The mod 2 Margolis homology of the Dickson algebra

Abstract

We completely compute the mod 2 Margolis homology of the Dickson algebra Dn , i.e. the homology of Dn with the differential to be the Milnor operation Q j , for every n and j . The motivation for this problem is that, the Margolis homology of the Dickson algebra plays a key role in study of the Morava K-theory K ( j )∗(BSm ) of the symmetric group on m letters Sm . We show that Pengelley–Sinha’s conjecture on H∗(Dn ;Q j ) for n ≤ j is true if and only if n = 1 or 2. For 3 ≤ n ≤ j , our result proves that this conjecture turns out to be false since the occurrence of some “critical elements” hs1 ,...,sk ’s of degree (2 j+1 −2n )+∑ki=1(2n −2si ) in this homology for 0 < s1 < ·· · < sk < n and k > 1. Résumé. Dans cette note on calcule entièrement l’homologie de Margolis modulo 2 de l’algèbre de Dickson Dn , i.e. l’homologie de Dn en choisissant pour différentielles les opérations de Milnor Q j , pour tous n et j . La motivation pour cette étude est le rôle clé joué par cette homologie dans l’étude de la K-théorie de Morava K ( j )∗(BSm ) du groupe symétrique Sm en m lettres. Nous montrons que la conjecture de Pengelley–Sinha sur H∗(Dn ;Q j ) pour n ≤ j est vraie si et seulement si n = 1,2. Pour 3 ≤ n ≤ j notre résultat montre que la conjecture est fausse à cause de l’occurence d’éléments « critiques » hs1 ,...,sk de degré (2 j+1 − 2n )+∑ki=1(2n − 2si ) dans cette homologie pour 0 < s1 < ·· · < sk < n et k > 1. Mathematical subject classification (2010). 55S05, 55S10, 55N99. Funding. This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2019.300. Manuscript received 24th February 2020, revised 2nd May 2020, accepted 4th May 2020. Let A be the mod 2 Steenrod algebra, genenated by the cohomology operations Sq j with j ≥ 0 and subject to the Adem relation with Sq0 = 1. Further A is a Hopf algebra, whose coproduct is given by the formula ∆(Sq j ) =∑ j i=0 Sq i ⊗Sq j−i . Let A∗ be the Hopf algebra, which is dual to A . Let ξ j = (Sq2 j · · ·Sq2Sq1)∗ be the Milnor element of degree 2 j+1−1 in A∗, for j ≥ 0, where the duality is taken with respect to the admissible basis of A . According to Milnor [4], as an algebra, A∗ ∼= F2[ξ0,ξ1, . . . ,ξ j , . . . ], the polynomial algebra in infinitely many generators ξ0,ξ1, . . . ,ξ j , . . . . Let Q j , for j ≥ 0, be the Milnor operation (see [4]) of degree (2 j+1 −1) in A , which is dual to ξ j with respect to the basis of A∗ consisting of all monomials in the generators ξ0,ξ1, . . . ,ξ j , . . . . ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 506 Nguyễn H. V. Hung Remarkably, Q j is a differential, that is Qj = 0 for every j . In fact, Q0 = Sq1, Q j = [Q j−1,Sq j ], the commutator of Q j−1 and Sq2 j in the Steenrod algebra A , for j > 0. In the article, we compute the Margolis homology of the Dickson algebra Dn , i.e. the homology of Dn with the differential to be the Milnor operation Q j . The real goal that we persue is to compute the Morava K -theory K ( j )∗(BSm) of the symmetric group Sm on m letters. It was well known that, the Milnor operation is the first non-zero differential, Q j = d2 j+1−1, in