The mod p Margolis homology of the Dickson–Mùi algebra

Abstract

Let En = (Z/p)n be regarded as the translation group on itself. It is considered as a subgroup of the symmetric group Spn on p n letters. We completely compute the mod p Margolis homology of the Dickson– Mùi algebra, i.e. the homology of the image of the restriction Res(Spn ,E n ) : H∗(Spn ;Fp ) → H∗(En ;Fp ) with the differential to be the Milnor operation Q j , for p an odd prime and for any n, j . The motivation for this problem is that, the Margolis homology of the Dickson–Mùi algebra plays a key role in study of the Morava K-theory K ( j )∗(BSm ) of the symmetric group Sm on m letters. The main tool of our work is the notion of “critical” elements. The mod p Margolis homology of the Dickson–Mùi algebra concentrates on even degrees. It is analogous to the mod 2 Margolis homology of the Dickson algebra. Résumé. Soit En = (Z/p)n le groupe agissant sur lui même par les translations. On le considère comme sousgroupe du groupe symétrique Spn en p n lettres. Dans cette note on calcule entièrement l’homologie de Margolis modulo p de l’algèbre de Dickson–Mùi, i.e. l’homologie de l’image de la restriction Res(Spn ,E n ) : H∗(Spn ;Fp ) → H∗(En ;Fp ) en choisissant pour différentielles les opérations de Milnor Q j , pour p un nombre premier impair et pour tout n, 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. L’outil principal de notre travail est la notion d’éléments « critiques ». L’homologie de Margolis mod p de l’algèbre de Dickson– Mùi concentre en degrés pairs. Elle est analogue à l’homologie de Margolis mod 2 de l’algèbre de Dickson. 2020 Mathematics Subject Classification. 55S05, 55S10, 55N99. Manuscript received 22nd July 2020, revised 5th October 2020 and 16th November 2020, accepted 15th November 2020. A key step toward the determination of the symmetric group’s cohomology is to apply the Quillen restriction from this cohomology to the cohomologies of all maximal elementary abelian subgroups of the symmetric group. Let E n = (Z/p)n be regarded as the translation group on itself. So it is considered as a subgroup of the symmetric group Spn on pn letters. This is the “generic” maximal elementary abelian p-subgroup of the symmetric group Spn , where the terminology “generic” means that the set {E n |n ≥ 1} has been used to describe all maximal elementary abelian p-subgroups of ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 230 Nguyễn H. V. Hung any symmetric groups. (See Mùi [7, Prop. II.2.3].) Let us study the restriction Res(Spn ,E n) : H∗(Spn , ;Fp ) → H∗(E n ;Fp ) induced in cohomology by the canonical inclusion E n ⊂Spn . We have H∗(E n ;Fp ) = { Fp [y1, . . . , yn], p = 2, E(x1, . . . , xn)⊗Fp [y1, . . . , yn], p > 2, where deg(yi ) = 1 for p = 2, and deg(xi ) = 1,deg(yi ) = 2 for p an odd prime (1 ≤ i ≤ n). Here E(x1, . . . , xn) and Fp [y1, . . . , yn] denote respectively the exterior algebra and the polynomial algebra on the given generators. The Weyl group, which is the quotient of the normalizer by the centralizer, of the maximal elementary abelian subgroup E n in Spn is the general linear group GLn = GL(n,Fp ). It is wellknown that the image of the restriction Res(Spn ,E n) is a subalgebra of the invariant algebra under the Weyl group action H∗(E n ;Fp )GLn . According to H. Mùi [7, Thm. II.6.1 and Thm. II.6.2], the image of the restriction Res(Spn ,E n) is the Dickson algebra Dn = Fp [y1, . . . , yn]n for p = 2, and a subalgebra