The mod 2 Margolis homology of the Dickson algebra

Pub Date : 2020-07-28 DOI:10.5802/crmath.68

Nguyễn H. V. Hưng

{"title":"The mod 2 Margolis homology of the Dickson algebra","authors":"Nguyễn H. V. Hưng","doi":"10.5802/crmath.68","DOIUrl":null,"url":null,"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 )","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.5802/crmath.68","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 1

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 )

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Dickson代数的mod2 Margolis同调

我们完全计算了Dickson代数Dn的mod2 Margolis同调，即对于每一个n和j, Dn与微分为米尔诺运算Q j的同调。这个问题的动机是Dickson代数的Margolis同调在研究m个字母Sm上的对称群的Morava K理论K (j)∗(BSm)中起着关键的作用。证明了当且仅当n = 1或2时，关于n≤j时H * (Dn;Q j)的Pengelley-Sinha猜想为真。对于3≤n≤j，我们的结果证明，由于某些“关键元素”hs1，…的出现，这个猜想是错误的。，当0 < s1 <···< sk < n且k > 1时，sk的阶数(2j +1−2n)+∑ki=1(2n−2si)。的简历。Dans对计算单位l 'homologie de Margolis modulo 2 de l ' alg<s:1> de Dickson Dn，即l 'homologie de Dn en choisissant pour diffrentielles les opde Milnor Q j, pour tous n et j作了注释。La motivation pour cete samsamte est le rôle clclous joujoue par cetehomologie dans l ' samsamede La K- thsamriede Morava K (j) * (BSm) du group symsamtrique Sm en m字母。pengelley猜想的Nous montrons - sinha sur H∗(Dn;Q j) pour n≤j . est vraie si et沉降si n = 1,2。Pour 3≤n≤j notre - resamusult montre que la conjecture est fausse comcause de l ' incident d ' samusults«critique»hs1，…，sk de degr分数(2j +1−2n)+∑ki=1(2n−2si) =1，表明同源性为0 < s1 <···< sk < n et k > 1。数学学科分类(2010)。55s05, 55s10, 55n99。资金。本研究由越南国家科学技术发展基金会(NAFOSTED)资助，批准号为101.04-2019.300。2020年2月24日收稿，2020年5月2日改稿，2020年5月4日收稿。设A为模2 Steenrod代数，由j≥0时的上同调运算Sq j生成，且服从当Sq0 = 1时的Adem关系。更进一步，A是一个Hopf代数，其协积由公式∆(Sq j) =∑j i=0 Sq i⊗Sq j−i给出。设A *是A的对偶的Hopf代数。设ξ j = (Sq2 j···Sq2Sq1)∗为A *中阶为2j +1−1的Milnor元素，当j≥0时，其中对A的可容许基取对偶性。根据Milnor[4]，作为代数，A∗~ = F2[ξ0，ξ1，…]，ξ j，…，无穷多个发生器中的多项式代数ξ0，ξ1，…ξ j . . . .设Q j，当j≥0时，是A中阶(2j +1−1)的密尔诺运算(见[4])，它对由发生器ξ0，ξ1，…中所有单项式组成的A *的基对偶。ξ j . . . .值得注意的是，Qj是一个微分，即对于每一个j, Qj = 0。事实上，在Steenrod代数A中，对于j > 0, Q0 = Sq1, qj = [Q j−1,Sq j]， qj−1和sq2j的对易子。在本文中，我们计算了Dickson代数Dn的Margolis同调，即Dn与微分为Milnor运算qj的同调。我们追求的真正目标是计算m个字母上对称群Sm的Morava K -理论K (j)∗(BSm)。众所周知，在空间X的Morava K理论中，用于计算K (j)∗(X)的Atiyah-Hirzebruch谱序列中，Milnor运算是第一个非零微分Q j = d2 j+1−1。因此，H * (X)的Q j -同源性是K (j) * (X)的Atiyah - Hirzebruch谱序列中的E2 j+1 -页。(参见Yagita[10，§2]，尽管在这篇文章之前这个事实是众所周知的。)确定对称群的上同调的关键步骤是将此上同调中的Quillen限制应用于对称群的所有初等阿贝尔子群的上同调。对于m = 2n和对称群S2n的“一般”初等阿贝尔2-子群(Z/2)n，限制H∗(BS2n)→H∗(B(Z/2)n)的像正是Dickson代数Dn(见Mùi [5, Thm])。II.6.2])。因此，K (j)∗(BS2n)的Atiyah-Hirzebruch谱序列中的E2 j+1 -页映射到Margolis同源H∗(Dn;Q j)。这就是为什么要考虑Dickson代数的Margolis同调。让我们研究不变量Dn = F2[x1，…]的值域n Dickson代数。， xn]2，其中每个发生器xi的阶为1，一般线性群GL(n,F2)正则作用于F2[x1，…]。, xn]。根据Dickson[1]，让我们考虑行列式[e1，…]。， zh] = detx2 e1 1…x2e1 n .. .. . ..X2 en 1…x2en n对于非负整数e1，…,在。然后ω[e1，…]，en] = det(ω)[e1，…]，en]，对于ω∈GL(n,F2)(见[1])。设Ln,s =[0,1，…]，我…，n]，(0≤s≤n)，其中，i表示省略s, Ln = Ln,n。次为2n−2s的Dickson不变量cn,s最初定义为:cn,s = Ln,s /Ln，(0≤s < n). Dickson在[1]中证明了Dn是Dickson不变量Dn = F2上的多项式代数[cn,0，…]。cn, n−1]。明确地说，Dickson不变量可以表示为Hưng-Peterson[3，§2]:cn,s =∑i1+···+in=2n - 2s x1····xn n，(0≤s < n)，其中对所有序列i1，…， k可以是0或2的幂。我们对Dickson代数Dn中的下列元素感兴趣:A j,n,s =[0，…]，我…，n−1,j]/Ln，对于0≤s < n≤j。按照惯例，A j,n，−1 = 0。在本文中，当j和n固定时，元素cn,s和元素A j,n,s分别用cs和As表示，简称为c。洪洪辉。数学学报，2020,358,n4,505 -510 Nguyễn对于0≤j, 0≤s < n, Q j (cs) =c0, 0≤j < n−1,j = s−1,0,0≤j < n−1,j6 = s−1,c0, j = n−1,c0 (cs An−1 + As−1)，0≤s < n≤j。Steenrod代数对Dickson代数的作用基本在[2]中计算。关于引理的相关和部分结果可以在[7-9]中看到。接下来的两个定理在Sinha[6]中得到了说明。根据引理1，他们的证明很简单。定理2。对于0≤j < n−1 H∗(Dn,问j)∼= F2 (cj + 1)⊗F2 (c1,。， j+1，…，cn−1]，其中，j+1表示省略j+1。设F2[c1，…]，cn−1]可能是F2[c1，…]的F2子模块。，cn−1]由所有单项式c i1 1···c in−1 n−1与i1 +··+ in−1偶生成。定理3。H * (Dn;Qn−1)~ = F2[c1，…]。cn−1)电动汽车。命题4。对于0≤s1，…， sk < n≤j, Q j (cs1···csk) = c0 (kcs1···csk) An−1 + k∑i=1 cs1…ĉsi . . .csk A 2 si−1)

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