Modular coinvariants and the mod p homology of QSk

In this paper, we use the modular coinvariants theory to establish a complete set of relations of the mod p homology of {QSk }k≥0, for p odd, as a ring object in the category of coalgebras, so called a coalgebraic ring or a Hopf ring. Beside, we also describe the action of the mod p DyerLashof algebra as well as one of the mod p Steenrod algebra on the coalgebraic ring

Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Modular coinvariants and the mod p homology of QS k Phan Hoàng Chơn Abstract In this paper, we use the modular coinvariants theory to establish a complete set of relations of the mod p homology of {QS k }k≥0 , for p odd, as a ring object in the category of coalgebras, so called a coalgebraic ring or a Hopf ring. Beside, we also describe the action of the mod p Dyer-Lashof algebra as well as one of the mod p Steenrod algebra on the coalgebraic ring. 1. Introduction Let G∗ (−) be an unreduced multiplicative cohomology theory. Then, G∗ (−) can be represented unstably by the infinite loop spaces Gn of its associated Ω-spectrum (i.e. Gk (X) ∼ = [X, Gk ] naturally and ΩGk+1 Gk , where we denote by [X, Y ] the homotopy classes of unbased maps from X to Y ). The collection of these spaces G∗ = {Gk }k∈Z is considered as a graded ring space with the loop sum m : Gk × Gk → Gk and the composition product µ : Gk × G → Gk+ . Therefore, the homology of {Gk }k∈Z (beside the usual addition and coproduct) has two operations, which are denoted by and ◦, respectively, induced by m and µ. These operations make the homology of {Gk }k∈Z a ring object in the category of coalgebras, which is called a Hopf ring or coalgebraic ring (see Ravenel-Wilson [26], and Hunton-Turner [9]). The Hopf ring structure actually becomes an important tool to study the homology of Ω-spectrum as well as the unreduced generalized multiplicative cohomology theory, and it is of interest in study of algebraic topologists. For example, the Hopf ring for complex cobordism M U is studied by Ravenel-Wilson [26], the Hopf ring for Morava K-theory is studied by Wilson [28] and for connective Morava K-theory by Kramer[19], Boardman-Kramer-Wilson [2]. Recently, the Hopf ring structure for BP and KO, KU are respectively investigated by Kashiwabara [11], Kashiwabara-Strickland-Turner [16] and Mortion-Strickland [23]. Let QS k = limΩn Σn S k be the infinite loop space of the sphere S k . Then {QS k }k≥0 is an −→ Ω-spectrum, called the sphere spectrum, therefore, the mod p homology of {QS k }k≥0 also has a Hopf ring structure. Moreover, it is well known that all spectra are module spectra over the sphere spectrum, so the mod p homology of any infinite loop space becomes an H∗ QS 0 -module or {H∗ QS k }k≥0 -module object in the category of coalgebras, which is called a coalgebraic module. As is well-known, (see Kashiwabara [14]) the mod p homology of an infinite loop space has an A-H∗ QS 0 -coalgebraic module structure. Beside, from the result of May [3], the mod p homology of an infinite loop space also has a so-called A-R-allowable Hopf algebra, i.e., it is a Hopf algebra on which both the Steenrod and the Dyer-Lashof algebra act satisfying some compatibility conditions. Thus, understanding the coalgebraic ring structure of H∗ QS 0 2000 Mathematics Subject Classification 55P47, 55S12 (Primary), 55S10, 20C20 (Secondary). This work is partial supported by a NAFOSTED gant. Page 2 of 22 PHAN HOÀNG CHƠN plays important role in the study of homology of infinite loop spaces as well as in the study of the category of A-H∗ QS 0 -coalgebraic modules and one of A-R-allowable Hopf algebras, and relationship between them. By the results of Araki-Kudo [1], Dyer-Lashof [5] and May [3], the mod p homology of {QS k }k≥0 is generated as a Hopf ring by Qi [1], i ≥ 0, σ (for p = 2) and by Qi [1], i ≥ 0, βQi [1], i ≥ 1, σ (for p odd), where Qi is the ith homology operation (which is called the DyerLashof operation), [1] ∈ H∗ QS 0 is the image of the non-base point generator of H0 S 0 under the homomorphism H0 S 0 → H0 QS 0 induced by the inclusion S 0 → QS 0 and σ is the image of the basis element of H1 S 1 under the homomorphism H1 S 1 → H1 QS 1 induced by the inclusion S 1 → QS 1 . This actually corresponds the fact that the Quillen’s approximation map of finite groups by elementary abelian subgroups is a monomorphism [25]. However, a long time, no one undertook to sudy the relations until the importance of the coalgebraic ring structure of H∗ QS k is clearly made again from works of Hunton-Turner [9] and Kashiwabara [13] (which develop the homological algebra for the category of modules over a Hopf ring). These works are maybe the main motivation for study in [27] and [6], which give a description of a complete set of relations as a Hopf ring of H∗ QS k for p = 2. Later, it was discovered in [12] that the nice description of the complete set of relations comes from the fact that the Quillen’s map for the symmetric groups is actually an isomorphism at the prime 2 (see [7]). Also according to [7], the map is no longer an isomorphism for odd primes, therefore, it is difficult to generalize the results in [27] and [6] for odd primes. However, in the Brown-Peterson cohomology theory, the Quillen’s map of the symmetric groups is also an isomorphism [8]. This fact allows to generalize the results in [27] and [6] for the Bockstein-nil homology of H∗ QS k [15]. Thus, the describing of a complete set of relations as a Hopf ring for {H∗ QS k }k≥0 is not only important but also difficult. In this work, we discover that the isomorphism between the dual of R[n] and the image of the restriction map from the cohomology of the symmetric group Σpn to the elementary abelian p-group of rank n, Vn , is the main key to establish the nice description of the complete set of relations as above discussion, where R[n] denote the subspace of the Dyer-Lashof algebra spanned by all monomials of length n. Using this idea and modifying the framework in [27] allows us to obtain a nice description of the complete set of relations as a Hopf ring of {H∗ QS k }k≥0 for p odd. In more detail, we construct a new basis for B[n]∗ , which is the dual of the image of the restriction map from the cohomology of the symmetric group Σpn to the cohomology of Vn [24]. Using the basis and combining with the fact that the induced in (n) homology of the Kahn-Priddy transfer, tr∗ , is multiplicative and GLn -invariant to investigate, we obtain an analogous description