the Atiyah–Hirzebruch spectral sequence for computing K ( j )∗(X ), the Morava K -theory of a space X . So, the Q j -homology of H∗(X ) is the E2 j+1 -page in the Atiyah– Hirzebruch spectral sequence for K ( j )∗(X ). (See e.g. Yagita [10, §2], although the fact was well known before this article.) A key step in the determination of the symmetric group’s cohomology is to apply the Quillen restiction from this cohomology to the cohomologies of all elementary abelian subgroups of the symmetric group. For m = 2n and the “generic” elementary abelian 2-subgroup (Z/2)n of the symmetric group S2n , the image of the restriction H∗(BS2n ) → H∗(B(Z/2)n) is exactly the Dickson algebra Dn (see Mùi [5, Thm. II.6.2]). So, the E2 j+1 -page in the Atiyah–Hirzebruch spectral sequence for K ( j )∗(BS2n ) maps to the Margolis homology H∗(Dn ;Q j ). This is why the Margolis homology of the Dickson algebra is taken into account. Let us study the range n Dickson algebra of invariants Dn = F2[x1, . . . , xn]2, where each generator xi is of degree 1, and the general linear group GL(n,F2) acts canonically on F2[x1, . . . , xn]. Following Dickson [1], let us consider the determinant [e1, . . . ,en] = det  x2 e1 1 . . . x 2e1 n .. . . . .. x2 en 1 . . . x 2en n  for non-negative integers e1, . . . ,en . Then ω[e1, . . . ,en] = det(ω)[e1, . . . ,en], for ω ∈ GL(n,F2) (see [1]). Set Ln,s = [0,1, . . . , ŝ, . . . ,n], (0 ≤ s ≤ n), where ŝ means s being omitted, and Ln = Ln,n . The Dickson invariant cn,s of degree 2n − 2s is originally defined as follows: cn,s = Ln,s /Ln , (0 ≤ s < n). Dickson proved in [1] that Dn is a polynomial algebra on the Dickson invariants Dn = F2[cn,0, . . . ,cn,n−1]. To be explicit, the Dickson invariant can be expressed as in Hưng–Peterson [3, §2]: cn,s = ∑ i1+···+in=2n−2s x1 1 · · ·xn n , (0 ≤ s < n). where the sum is over all sequences i1, . . . , in with ik either 0 or a power of 2. We are interested in the following element of the Dickson algebra Dn : A j ,n,s = [0, . . . , ŝ, . . . ,n −1, j ]/Ln , for 0 ≤ s < n ≤ j . By convention, A j ,n,−1 = 0. In this article, when j and n are fixed, the elements cn,s and A j ,n,s will respectively be denoted by cs and As for abbreviation. C. R. Mathématique, 2020, 358, n 4, 505-510 Nguyễn H. V. Hung 507 Lemma 1. For 0 ≤ j , 0 ≤ s < n, Q j (cs ) =  c0, 0 ≤ j < n −1, j = s −1, 0, 0 ≤ j < n −1, j 6= s −1, c0cs , j = n −1, c0 ( cs An−1 + As−1 ) , 0 ≤ s < n ≤ j . The action of the Steenrod algebra on the Dickson one is basically computed in [2]. Related and partial results concerning the lemma can be seen in [7–9]. The next two theorems are stated in Sinha [6]. Their proofs are straightforward from Lemma 1. Theorem 2. For 0 ≤ j < n −1, H∗(Dn ,Q j ) ∼= F2[cj+1]⊗F2[c1, . . . , ĉ j+1, . . . ,cn−1], where ĉ j+1 means c j+1 being omitted. Let F2[c1, . . . ,cn−1]ev be the F2-submodule of F2[c1, . . . ,cn−1] generated by all the monomials c i1 1 · · ·c in−1 n−1 with i1 +·· ·+ in−1 even. Theorem 3. H∗(Dn ;Qn−1) ∼= F2[c1, . . . ,cn−1]ev. Proposition 4. For 0 ≤ s1, . . . , sk < n ≤ j , Q j (cs1 · · ·csk ) = c0 ( kcs1 · · ·csk An−1 + k ∑ i=1 cs1 . . . ĉsi . . .csk A 2 si−1 )