 implies that we have "E(qs) = "E(s). As qi = 1 if and only if 2(p - 1)|i this means E"(s) = E(s(p-1)). The relation (ii) in a(W ) in general is a pull-back of the particular case u = u0, v = v0, W = Z=p x Z=p, where u0 and v0 are the projections to the first and the second factor Z=p. This particular relation can be written as T *(au0av0) = au0av0, where T : Z=p x Z=p ! Z=p x Z=p is the homomorphism given by T (x, y) = (x + y, y). Therefore Theorem (3.4) implies "E(s) O "E(t) = "E(s + t) O "E(t). combining with the precedent equality, we get E(sp-1) O E(tp-1) = E((s + t)p-1) O E(tp-1) which is the second defining relation in HT*(CS0). The relation a0 = p in the general case is the pull-back of the same relation in a(0) by the map W ! 0. The particular case can be written as j*aid:Z=p!Z=p= paid:0!0 which corresponds to the relation E0 = [p]. As there are no other relations in a(V ), we get thus a complete set of relations in c(a), which proves the isomorphism c(a) ~=HT*(CS0; Z=p). 14 The second step : we show that c(A) is isomorphic to H*(CS0; Z=p)fiN. Note first that we have a natural isomorphism A(V ) ~= [BV, CS0]. This gives rise to an equivalence of categories BV =CS0 ~= V=A. On the other hand, we have natural isomorphisms H*(BV )fiN~= *(V ), so we obtain an isomorphism c(A) ~=colimBV =CS0H*(B(-); Z=p)fiN. We have already seen that this latter is isomorphic to H*(CS0; Z=p)fiN. The third step : we show that the functors A and a are naturally iso- morphic. Let u be an element of V *. Note by ffu Z=p considered as a V - set via the homomorphism u : V ! Z=p. Define a semi-ring map OE : a(V ) ! A(V ) by OE(au) = ffu. We need to show OE is well-defined. First of all we have ff0 = p. To show that ffuffv = ffuffu+v, it suffices to prove it when V = Z=p x Z=p and u, v are projections. (The general case is just a pull-back of this particular case.) In this case ffuffv is V considered as a V -set by the identity homomorphism, and ffu+vffu is V considered as a V -set by the homomorphism T . Since both homomorphisms are injec- tive, the resulting V -sets are isomorphic. Thus OE is well-defined. Next we construct a monoid homomorphism _ : A(V ) ! a(V ) such that _(1) = 1, _(V=W ) = au1. .a.ur, where ui's are linearly independent elements of V * such that W = [iKer(ui). Now the relations auav = auau+v and aau = au guarantee that this is a well-defined map. It is now easy to show that OE and _ are inverse to each other. Now it remains to prove the case for M1,nwith n > 0. The relation c) comes from the property of Dyer-Lashof operations and that the fact that circling with oe1 is just suspending, noting that Ei = Q2(p-1)i[1]. Further- more Theorem 4.4 ii) shows that in positive spacelike degrees, there is no relation other than the relation c). 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