of a complete set of relation of {H∗ QS k }k≥0 as a coalgebraic ring for odd primes. This fact again confirms the closely correspondence between the Hopf ring structure of {H∗ QS k }k≥0 and the Quillen’s map of the symmetric groups. The results in [27], [6] as well as in [15] can be deduce from our results by letting p = 2 or killing the action of the Bockstein operation for p odd. It should be noted that much of our work rests on previous results with a suitable modifying. For example, relations (4.1)-(4.3) (see Proposition 4.7) can be (2) followed from the multiplicativity and the GL2 -invariant of tr∗ as the case of p = 2. However, the relation (4.4) is here difference. In deed, for p = 2 or for the Bocktein-nil homology, the general case of the relation can be simple implied from the case of the length 1 and other relations, but here it is impossible because of the action of Bocktein operation. In this paper, additive base of (H ∗ BVn )GLn , (H∗ BVn )GLn as well as the cokernel of / (H ∗ BVn )GLn are also established. Beside, using the similar the restriction map H ∗ BΣpn method of Turner [27], we give descriptions of the action of the mod p Steenrod algebra A and the action of the mod p Dyer-Lashof algebra R on the Hopf ring as relative results. The paper is divided into five sections. The first two sections are preliminaries. In Section 2, we review some main points of the Dickson-Mùi algebra, the image of the restriction from MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k Page 3 of 22 the cohomology of the symmetric group Σpn to the cohomology of the elementary abelian pgroup of rank n, Vn , as well as the mod p Dyer-Lashof algebra. In Section 3, we construct new additive base for (H ∗ BVn )GLn , (H∗ BVn )GLn , the cokenel as well as the dual of the image of / (H ∗ BVn )GLn . By the results of May [3], the new basis of the the restriction map H ∗ BΣpn dual of the image of the restriction map is considered as an additive basis of the subspace R[n] of the mod p Dyer-Lashof algebra. The Hopf ring for {H∗ QS k }k≥0 as well as the actions of the Steenrod algebra and the Dyer-Lashof algebra on {H∗ QS k }k≥0 are respectively presented in two final sections. 2. Preliminaries In this section, we review some main points of the Dickson-Mùi algebra and the image of the restriction from the cohomology of the symmetric group Σpn to the cohomology of the elementary abelian p-group of rank n, Vn . We also review some basic properties of the mod p Dyer-Lashof algebra. 2.1. Modular invariant Let Vn be an n-dimensional Fp -vector space, where p is an odd prime number. It is well-known that the mod p cohomology of the classifying space BVn is given by H ∗ BVn = E(e1 , · · · , en ) ⊗ Fp [x1 , · · · , xn ], where (e1 , · · · , en ) is a basis of H 1 BVn = Hom(Vn , Fp ), xi = β(ei ) for 1 ≤ i ≤ n with β the Bockstein homomorphism, E(e1 , · · · , en ) is the exterior algebra generated by ei ’s and Fp [x1 , · · · , xn ] is the polynomial algebra generated by xi ’s. Let GLn denote the general linear group GLn = GL(Vn ). The group GLn acts on Vn and then on H ∗ BVn according to the following standard action ais xi , (aij )xs = (aij ) ∈ GLn . ais ei , (aij )es = i i The algebra of all invariants of H ∗ BVn under the actions of GLn is computed by Dickson [4] and Mùi [24]. We briefly summarize rtheir results. For any n-tuple of non-negative integers j (r1 , . . . , rn ), put [r1 , · · · , rn ] := det(xpi ), and define Ln,i := [0, · · · , ˆi, . . . , n]; Ln := Ln,n ; qn,i := Ln,i /Ln , for any 1 ≤ i ≤ n. In particular, qn,n = 1 and by convention, set qn,i = 0 for i < 0. The degree of qn,i is 2(pn − pi ). Define Vn := Vn (x1 , · · · , xn ) := (λ1 x1 + · · · + λn−1 xn−1 + xn ). λj ∈Fp Another way to define Vn is that Vn = Ln /Ln−1 . Then qn,i can be inductively expressed by the formula p qn,i = qn−1,i−1 + qn−1,i Vnp−1 . For non-negative integers k, rk+1 , . . . , rn , set [k; rk+1 , · · · , rn ] := 1 k! e1 · e1 xp1 rk+1 · rn xp1 ··· ··· ··· ··· ··· ··· en · en rk+1 xpn · rn xpn . Page 4 of 22 PHAN HOÀNG CHƠN For 0 ≤ i1 < · · · < ik ≤ n − 1, we define Mn;i1 ,...,ik := [k; 0, · · · , ˆi1 , · · · , ˆik , · · · , n − 1], Rn;i1 ,··· ,ik := Mn;i1 ,...,ik Lnp−2 . The degree of Mn;i1 ,··· ,ik is k + 2((1 + · · · + pn−1 ) − (pi1 + · · · + pik )) and then the degree of Rn;i1 ,··· ,ik is k + 2(p − 1)(1 + · · · + pn−1 ) − 2(pi1 + · · · + pik ). We put Pn := Fp [x1 , · · · , xn ]. The subspace of all invariants of H ∗ BVn under the action of GLn is given by the following theorem. Theorem 2.1 (Dickson [4], Mùi [24]). (i) The subspace of all invariants of Pn under the action of GLn is given by D[n] := PnGLn = Fp [qn,0 , · · · , qn,n−1 ]. (ii) As a D[n]-module, (H ∗ BVn )GLn is free and has a basis consisting of 1 and all elements of {Rn;i1 ,··· ,ik : 1 ≤ k ≤ n, 0 ≤ i1 < · · · < ik ≤ n − 1}. In other words, n ∗ (H BVn ) GLn = PnGLn Rn;i1 ,··· ,ik PnGLn . ⊕ k=1 0≤i1 ps; pi − r (−1)r+i i er βes − (p − 1)(i − s) βer+s−i ei pi − r (−1)r+i i (−1)r+i + i (p − 1)(i − s) − 1 r+s−i i e βe , r ≥ ps. pi − r − 1 These elements are called Adem relations. The quotient algebra R = T /IAdem is called the Dyer-Lashof algebra. We denote the image of eI,ε by QI,ε , then Qi and βQi satisfy the Adem relations: (p − 1)(i − s) − 1 Qr Qs = (−1)r+i Qr+s−i Qi , r > ps; (2.1) pi − r i Qr βQs = (p − 1)(i − s) βQr+s−i Qi pi − r (−1)r+i i − (−1) r+i i (p − 1)(i − s) − 1 Qr+s−i βQi , r ≥ ps. pi − r − 1 (2.2) Let P∗r be the dual to the Steenrod cohomology operation P r , then the Nishida relations hold: (p − 1)(s − r) P∗r Qs = (−1)r+i Qs−r+i P∗i ; r − pi i P∗r βQs = (−1)r+i i (p − 1)(s − r) − 1 βQs−r+i P∗i r − pi (−1)r+i + i I,ε (p − 1)(s − r) − 1 Qs−r+i P∗i β. r − pi − 1 A monomials Q is called admissible if (I, ε) is admissible (i.e. a string (I, ε) = ( 1 , i1 , · · · , n , in ) is admissible if pik − k ≥ ik−1 for 2 ≤ k ≤ n). Let R[n] be the subspace of R spanned by all monomials of length n. Due to the form of the Adem relations, R[n] has an additive basis consisting of all admissible monomials of length n and non-negative excess, which is called the admissible basis. Next, we recall the structure of the dual of the Dyer-Lashof algebra. For p = 2, the structure is studied by Madsen [20]. He shows that R[n]∗ is isomorphic to the Dickson algebra. For p odd, May [3] shows that R[n]∗ is isomorphic to a proper subalgebra of the Dickson-Mùi algebra (see also Kechagias [18]). For convenience we shall write I instead of (I, ε). Page 6 of 22 PHAN HOÀNG CHƠN Let In,i , Jn;i , Kn;s,i be admissible sequences of non-negative excess and length n as follows In,i = (pi−1 (pn−i − 1), · · · , pn−i − 1, pn−i−1 , · · · , 1); Jn;i = (pi−1 (pn−i − 1), · · · , pn−i − 1, (1, pn−i−1 ), · · · , 1); Kn;s,i = (pi−1 (pn−i − 1) − ps−1 , · · · , pi−s (pn−i − 1) − 1), (1, pi−s−1 (pn−i − 1)), pi−s−2 (pn−i − 1), · · · , p(pn−i − 1), (1, pn−i − 1), pn−i−1 , · · · , 1). Then the excess of QIn,i is 0 if 0 < i ≤ n − 1 and 2 if i = 0; and exc(QJn;i ) = 1, 0 ≤ i ≤ n − 1; exc(QKn;s,i ) = 0, 0 ≤ s < i ≤ n − 1. Let ξn,i = (QIn,i )∗ , 0 ≤ i ≤ n − 1, τn;i = (QJn;i )∗ , 0 ≤ i ≤ n − 1, and σn;s,i = (QKn;s,i )∗ , 0 ≤ s < i ≤ n − 1, with respect to the admissible basis of R[n]. The following theorem gives the structure of the dual of the Dyer-Lashof algebra. Theorem 2.2 (May [3], see also Kechagias [18]). As an algebra, R[n]∗ is isomorphic to the free associative commutative algebra over Fp generated by the set {ξn,i , τn;i , σn;s,i : 0 ≤ i ≤ n − 1, 0 ≤ s < i}, subject to relations: 2 (i) τn,i = 0, 0 ≤ i ≤ n − 1; (ii) τn;s τn;i = σn;s,i ξn,0 , 0 ≤ s < i ≤ n − 1; (iii) τn;s τn;i τn;j = τn;s σn;i,j ξn,0 , 0 ≤ s < i < j ≤ n − 1; 2 (iv) τn;s τn;i τn;j τn;k = σn;s,i σn;j,k ξn,0 , 0 ≤ s < i < j < k ≤ n − 1. The relationship between the dual of the Dyer-Lashof algebra and the modular invariants is given by the following theorem. Theorem 2.3 (Kechagias [17], [18]). As algebras over the Steenrod algebra, R[n]∗ is isomorphic to B[n] via the isomorphism Φ given by Φ(ξn,i ) = −qn,i , Φ(τn;i ) = Rn;i , 0 ≤ i ≤ n − 1 and Φ(σn;s,i ) = Rn;s,i , 0 ≤ s < i ≤ n − 1. 3. Additive base of modular (co)invariants In this section, we construct a new basis for B[n]∗ , which is a useful tool for the Section 4. Since R[n] ∼ = B[n]∗ , the basis can be considered is a basis of R[n]. Beside, some additive base of the Dickson-Mùi invariants (H ∗ BVn )GLn , the Dickson-Mùi coinvariants (H∗ BVn )GLn as well / (H ∗ BVn )GLn are as the cokernel of the restriction map of the symmetric group H ∗ BΣpn established. We order the set of tuples I = ( 1 , i1 , · · · , n , in ) by the ordering defined inductively as follows (1) ( 1 , i1 ) < (ω1 , j1 ) if 1 + i1 < ω1 + j1 or i1 + 1 = j1 + ω1 , 1 < ω1 ; (2) ( 1 , i1 , · · · , k , ik ) < (ω1 , j1 , · · · , ωk , jk ) if: (a) I = ( 1 , i1 , · · · , k−1 , ik−1 ) < (ω1 , j1 , · · · , ωk−1 , jk−1 ) = J or (b) I = J, ik + pk−1 k < jk + pk−1 ωk or (c) I = J, ik + pk−1 k = jk + pk−1 ωk and k < ωk . It should be noted that, when k = ωk = 0 for all k, the above ordering coincides with the lexicographic ordering from the left. in i1 1 n (respect, V I , xI ) is called less than q J (respect, A monomial q I = Rn;0 qn,0 · · · Rn;n−1 qn,n−1 J J V , x ) if I < J. MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k Page 7 of 22 Then we obtain the following lemmas. Lemma 3.1. For is ≥ 0, we have (p−1)i1 p(p−1)(i1 +i2 ) x2 in i1 = x1 · · · qn,n−1 qn,0 n−1 · · · xnp (p−1)(i1 +···+in ) + greater. Proof. For 0 ≤ s ≤ n − 1, using the inductive formula p qn,s = qn−1,s−1 + qn−1,s Vnp−1 , we can express qn,s in Vi ’s as follows qn,s = (Vs · · · Vn )p−1 + greater. It implies (p−1)i1 i1 in qn,0 · · · qn,n−1 = V1 · · · Vn(p−1)(i1 +···+in ) + greater Moreover, by the definition n−1 (λ1 x1 + · · · + λn−1 xn−1 + xn ) = xpn Vs = + greater. λi ∈Fp So that, we have (p−1)i1 p(p−1)(i1 +i2 ) x2 i1 in qn,0 · · · qn,n−1 = x1 n−1 · · · xnp (p−1)(i1 +···+in ) + greater. The proof is complete. For any string of integers I = ( 1 , i1 , . . . , n , in ), with i1 ∈ Z, is ≥ 0, 2 ≤ s ≤ n, and {0, 1}, we put b(I) = s s and m(I) = max{ s : 1 ≤ s ≤ n}. Lemma 3.2. For I = ( 1 , i1 , . . . , m(I) + b(I) ≥ 0, we have n , in ), with i1 ∈ Z, is ≥ 0, 2 ≤ s ≤ n, s (p−1)(i1 +b(I))− 1 n−1 · · · enn xnp (p−1)(i1 +···+in +b(I))−pn−1 n Proof. From the proof of Lemma 3.1, we have (p−1)i1 i1 in qn,0 · · · qn,n−1 = V1 (p−1)i1 = L1 · · · Vn(p−1)(i1 +···+in ) + greater (p−1)(i1 +···+in ) (p−1)(i1 +i2 ) L2 (p−1)(i1 +i2 ) ··· L1 Ln (p−1)(i1 +···+in ) + greater Ln−1 (p−1)(i1 +···+in ) Ln = (p−1)i2 L1 (p−1)in · · · Ln−1 + greater. Since Rn;s = Mn;s Lp−2 n , for 0 ≤ s ≤ n − 1, we obtain i1 in 1 n Rn;0 qn,0 · · · Rn;n−1 qn,n−1 (p−1)(i1 +···+in +b(I))−b(I) 1 n = Mn;0 · · · Mn;n−1 Ln (p−1)i2 L1 (p−1)in · · · Ln−1 (p−1)(i1 +b(I))−b(I) 1 n = Mn;0 · · · Mn;n−1 V1 ∈ ∈ {0, 1}, and i1 − i1 in 1 n Rn;0 qn,0 · · · Rn;n−1 qn,n−1 =(−1) 2 +···+(n−1) n e11 x1 + greater. s + greater · · · Vn(p−1)(i1 +···+in +b(I))−b(I) + greater. Page 8 of 22 PHAN HOÀNG CHƠN Since is ≥ 0, 2 ≤ s ≤ n and i1 − m(I) + b(I) ≥ 0, applying the proof of Lemma 3.1, we get i1 in 1 n Rn;0 qn,0 · · · Rn;n−1 qn,n−1 (p−1)(i1 +b(I))−b(I) 1 n = Mn;0 · · · Mn;n−1 x1 n−1 · · · xnp (p−1)(i1 +···+in +b(I))−pn−1 b(I) + greater. Moreover, for 0 ≤ s ≤ n − 1, s−1 Mn;s = (−1)s x1 xp2 · · · xsp s−1 s+1 s+1 n−1 es+1 xps+2 · · · xpn + greater, n−1 in other words, x1 xp2 · · · xsp es+1 xps+2 · · · xpn is the least monomial occurring non-trivially in Mn;s . Indeed, it is sufficient to compare the order of n following monomials. e1 x2 xp3 · · · xsp s−2 s−1 s+1 n−1 xps+1 xps+2 · · · xpn , ···································· 2 x1 xp2 xp3 · · · xsp s−1 s+1 n−1 es+1 xps+2 · · · xpn , ···································· 2 x1 xp2 xp3 · · · xps s−1 s+1 n−1 xps+1 · · · xpn−1 en . By directly checking, we have the assertion. Combining these facts, we have the assertion of the lemma. Proposition 3.3. For any n ≥ 1, as an Fp -vector space, (H ∗ BVn )GLn has a basis i1 in 1 n consisting of all elements q I = Rn;0 qn,0 · · · Rn;n−1 qn,n−1 for i1 ∈ Z, is ≥ 0, 2 ≤ s ≤ n, s ∈ {0, 1} and i1 − m(I) + b(I) ≥ 0. i1 in 1 n Proof. From Theorem 2.1, {q I = Rn;0 qn,0 · · · Rn;n−1 qn,n−1 : i1 − m(I) + b(I) ≥ 0} is a set of generators of (H ∗ BVn )GLn . Moreover, from Lemma 3.2, this set is linear independent. i1 in 1 n Proposition 3.4. For any n ≥ 1, the set of elements q I = Rn;0 qn,0 · · · Rn;n−1 qn,n−1 for i1 ∈ Z, is ≥ 0, 2 ≤ s ≤ n, s ∈ {0, 1} and 2i1 + b(I) ≥ 0, provides an additive basis for B[n]. Proof. From Proposition 3.3, we see that the set in the proposition is the subset of a basis of (H ∗ BVn )GLn , therefore, it is linear independent. Moreover, since, for 0 ≤ s < t ≤ n − 1, −1 Rn;s,t = Rn;s Rn;t qn,0 , every elements in B[n] can be written as a linear combination of elements of the set. Corollary 3.5. For any n ≥ 1, as an Fp -vector space, the cokernal of the restriction map / H ∗ (BVn )GLn has a basis consisting of all elements that are the images under H ∗ (BΣpn ) i1 in 1 n the quotient map of all elements of the form Rn;0 qn,0 · · · Rn;n−1 