of the algebra (E(x1, . . . , xn)⊗Fp [y1, . . . , yn])n for p an odd prime, where GLn acts canonically on Fp [y1, . . . , yn] and on E(x1, . . . , xn)⊗Fp [y1, . . . , yn]. For p an odd prime, let us denote DMn := ImRes(Spn ,E n) and call it the n-th Dickson–Mùi algebra. It should be noted that DMn 6= (E(x1, . . . , xn)⊗ Fp [y1, . . . , yn])n (see H. Mùi [7, I.4.17 & II.6.1] or Theorem 1 below). Let Q j be the Milnor operation (see [6]) of degree 2p j − 1 in the mod p Steenrod algebra A inductively defined for j ≥ 0 as follows Q0 =β, Q j+1 = P p j Q j −Q j P p j , where β denotes the Bockstein operation. In the article, for p an odd prime, we completely compute the mod p Margolis homology of the Dickson–Mùi algebra DMn , i.e. the homology of DMn with the differential to be the Milnor operation Q j , for every n and j . The solution for the similar problem on the mod 2 Margolis homology of the Dickson algebra has been announced in [4] and published in detail in [2], where we denied the Pengelley–Sinha conjecture on the problem. This conjecture turns out to be false because of the occurence of the so-called critical elements, which are our main creation in the study. The Dickson–Mùi algebra DMn is not free in the category of graded commutative algebras. Therefore, its Margolis homology is completly different and requires new techniques, more care and details than the case of p = 2. In particular, Definition 10 of critical elements is distinguished from the one for p = 2. The real goal that we persue in the near future 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 = d2p j −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 E2p j -page in the Atiyah–Hirzebruch spectral sequence for K ( j )∗(X ). (See e.g. Yagita [9, §2], although the fact was well known before this article.) Particularly, the E2p j -page in the Atiyah–Hirzebruch spectral sequence for K ( j )∗(BSpn ) maps to H∗(DMn ;Q j ). This is why the Margolis homology of the Dickson–Mùi algebra is taken into account. Let us study the n-th Dickson algebra of invariants Dn = Fp [y1, . . . , yn]n . Following Dickson [1], we set [e1, . . . ,en] = det ( y p ek ` ) 1≤k,`≤n , for non-negative integers e1, . . . ,en . The right action ofω= (ωi j )n×n ∈GLn sends g ∈ Fp [y1, . . . , yn] to (gω)(y1, . . . , yn) = g (∑ni=1ωi 1 yi , . . . ,∑ni=1ωi n yi ), while its left action sends g to (ωg )(y1, . . . , yn) = g ( ∑n j=1ω1 j y j , . . . , ∑n j=1ωn j y j ). Since ωg = gωt , where ωt is the transposed matrix of ω, a polynomial is a right GLn-invariant if and only if it is a left GLn-invariant. By Fermat’s little theorem, [e1, . . . ,en]ω= det(ω)[e1, . . . ,en] forω ∈GLn (see [1]). Set Ln,s = [0,1, . . . , ŝ, . . . ,n] (0 ≤ s ≤ n), where ŝ C. R. Mathématique — 2021, 359, n\uf6f0 3, 229-236 Nguyễn H. V. Hung 231 means s being omitted, and Ln =Ln,n . The Dickson invariant, defined by cn,s =Ln,s /Ln (0≤ s <n), is a GLn-invariant. It is of degree 2n − 2s for p = 2 and degree 2(pn − p s ) for p an odd prime. Dickson proved in [1] that Dn is a polynomial algebra on the Dickson invariants Dn = Fp [cn,0, . . . ,cn,n−1]. Let A = (ai j )n×n be an n × n matrix with entries ai j ’s in the graded commutative algebra E(x1, . . . , xn)⊗Fp [y1, . . . , yn]. The determinant of A is defined by det A = ∑ σ∈Sn sgn(σ)a1σ(1) · · ·anσ(n). Remark. As x1, . . . , xn are of odd degree, xk x` =−x`xk for any k and `, we have det \uf8ec\uf8edx1 . . . xn .. . . . .. x1 . . . xn \uf8f7\uf8f8= n!x1 · · ·xn ,