qn,n−1 for i1 ∈ Z, is ≥ 0, 2 ≤ s ≤ n, s ∈ {0, 1} and m(I) − b(I) ≤ i1 < −b(I)/2. in i1 n 1 qn,0 · · · Rn;n−1 qn,n−1 : 2i1 + b(I) ≥ k} is For k ≥ 0, the subspace of B[n] generated by {Rn;0 a subalgebra of B[n], which is denoted by Bk [n]. It is immediate that B0 [n] = B[n]. MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k Page 9 of 22 Let ui ∈ H1 BVn be the dual of ei and let vi ∈ H2 BVn be the dual of xi . Then the homology of Vn , H∗ BVn , is the tensor product of the exterior algebra generated by ui ’s and the divided [t] power algebra generated by vi ’s. We denote by vi the t-th divided power of vi . Since R[n] is isomorphic to B[n]∗ , R[n] is considered the quotient algebra of (H∗ BVn )GLn . The following theorem provides an additive basis for B[n]∗ and then for R[n]. Theorem 3.6. For k ≥ 0, the set of all elements [(p−1)(i1 +b(I))− [u11 v1 for i1 ∈ Z, is ≥ 0, 2 ≤ s ≤ n, Bk [n]∗ , the dual of Bk [n]. s 1] · · · unn vn[p n−1 (p−1)(i1 +···+in +b(I))−pn−1 n] ], ∈ {0, 1}, and 2i1 + b(I) ≥ k provides an additive basis for Proof. Denote q( 1 , i1 , · · · , (−1) n , in ) 2 +···+(n−1) n = [(p−1)(i1 +b(I))− u11 v1 1] · · · unn vn[p n−1 (p−1)(i1 +···+in +b(I))−pn−1 n] . From Lemma 3.2, we see that i1 in 1 n Rn;0 qn,0 · · · Rn;n−1 qn,n−1 , q(ω1 , s1 , · · · , ωn , sn ) = 0, (ω1 , s1 , · · · , ωn , sn ) < ( 1 , i1 , · · · , 1, (ω1 , s1 , · · · , ωn , sn ) = ( 1 , i1 , · · · , n , in ); n , in ). Therefore, the set of all [q( 1 , i1 , · · · , n , in )] satisfying the condition in the theorem provides a basis of Bk [n]∗ . Moreover, since Bk [n]∗ is a quotient algebra of (H∗ BVn )GLn , [q( 1 , i1 , · · · , = n , in )] [(p−1)(i1 +b(I))− [u1 v1 1 1] · · · unn vn[p n−1 (p−1)(i1 +···+in +b(I))−pn−1 n] ]. Hence, we have the assertion of the theorem. It should be noted that, when k = 0, the basis mentioned in Theorem 3.6 is not the dual basis of the one in Proposition 3.4. Using the proof is similar to the proof of Theorem 3.6, we have the following proposition. Proposition 3.7. For n ≥ 1, the set of all elements [(p−1)(i1 +b(I))− [u11 v1 for i1 ∈ Z, is ≥ 0, 2 ≤ s ≤ n, (H∗ BVn )GLn . s 1] · · · unn vn[p n−1 (p−1)(i1 +···+in +b(I))−pn−1 n] ], ∈ {0, 1}, and i1 + b(I) − m(I) ≥ 0 provides an additive basis for Let Ik be the ideal of R generated by all monomials of excess less than k. The quotient algebra R/Ik is denoted by Rk . And we also denote by Rk [n] the subspace of Rk spanned by all monomial of length n. Then, we have the following proposition. Proposition 3.8. As algebras over the Steenrod algebra, Rk [n]∗ ∼ = Bk [n] via the isomorphism give in Theorem 2.3. Page 10 of 22 PHAN HOÀNG CHƠN Proof. For a string of integers e = (e1 , · · · , ej ) such that 1 ≤ e1 < · · · < ej ≤ n, we put Kn;e1 ,e2 + · · · + Kn;ej−1 ,ej , if j is even, Kn;e1 ,e2 + · · · + Kn;ej−2 ,ej−1 + Jn;ej , if j is odd, Ln;e = and Ln;e is the string of all zeros if e is empty. Here we mean ( 1 , i1 , · · · , n , in ) + ( 1 , j1 , · · · , n , jn ) to be the string (ω1 , t1 , · · · , ωn , tn ) with ts = is + js and ωs = s + s (mod 2). In [3, p.38], May shows that for any string I of non-negative excess, it can be uniquely expressed in the form n−1 I= ti In,i + Ln;e , i=0 for some string e, and exc(I) = 2t0 + exc(Ln;e ). By the same argument of the proof of Theorem 3.7 in [3, p.29], we obtain that the set of all monomials i1 in 2 ξn,0 · · · ξn,n−1 , (σn;e1 ,e2 · · · σn;ej−2 ,ej−1 ) 1 τn;e j 2i1 + 2 ≥k ∗ provides an additive basis of Rk [n] . Using relation (ii) in Theorem 2.2, above monomials can be written in the form (up to a sign) i1 in 1 n τn;0 ξn,0 · · · τn;n−1 ξn,n−1 , 2i1 + b(I) ≥ k. i1 1 τn;0 ξn,0 in n It implies that the set of all monomials · · · τn;n−1 ξn,n−1 , 2i1 + b(I) ≥ k is a basis of Rk [n]∗ . By the definition of Bk [n] and Theorem 2.3 we have the assertion of the proposition. 4. The Hopf ring structure of H∗ QS k In this section, we use results of the modular (co)invariants in above sections to describe a complete set of relations for {H∗ QS k }k≥0 as a Hopf ring. Let [1] ∈ H∗ QS 0 be the image of non-base point generator of H0 S 0 under the map induced by the canonical inclusion S 0 → QS 0 and let σ ∈ H∗ QS 1 be the image of the generator of H1 S 1 under the homomorphism induced by the inclusion S 1 → QS 1 . Note that the element σ is usually known as the homology suspension element because σ ◦ x is the homology suspension of x. From the results of Dyer-Lashof [5] and May [3], we have Theorem 4.1 (Dyer-Lashof [5], May [3]). The mod p homology of {QS k }k≥0 is given by H∗ QS 0 = P [QI [1] : I admissible, exc(I) + k I ◦k 1 H∗ QS = P [Q (σ ) : I admissible, exc(I) + > 0] ⊗ Fp [Z], 1 > k], k > 0. Some basic properties are given in the following theorem. Theorem 4.2 (May [3], [22]). For b, f ∈ H∗ QS k , (i) P∗k (b ◦ f ) = i P∗i (b) ◦ P∗k−i (f ) and β(b ◦ f ) = β(b) ◦ f + (−1)degb b ◦ β(f ). (ii) Qk (b) ◦ f = i Qk+i (b ◦ P∗i (f )). (iii) βQk (b) ◦ f = i βQk+i (b ◦ P∗i (f )) − i (−1)degb Qk+i (b ◦ P∗i β(f )). In [10], Kahn and Priddy constructed the transfer tr(1) : (BV1 )+ → QS 0 . MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k Page 11 of 22 (1) The induced transfer tr∗ : H∗ (BV1 )+ → H∗ QS 0 sends u v [i(p−1)− ] to β Qi [1] and others to zero. Let ψ : Σm × Σn → Σmn be the permutation product of symmetric groups; and let In : Vn → Σpn be the composition ψ×···×ψ Vn = V1 × · · · × V1 → Σp × · · · × Σp −−−−−→ Σpn . By the results of Madsen and Milgram [21, Theorem 3.10], we have the following commutative diagram / BΣpn BIn BVn tr (1) ×···×tr (1) i  QS 0 × · · · × QS 0  / QS 0 µ where µ is the composition product in QS 0 . Therefore, we get the Kahn-Priddy’s transfer tr(n) = µ ◦ (tr(1) × · · · × tr(1) ) : BVn → QS 0 . (n) The induced transfer in homology tr∗ : H∗ BVn → H∗ QS 0 sends the “external product” in H∗ BVn (with respect to the decomposition BVn BVr × BVn−r ) to the circle product in H∗ QS 0 . In other words, we have (n) [i (p−1)− tr∗ (u11 v1 1 = 1] · · · unn vn[in (p−1)− (1) [i (p−1)− tr∗ (u11 v1 1 1] n] ) (1) ) ◦ · · · ◦ tr∗ (unn vn[in (p−1)− n] ). (n) Since tr = i ◦ BIn and GLn is the “Weyl group” of the inclusion Vn ⊂ Σpn , we have an (n) important feature of the map tr∗ is that they factor through the coinvariant of the general linear group. In other words, the diagram (n) tr∗ H∗ BVn / H∗ QS 0 ? p  (H∗ BVn )GLn is commutative. Moreover, we have the following proposition. (n) Proposition 4.3. The transfer tr∗ factors through (B[n])∗ . In other words, the diagram (n) tr∗ H∗ BVn p  ∗ B[n] / H∗ QS 0 ? ϕn commutes. ∗ ∗ = (BIn )∗ ◦ i∗ , the image of tr(n) is Proof. Since the induced transfer on cohomology tr(n) ∗ ∗ ∗ contained in the image of the restriction (BIn ) : H BΣpn → H BVn . Moreover, from Mùi Page 12 of 22 PHAN HOÀNG CHƠN [24, Chapter 2, Theorem 6.1], the image of the restriction (BIn )∗ is B[n] ⊂ (H ∗ BVn )GLn . Therefore, the assertion of the proposition is immediate from the dual. For any I = ( 1 , i1 , · · · , n , in ), with i1 ∈ Z, is ≥ 0, 2 ≤ s ≤ n, s ∈ {0, 1}, and i1 + b(I) − in i1 n 1 with respect to the qn,n−1 , · · · , Rn;n−1 qn,0 m(I) ≥ 0, let E( 1 ,i1 ,··· , n ,in ) is the dual of Rn;0 monomials basis given in Proposition 3.3; and we use the same notation E( 1 ,i1 ,··· , n ,in ) to (n) denote its image under the transfer tr∗ . In particular, E( ,k) = β Qk [1]. We have another description of the homology of {QS k }k≥0 as follows. Theorem 4.4. The homology of QS k is given by H∗ QS 0 = P [E( 1 ,i1 +b(I)) ◦ · · · ◦E( n−1 (i +···+i +b(I))−∆ n ,p 1 n n n) n ≥ 1, 2i1 + b(I) + : 1 > 0] ⊗ Fp [Z], and for k > 0, H∗ QS k = P [σ ◦k ◦ E( 1 ,i1 +b(I)) ◦ · · · ◦E( n−1 (i +···+i +b(I))−∆ n ,p 1 n n n) n ≥ 1, 2i1 + b(I) + where ∆s = ps−1 −1 p−1 : 1 > k], = 1 + · · · + ps−2 , s ≥ 2, and ∆1 = 0. In order to prove the theorem, we need two following lemmas. Lemma 4.5. For n ≥ 1, σ ◦k ◦ E( 1 ,i1 +b(I)) ◦ · · · ◦E( 2 +···+(n−1) n = (−1) n−1 (i +···+i +b(I))−∆ n ,p 1 n n n) β Qj1 · · · β n Qjn (σ ◦k ) + 1 QK (σ ◦k ), where n−s−1 js = pn−s (i1 + · · · + is + b(I)) + p (pn−s− − 1)is+ +1 − δn (s), =0 δn (s) = pn−s−1 n + ··· + s+1 , exc(K) < exc(J) = 2i1 + b(I). Proof. The n = 1 case is immediate. Using Theorem 4.2 and Nishida’s relations, we obtain the assertion of the lemma for n = 2. We shall prove the case n ≥ 3 by induction. It is sufficient to prove in the case n = 3. By the inductive hypothesis, the element E( 2 ,p(i1 +i2 +b(I))− 2 ) ◦ E( p2 −1 2 3 ,p (i1 +i2 +i3 +b(I))− p−1 3 ) = β 1 Qp(i1 +t2 +b(I))− 2 [1] ◦ β 3 Qp 2 2 −1 (i1 +i2 +t3 +b(I))− pp−1 can be written as follows (−1) 3 β 2 Qp 2 (i1 +i2 +b(I))+p(p−1)i3 −( 2 +p 3 ) β 3 Qp(i1 +i2 +i3 +b(I))− 3 [1] + other terms of smaller excess. 3 [1] MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k Therefore, y = E( ten as (−1) 1 ,i1 +b(I)) ◦ E( 2 ,p(i1 +i2 +b(i))− 2 ) ◦ E( Page 13 of 22 p2 −1 2 3 ,p (i1 +i2 +i3 +b(I))− p−1 3 ) can be writ- β 1 Qi1 +b(I)+k 3 k (P∗k (β 2 Qp 2 (i1 +i2 +b(I))+p(p−1)i3 −( 2 +p 3 ) β 3 Qp(i1 +i2 +i3 +b(I))− 3 [1])) + other terms of smaller excess. We observe that, for k ≥ pi, P∗k (β 2 Qp 2 (i1 +i2 +b(I))+p(p−1)i3 −( β 3 Qp(i1 +i2 +i3 +b(I))− 3 [1]) = (p − 1)[p (i1 + i2 + b(I)) + p(p − 1)i3 − ( 2 + p 3 ) − k] − k − pi 2 +p 3 ) 2 (−1)k i (p − 1)[p(i1 + i2 + i3 + b(I)) − i β 2 Qp 2 (i1 +i2 +b(I))+p(p−1)i3 −( = (−1)k β 2 Qp 2 3 − i] − 3 2 × × β 3 Qp(i1 +i2 +i3 +b(I))− 3 −i [1] + others (p − 1)[p2 (i1 + i2 + b(I)) + p(p − 1)i3 − ( 2 + p 3 ) − k] − 2 × k − p(p − 1)(i1 + i2 + i3 + b(I)) − p 3 (i1 +i2 +b(I))+p(p−1)i3 −( for i = (p − 1)(i1 + i2 + i3 + b(I)) − Therefore, y = (−1) 2 +p 3 )−k+i 2 +p 3 )−k+i β 3 Qi1 +i2 +i3 +b(I) [1] + others, 3. 2 +2 3 β 1 Qj1 β 2 Qj2 β 3 Qj3 [1] + others. As σ ◦k ◦ β Qj [1] = β Qj (σ ◦k ), then σ ◦k ◦ y can be written in the needed form. Lemma 4.6. The function mapping I = ( 1 , i1 + b(I), · · · , n , pn−1 (i1 + · · · + in + b(I)) − ∆n n ), 2i1 + b(I) > k, to admissible string J = ( 1 , j1 , · · · , n , jn ), with exc(J) > k, given as in Lemma 4.5, is a bijection. Proof. It is immediate. From Lemma 4.5, the set of elements σ ◦k ◦ E( 1 ,i1 +b(I)) ◦ · · · ◦ E( n ,pn−1 (i1 +···+in +b(I))−∆n n ) belongs to the indecomposable quotient (with respect to the star product) QH∗ QS k and it is linear independent. Moreover, the degree of Proof of Theorem 4.4. σ ◦k ◦ E( 1 ,i1 +b(I)) ◦ · · · ◦ E( n−1 (i +···+i +b(I))−∆ n ,p 1 n n n) is equal to the degree of β 1 Qj1 · · · β n Qjn (σ ◦k ). Finally, from Lemma 4.6, we obtain that this set generates QH∗ QS k in each degree. Thus, the elements E( ,s) = β Qs [1] and σ generate {H∗ QS k }k≥0 as a Hopf ring. The problem is to find a complete set of relations. It is solved by investigating the structure of the dual of Bk [n]. Let E (s) ∈ H∗ QS 0 [[s]], = 0, 1, be defined by E(0,k) sk , E 0 (s) = k≥0 E(1,k) sk . E 1 (s) = k≥1 Page 14 of 22 PHAN HOÀNG CHƠN Since the coproduct on the E( H2k(p−1)− k (BV1 )GL1 , arises from the coproduct on [u k v [k(p−1)− k ,k) k] ] in E(0,i) ⊗ E(0,j) , ψ(E(0,k) ) = i+k=j (E(0,i) ⊗ E(1,j) + E(1,i) ⊗ E(0,j) ). ψ(E(1,k) ) = i+j=k Therefore, ψ(E 0 (s)) = E 0 (s) ⊗ E 0 (s); ψ(E 1 (s)) = E 0 (s) ⊗ E 1 (s) + E 1 (s) ⊗ E 0 (s). For x ∈ H∗ QS k we define Q0 (s)x, Q1 (s)x ∈ H∗ QS k [[s]] as follows Qk xsk ; Q0 (s)x = k≥0 0 βQk xsk . Q1 (s)x = k≥1 0 1 1 Then we obtain that E (s) = Q (s)[1] and E (s) = Q (s)[1]. A complete set of algebraic relations for {H∗ QS k }k≥0 is given in the following proposition. Proposition 4.7. For s, t are formal variables, we have relations E 0 (sp−1 ) ◦ E 0 (tp−1 ) = E 0 (sp−1 ) ◦ E 0 ((s + t)p−1 ); (4.1) t ; s+t t E 1 (sp−1 ) ◦ E 1 (tp−1 ) = E 1 (sp−1 ) ◦ E 1 ((s + t)p−1 ) ; s+t E 0 (sp−1 ) ◦ E 1 (tp−1 ) = E 0 (sp−1 ) ◦ E 1 ((s + t)p−1 ) When k = 2i1 + b(I) + σ ◦k ◦ E( 1, (4.2) (4.3) b(I) > 0 and n ≥ 1, 1 ,i1 +b(I)) ◦ · · · ◦ E( n−1 (i +···+i +b(I))−∆ n ,p 1 n n n) = (1 − 1 )y p (4.4) for some y ∈ {H∗ QS k }k≥0 , where σ ◦0 = [1]. In particular, E(0,0) = [p]; σ ◦2k ◦ E( ,k) (4.5) = (1 − )(σ ◦2k p ) . (4.6) Remark 4.8. In the case p = 2 (see [27]) as well as in the Bockstein-nil homology for p odd (see [15]), the relations (4.2) and (4.3) omit because these relations come from the action of the Bockstein operation. In addition, when p = 2 or b(I) = 0 for p odd, the general case of relation (4.4) follows from n = 1 case, i.e., from relation (4.6). Indeed, using relation (4.6) we get σ ◦2i1 ◦ E(0,i1 ) ◦ · · · ◦ E(0,pn−1 (i1 +···+in )) = (σ ◦2i1 ) Using the distributivity between the obtain (σ ◦2i1 ) p p ◦ E(0,p(i1 +i2 )) ◦ · · · ◦ E(0,pn−1 (i1 +···+in )) . product and the ◦ product (see [26, Lemma 1.12]), we ◦ E(0,p(i1 +i2 )) ◦ · · · ◦E(0,pn−1 (i1 +···+in )) = (σ ◦2i1 ◦ E(0,i1 +i2 ) ◦ · · · ◦ E(0,pn−2 (i1 +···+in )) ) p . However, for b(I) > 0, the general case of relation (4.4) does not follow from n = 1 case and relations (4.1)-(4.3). For example, for I = (0, 0, 1, p), in order to prove the relation σ ◦ E(0,1) ◦ E(1,p−1) = y p , for some y ∈ {H∗ (QS k )}k≥0 , MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k Page 15 of 22 we must use relation (4.2) to write E(0,1) ◦ E(1,p−1) as a sum of E(0,i) ◦ E(1,j) for i < 1, before applying relation (4.6). But from relation (4.2), we obtain E(0,i) ◦ E(1,j) = m≥0 (p − 1)(i + j − m) − 1 E(0,m) ◦ E(1,i+j−m) . (p − 1)(i − m) Applying the relation, we can write E(0,1) ◦ E(1,p−1) = E(0,0) ◦ E(1,p) + E(0,1) ◦ E(1,p−1) . It implies E(0,0) ◦ E(1,p) = 0 and E(0,1) ◦ E(1,p−1) is not expressed as a sum of E(0,i) ◦ E(1,j) for i < 1. In other words, the relation σ ◦ E(0,1) ◦ E(1,p−1) = y p can not follow from (4.6) and (4.2). It should be also noted that, for k = 0, if b(I) = 1 = 1, then relation (4.4) becomes to trivial relation; but if b(I) > 1 = 1 or 1 = 0, the relation is nontrivial. Proof of Proposition 4.7. It should be noted that the formulas (4.1)-(4.3) can be proved by using the method in Turner [27], of course, it is more complicated. (n) Here we use the multiplicativity of the transfer and the fact that the transfer tr∗ is GLn invariant to show these relations. (1) First, we consider the first transfer tr∗ as the element tr∗ ∈ HomFp (H∗ BV1 , H∗ QS 0 ) ∼ = H∗ QS 0 [[s]] ⊗ E( ). (1) (1) Because tr∗ sends the generator in the degree 2(p − 1)i to E(0,i) , that in the degree 2(p − 1)i − 1 to E(1,i) , and the rest to zero, it is equal to E 0 (sp−1 ) + s−1 E 1 (sp−1 ). (2) Next, the second transfer tr∗ (2) tr∗ can be considered as the element ∈ HomFp (H∗ BV2 , H∗ QS 0 ) ∼ = H∗ QS 0 [[s, t]] ⊗ E( , ). By the multiplicativity of the transfer, this element has to be (E 0 (sp−1 ) + s−1 E 1 (sp−1 )) ◦ (E 0 (tp−1 ) + t−1 E 1 (tp−1 )). Since the transfer factors through the coinvariant of H∗ BV2 under the action of the general (2) linear group GL2 , acting ( 10 11 ) on tr∗ , we obtain (E 0 (sp−1 ) + s−1 E 1 (sp−1 )) ◦ (E 0 (tp−1 ) + t−1 E 1 (tp−1 )) = (E 0 (sp−1 ) + s−1 E 1 (sp−1 )) ◦ (E 0 ((s + t)p−1 ) + ( + )(s + t)−1 E 1 ((s + t)p−1 )). Expanding this equality and comparing the coefficients of “1”, and follows the formulas (4.1), (4.2) and (4.3). From above proof, we observe that the formulas (4.1), (4.2) and (4.3) also hold in colimBV /CS 0 H∗ (−)[[s, t]], where BV /CS 0 is the category whose objects are homotopy classes of maps from a classifying space of an elementary abelian p-group to CS 0 , whose morphism are commutative triangles, and CS 0 denotes the combinatorial model of QS 0 , that is, the disjoint union of BΣn ’s (see [15, Section 5]). From Lemma 4.5, for n ≥ 1, σ ◦k ◦ E( 1 ,i1 +b(I)) = (−1) ◦ · · · ◦E( 2 +···+(n−1) n n−1 (i +···+i +b(I))−∆ n ,p 1 n n n) β 1 Qj1 · · · β n Qjn (σ ◦k ) + QK (σ ◦k ), where exc(K) < exc(I) = 2i1 + b(I). Since k = 2i1 + b(I) + 1 , the second sum of the formula is trivial. Page 16 of 22 PHAN HOÀNG CHƠN If 1 = 1, then exc(I) < k, therefore, the first item is also trivial. Otherwise, if 1 = 0, then 2j1 = deg(β 2 Qj2 · · · β n Qjn (σ ◦k )), therefore, the first item is the p-th power of an element. Thus, the formula (4.4) is proved. Since σ is primitive elements with respect to the For n ≥ 1 and 2i1 + b(I) < k, Corollary 4.9. σ product, we have the following corollary. ◦k ◦ E( 1 ,i1 +b(I)) ◦ · · · ◦ E( n−1 (i +···+i +b(I))−∆ n ,p 1 n n n) = 0, where σ ◦0 = [1]. Let us put E (sp−1 ) = s−1 E (sp−1 ) ∈ H∗ Q0 [[s]], qualities (4.1)-(4.3) can be reduced as follows. For s, t are formal variable, then we have relation Corollary 4.10. p−1 E (s 1 ) ◦ E 2 (tp−1 ) = E 1 (sp−1 ) ◦ E 2 ((s + t)p−1 ), For A ∈ GL , B ∈ GLk , denote A ⊕ B = ( A0 B0 ) ∈ GL we have the lemma. +k 1 ≤ 2. and a ⊕ A = ( a0 A0 ) ∈ GL (4.7) +1 . Then Lemma 4.11. For n ≥ 2, the general linear group GLn = GLn (Fp ) is generated by {T, Σn , Ta : a ∈ F∗p }, where T = ( 10 11 ) ⊕ In−2 , Ta = a ⊕ In−1 . Theorem 4.12. The homology {H∗ QS k }k≥0 is the coalgebraic ring in Fp [Z] generated by E(0,i) (i ≥ 0), E(1,j) (j ≥ 1) and σ modulo all the relations implied by Proposition 4.7. The coproduct is specified by ψ(σ) = 1 ⊗ σ + σ ⊗ 1; ψ(E 0 (s)) = E 0 (s) ⊗ E 0 (s); ψ(E 1 (s)) = E 0 (s) ⊗ E 1 (s) + E 1 (s) ⊗ E 0 (s); ψ(a ◦ b) = ψ(a) ◦ ψ(b). The theorem can be show by using the framework of Turner [27] and Eccles. et.al [6], we mean that we can use the method in [27] to show for k = 0, and then use the bar spectral sequence (as in [6]) to induct for k > 0. Here, we modify the method of Turner [27] to show the theorem directly (without using the bar spectral sequence). In order to do this, we need some notations. We define elements f 0 (vi , s), f 1 (ui , vi , s) ∈ H∗ BVn [[s]] for any ui , vi by [k] vi sk ; f 0 (vi , s) = [k−1] k f 1 (ui , vi , s) = k≥0 ui vi s , k≥1 f 0 (0, s) = f 0 (vi , 0) = 1. Then we have (1) tr∗ (f 0 (vj , s)) = E 0 (sp−1 ); (1) tr∗ (f 1 (ui , vi , s)) = E 1 (sp−1 ). Put f 0 (vi , s) = s−1 f 0 (vi , s) and f 1 (ui , vi , s) = s−1 f 1 (ui , vi , s), then (1) tr∗ (f 0 (vj , s)) = E 0 (sp−1 ); (1) tr∗ (f 1 (ui , vi , s)) = E 1 (sp−1 ). MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k Page 17 of 22 Proof. Let D∗,∗ be the coalgebra generated by E(0,i) ∈ D2i(p−1),0 (i ≥ 0), E(1,j) ∈ D2j(p−1)−1,0 (j ≥ 1) and σ ∈ D1,1 . Apply the Ravenel-Wilson free Hopf ring functor [26] to the coalgebra D∗,∗ to give H D∗,∗ , the free Fp [Z]-Hopf ring on D∗,∗ . There is a map of coalgebras D∗,∗ → {H∗ QS k }k≥0 mapping the element E( ,i) to the element E( ,i) ∈ {H∗ QS k }k≥0 . By the universality, the map extends to a unique map of Hopf rings h : H D∗,∗ → {H∗ QS k }k≥0 . Let A∗,∗ be the free Fp [Z]-Hopf ring on D∗,∗ subject to relations arising from Proposition 4.7. Since all relations defined in A∗,∗ hold in {H∗ QS k }k≥0 , the map h induces a unique map ¯ : A∗,∗ → {H∗ QS k }k≥0 . h Using Theorem 4.4, we get that this map is surjective. Therefore, it induces a surjection between indecomposable quotients (with respect to product) QA∗,k → QH∗ QS k . In order to prove A∗,∗ ∼ = {H∗ QS k }k≥0 , it is sufficient to prove the induced surjection between indecomposable quotients is an isomorphism. We now begin our proof of claim that QA∗,k → QH∗ QS k is an isomorphism. For s = (s1 , · · · , sn ) being a vector of formal variables and for = ( 1 , · · · , n ), i ∈ {0, 1}, we define u (s) = f 1 (u1 , v1 , s1 ) · · · f n (un , vn , sn ), where f 0 (ui , vi , si ) = f 0 (vi , si ), and we define E (sp−1 ) = E 1 (sp−1 ) ◦ · · · ◦ E n (sp−1 n ). 1 Let g : n≥1 H∗ BVn → A∗,0 be the map of Fp -algebras given by u (s) → E (sp−1 ). It is easy see that g is a surjection. From Corrollary 4.10 and Lemma 4.11, it is easy to check that g(u A(s)) = E (sp−1 ) = g(u (s)), for A ∈ GLn . Therefore, g factors through the coinvariants space of the general linear groups n≥1 (H∗ BVn )GLn . Moreover, from Proposition 3.4, elements E( 1 ,i1 ,··· , n ,in ) ∈ (H∗ BVn )GLn , for 2i1 + b(I) < 0, are trivial in B0 [n]∗ . Therefore, from Theorem 3.6 and Proposition 3.7, they can be written as a combination of elements of the form [(p−1)(j1 +ω)−ω1 ] 1 [uω 1 v1 n−1 n [p · · · uω n vn (p−1)(j1 +···+jn +ω)−pn−1 ωn ] ], for ωi = 0 or 1, ω = ω1 + · · · + ωn and 2j1 + ω < 0. Combining with the fact that g is an algebra homomorphism, we get that the image of E( 1 ,i1 ,··· , n ,in ) , 2i1 + b(I) < 0, under g can be written as a combination of the elements of the form E(ω1 ,j1 +ω) ◦ · · · ◦ E(ωn ,pn−1 (j1 +···+jn +ω)−∆n ωn ) , with 2j1 + ω < 0, ω = ω1 + · · · + ωn . It implies g(E( 1 ,i1 ,··· , n ,in ) ) = 0 for 2i1 + b(I) < 0. Hence, from Corollary 4.9, g factors through n≥1 B0 [n]∗ . In other words, the diagram n≥1 p / A∗,0 ? g H∗ BVn  n≥1 g ∗ B0 [n] is commutative. For any k ≥ 0, let gk be the composition g ¯ σ ◦k ◦− B0 [n] − → A∗,0 −−−−→ A∗,k . n≥1 Page 18 of 22 PHAN HOÀNG CHƠN When k = 0, g0 is just g¯. Since, from Corollary 4.9, in A∗,k , σ ◦k ◦ E( 1 ,i1 +b(I)) ◦ · · · ◦ E( n−1 (i +···+i +b(I))−∆ n ,p 1 n n n) = 0, 2i1 + b(I) < k, by the same above argument, the Fp -map gk factors through n≥1 Bk [n] and gk is also a surjection. For any n ≥ 1, let QA∗,k [n] be the subspace of QA∗,k spanned by all elements σ ◦k ◦ E( 1 ,i1 ) ◦ · · · ◦ E( n ,in ) and let QH∗ QS k [n] be the subspace of QH∗ QS k spanned by all elements β 1 Qj1 · · · β n Qjn (σ ◦k ). By Theorem 3.6, in Bk [n]∗ , we have : 2i1 + b(I) + 1 > k} = n−1 n−1 [(p−1)(i1 +b(I))− 1 ] Span{[u1 v1 · · · unn vn[p (p−1)(i1 +···+in +b(I))−p Span{E( 1 ,i1 ,··· , n ,in ) 1 2i1 + b(I) + 1 n] ]: > k}. Therefore, we have a surjection [(p−1)(i1 +b(I))− S = Span{[u11 v1 1] · · · unn vn[p n ≥ 1, 2i1 + b(I) + 1 n−1 (p−1)(i1 +···+in +b(I))−pn−1 n] ]: > k} → QA∗,k [n]. It implies that, in each degree d, dim(S) ≥ dim(QA∗,k [n]) ≥ dim(QH∗ QS k [n]). Finally, we observe that, for each degree d, n n , in )|2(i1 + 1 )(p n n−1 )− n = n )(p − p Card{( 1 , i1 , · · · , + 2(in + = Card{( 1 , i1 , · · · , + n , in )| 1 − 1) − 1 + ··· d} + 2((p − 1)(i1 + b(I)) − n−1 ((p − 1)(i1 + · · · + in + b(I)) − n + 2p 1) + ··· n ) = d}. ∼ QH∗ QS k . So dim(S) = dim(QA∗,k [n]) = dim(QH∗ QS k [n]). It implies QA∗,k = The proof is complete. 5. The actions of A and R on H∗ QS k In this section, using the same method of Turner [27], we describe the action of the mod p Dyer-Lashof operations as well as of mod p Steenrod operations on the Hopf ring. For convenience, we write P k instead of P∗k and write their action on the right. For x ∈ H∗ QS k and formal variable s, we define the formal series (xβ P k )sk , = 0, 1. xP (s) = k≥0 In order to prove the main theorem of this section, we need the following lemma. Lemma 5.1. There are the following relations: x ◦ Q (s)(y) = Q (s)(xP 0 (s−1 ) ◦ y) − (−1)degy Q0 (s)(xP 1 (s−1 ) ◦ y); (5.1) f 0 (vi , s)P 0 (t) = f 0 (vi , (s + sp t)); (5.2) f 0 (vi , s)P 1 (t) = f 1 (ui , vi , (s + sp t)); (5.3) f 1 (ui , vi , s)P 0 (t) = f 1 (ui , vi , (s + sp t)). (5.4) MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k Page 19 of 22 Proof. From Theorem 4.2, we obtain x ◦ Q (s)(y)  β Qk+i  =   xP i ◦ y  sk − i≥0 k≥ ≥ +i xβP i ◦ y  sk i≥1   −i xP i ◦ y  s β Q  = (−1)degy Qk+i  k≥    xβP i ◦ y  s (−1)degy Q  − ≥i i≥0 −i . i≥1 It should be noted that if xP i (respect, xβP i ) is nontrivial then the degree of xP i ◦ y (respect, xβP i ◦ y) is not less than 2i (respect, 2i + 1). It implies that, when < + i (respect, < i) then β Q (xP i ◦ y) (respect, Q (xβP i ◦ y)) is trivial. Therefore, the right hand side of above formula can be written as follows     (xP i ◦ y)s−i  s − β Q  ≥ ≥0 i≥0 0 −1 = Q (s)(xP (s (xβP i ◦ y)s−i  s (−1)degy Q  ) ◦ y) − (−1) i≥1 degy Q (s)(xP 1 (s−1 ) ◦ y). 0 Hence, the formula (5.1) is proved. From [n] vi β P k = n − (p − 1)k − k [n−(p−1)k− ] ui vi we have the formulas (5.2) and (5.3). Since P k acts trivially on ui for k > 0, then [n−1] u i vi Pk = n − (p − 1)k − 1 [n−(p−1)k−1] ui vi . k This implies the last formula. The main results of the section is the following theorem, which gives a description of the actions of the Dyer-Lashof algebra and the Steenrod algebra on the Hopf ring. Theorem 5.2. Let x, y ∈ H∗ QS k and let s, t, t1 , t2 , · · · be formal variables; tp−1 = k , tp−1 n ), = ( 1 , · · · , n ). The following hold in H∗ QS [[s, t, t1 , t2 , · · · ]]. (tp−1 ,··· 1 [n]P (s) = (1 − )[n]. 0 p−1 0 p−1 1 p−1 E (s E (s E (s (5.5) 0 0 p p−1 ). (5.6) 1 1 p p−1 ). (5.7) 0 1 p p−1 ). (5.8) )P (t) = E ((s − s t) )P (t) = E ((s − s t) )P (t) = E ((s − s t) E 1 (sp−1 )P 1 (t) = 0. p−1 Q (s 1 (5.9) (x y)P (s) = (−1) deg y (x ◦ y)P (s) = (−1) deg y 0 1 (5.10) 0 0 1 (5.11) xP (s) yP (s) + (xP (s)) yP (s). xP (s) ◦ yP (s) + (xP (s)) ◦ yP (s). Q (s)[n] = [n] ◦ E (s). )E ((st)p−1 ) = (1 − tˆp−1 )[E 2 0 (5.12) (stˆ)p−1 ◦ E 1 (sp−1 ) + 1 (1 − 2 )E 1 (stˆ)p−1 ◦ E 0 (sp−1 )]. 2 Q (s)(x y) = Q (s)x Q0 (s)y + (−1) Q (s)([n] ◦ y) = [n] ◦ Q (s)y. deg y Q0 (s)x Q1 (s)y. (5.13) (5.14) (5.15) Page 20 of 22 PHAN HOÀNG CHƠN p−1 Q (sp−1 )(E ((st)p−1 )) = (1 − tˆ )[E ((sˆt)p−1 ) ◦ E (sp−1 ) n (1 − i )E i ((sˆt)p−1 ) ◦ E 0 (sp−1 )]. + (5.16) i=1 k Here we denote by tˆ = k≥0 tp , tˆi = obtained from by replacing i by 1. pk k≥0 ti , p−1 , · · · , tˆp−1 tˆ = (tˆp−1 n ), and 1 i the vector Proof. The first equality is immediate by degree. (1) (1) Since tr∗ (v [n] P k ) = (−1)k tr∗ (v [n] )P k , then equalities (5.6)-(5.9) are implied from (5.2), (5.3) and (5.4). Since the coproduct of P (s) is given by ψ(P (s)) = P (s) ⊗ P 0 (s) + P 0 (s) ⊗ P 1 (s), the formulas (5.10) and (5.11) come from the Cartan formula. Letting y = [1] in (5.1) to obtain x ◦ Q (s)[1] = Q (s)(xP 0 (s−1 )) − Q0 (s)(xP 1 (s−1 )). (5.17) Letting x = [n] in above equality and combining with (5.5), we obtain (5.12). Replace x = E (up−1 ) in (5.17), we get E (up−1 ) ◦ Q (s)[1] = Q (s)(E (up−1 )P 0 (s−1 )) − Q0 (s)(E (up−1 )P 1 (s−1 )). Combining with (5.6)-(5.9), we give E (up−1 ) ◦ Q (s)[1] = (1 − up−1 t−1 )− [Q (s)(E (u − up s−1 )p−1 ) − (1 − )Q0 (s)(E 1 (u − up s−1 )p−1 )]. From (5.18), letting = 0 and (5.18) = 1, one gets E 1 (up−1 ) ◦ Q0 (s)[1] = (1 − up−1 t−1 )−1 Q0 (s)(E 1 (u − up s−1 )p−1 ). These formulas imply (replacing s by sp−1 ) Q 1 (sp−1 )E 2 ((u − up s1−p )p−1 ) = (1 − up−1 s1−p )[E 2 (up−1 ) ◦ Q 1 (sp−1 )[1] + By letting t = u/s − (u/s)p with noting that tˆ = in the form Q 1 (sp−1 )E 2 ((st)p−1 ) = (1 − tˆp−1 )[E 2 1 (1 − 2 )E 1 (up−1 ) ◦ Q0 (sp−1 )[1]]. k k≥0 (stˆ)p−1 ◦ E 1 (sp−1 ) + tp = u/s, it is easy to write the equality 1 (1 − 2 )E 1 (stˆ)p−1 ◦ E 0 (sp−1 )]. So (5.13) is proved. The equality (5.14) is just the Cartan formula. In order to prove (5.15), to replace x = [n] in (5.1) with noting that [n]P 1 (s) = 0, we obtain [n] ◦ Q (s)y = Q (s)([n]P 0 (s−1 ) ◦ y). Using (5.5) we have (5.15). Since (n − 1)-fold coproduct of P (s) is given by ψ n−1 (P 0 (s)) = P 0 (s) ⊗ · · · ⊗ P 0 (s), and ψ n−1 (P 1 (s)) = P 1 (s) ⊗ · · · ⊗ P 0 (s) + · · · + P 0 (s) ⊗ · · · ⊗ P 1 (s), MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k Page 21 of 22 the last formula follows from formula (5.13) and the Cartan formula. The proof is complete. As discussion in the introduction, the category of A-H∗ QS 0 -coalgebraic modules and the one of A-R-allowable Hopf algebra also play important role in the study of the mpd p homology of the infinite loop spaces. We will investigate these categories and the relationship between them elsewhere. Acknowledgements The author would like to thank Lê Minh Hà and J. Peter May for many fruitful discussion, and Takuji Kashiwabara for his comments on an earlier version of this paper. The paper was completed while the author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). He thanks the VIASM for support and hospitality. References 1. S. Araki and T. Kudo, Topology of Hn -spaces and H-squaring operations, Memoirs of the Faculty of Science 10 (1956), 85–120. 2. J. M. Boardman, R. L. Kramer, and W. S. Wilson, The periodic Hopf ring of connective Morava K-theory, Forum Math. 11 (1999), no. 6, 761–767. MR 1725596 (2000k:55009) 3. F. R. Cohen, T. J. Lada, and J. P. 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MR 764345 (86c:55008) Phan Hoàng Chơn Department of Mathematics and Application, Saigon University, 273 An Duong Vuong, District 5, Ho Chi Minh city, Vietnam. chonkh@gmail.com [...]... · · ⊗ P 0 (s) + · · · + P 0 (s) ⊗ · · · ⊗ P 1 (s), MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k Page 21 of 22 the last formula follows from formula (5.13) and the Cartan formula The proof is complete As discussion in the introduction, the category of A-H∗ QS 0 -coalgebraic modules and the one of A-R-allowable Hopf algebra also play important role in the study of the mpd p homology of the infinite.. .MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k Page 11 of 22 (1) The induced transfer tr∗ : H∗ (BV1 )+ → H∗ QS 0 sends u v [i (p 1)− ] to β Qi [1] and others to zero Let ψ : Σm × Σn → Σmn be the permutation product of symmetric groups; and let In : Vn → Σpn be the composition ψ×···×ψ Vn = V1 × · · · × V1 → p × · · · × p −−−−−→ Σpn By the results of Madsen and Milgram [21, Theorem 3.10],... describe the action of the mod p Dyer-Lashof operations as well as of mod p Steenrod operations on the Hopf ring For convenience, we write P k instead of P k and write their action on the right For x ∈ H∗ QS k and formal variable s, we define the formal series (xβ P k )sk , = 0, 1 xP (s) = k≥0 In order to prove the main theorem of this section, we need the following lemma Lemma 5.1 There are the following... AND THE MOD p HOMOLOGY OF QS k 22 J P May, Homology operations on infinite loop spaces, Algebraic topology (Proc Sympos Pure Math., Vol XXII, Univ Wisconsin, Madison, Wis., 1970), Amer Math Soc., Providence, R.I., 1971, pp 171–185 MR 0319195 (47 #7740) 23 D S C Morton and N Strickland, The Hopf rings for KO and KU , Journal of Pure and Applied Algebra 166 (2002), no 3, 247 – 265 24 H Mùi, Modular invariant... E 1 (sp−1 ) Put f 0 (vi , s) = s−1 f 0 (vi , s) and f 1 (ui , vi , s) = s−1 f 1 (ui , vi , s), then (1) tr∗ (f 0 (vj , s)) = E 0 (sp−1 ); (1) tr∗ (f 1 (ui , vi , s)) = E 1 (sp−1 ) MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k Page 17 of 22 Proof Let D∗,∗ be the coalgebra generated by E(0,i) ∈ D2i (p 1),0 (i ≥ 0), E(1,j) ∈ D2j (p 1)−1,0 (j ≥ 1) and σ ∈ D1,1 Apply the Ravenel-Wilson free Hopf ring... groups and the Quillen map at odd primes, J Pure Appl Algebra 190 (2004), no 1-3, 137–153 9 J R Hunton and P R Turner, Coalgebraic algebra, J Pure Appl Algebra 129 (1998), no 3, 297–313 MR 1631257 (99g:16048) 10 D S Kahn and S B Priddy, The transfer and stable homotopy theory, Math Proc Cambridge Philos Soc 83 (1978), no 1, 103–111 11 T Kashiwabara, Hopf rings and unstable operations, J Pure Appl Algebra... E( p2 −1 2 3 ,p (i1 +i2 +i3 +b(I))− p 1 3 ) = β 1 Qp(i1 +t2 +b(I))− 2 [1] ◦ β 3 Qp 2 2 −1 (i1 +i2 +t3 +b(I))− pp−1 can be written as follows (−1) 3 β 2 Qp 2 (i1 +i2 +b(I)) +p( p−1)i3 −( 2 +p 3 ) β 3 Qp(i1 +i2 +i3 +b(I))− 3 [1] + other terms of smaller excess 3 [1] MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k Therefore, y = E( ten as (−1) 1 ,i1 +b(I)) ◦ E( 2 ,p( i1 +i2 +b(i))− 2 ) ◦ E( Page 13 of. .. of 22 p2 −1 2 3 ,p (i1 +i2 +i3 +b(I))− p 1 3 ) can be writ- β 1 Qi1 +b(I)+k 3 k (P k (β 2 Qp 2 (i1 +i2 +b(I)) +p( p−1)i3 −( 2 +p 3 ) β 3 Qp(i1 +i2 +i3 +b(I))− 3 [1])) + other terms of smaller excess We observe that, for k ≥ pi, P k (β 2 Qp 2 (i1 +i2 +b(I)) +p( p−1)i3 −( β 3 Qp(i1 +i2 +i3 +b(I))− 3 [1]) = (p − 1) [p (i1 + i2 + b(I)) + p( p − 1)i3 − ( 2 + p 3 ) − k] − k − pi 2 +p 3 ) 2 (−1)k i (p − 1) [p( i1... VIASM for support and hospitality References 1 S Araki and T Kudo, Topology of Hn -spaces and H-squaring operations, Memoirs of the Faculty of Science 10 (1956), 85–120 2 J M Boardman, R L Kramer, and W S Wilson, The periodic Hopf ring of connective Morava K-theory, Forum Math 11 (1999), no 6, 761–767 MR 1725596 (2000k:55009) 3 F R Cohen, T J Lada, and J P May, The homology of iterated loop spaces, Lecture... (5.5) 0 0 p p−1 ) (5.6) 1 1 p p−1 ) (5.7) 0 1 p p−1 ) (5.8) )P (t) = E ((s − s t) )P (t) = E ((s − s t) )P (t) = E ((s − s t) E 1 (sp−1 )P 1 (t) = 0 p 1 Q (s 1 (5.9) (x y )P (s) = (−1) deg y (x ◦ y )P (s) = (−1) deg y 0 1 (5.10) 0 0 1 (5.11) xP (s) yP (s) + (xP (s)) yP (s) xP (s) ◦ yP (s) + (xP (s)) ◦ yP (s) Q (s)[n] = [n] ◦ E (s) )E ((st )p 1 ) = (1 − t p 1 )[E 2 0 (5.12) (stˆ )p 1 ◦ E 1 (sp−1 ) + 1 (1 ... MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k Page of 22 the cohomology of the symmetric group Σpn to the cohomology of the elementary abelian pgroup of rank n, Vn , as well as the mod p Dyer-Lashof... B[n] MODULAR COINVARIANTS AND THE MOD p HOMOLOGY OF QS k Page of 22 Let ui ∈ H1 BVn be the dual of ei and let vi ∈ H2 BVn be the dual of xi Then the homology of Vn , H∗ BVn , is the tensor product... the similar the restriction map H ∗ BΣpn method of Turner [27], we give descriptions of the action of the mod p Steenrod algebra A and the action of the mod p Dyer-Lashof algebra R on the Hopf