The mod \(p\) Margolis homology of the Dickson-Mùi algebra (Q2019407)

For \(p\) a prime, the \(n\)th Dickson-Mùi algebra \(DM_n\) is defined as the image of the restriction \[ H^* (B \mathfrak{S}_{p^n} ; \mathbb{F}_p) \rightarrow H^* (BE_n; \mathbb{F}_p) \] from the cohomology of the symmetric group \(\mathfrak{S}_{p^n}\) to the cohomology of the elementary abelian \(p\)-subgroup \(E_n\). For \(p\) odd and \(n>1\), this is a proper subalgebra of the invariants \(\big( \Lambda (x_1, \ldots ,x _n) \otimes \mathbb{F}_p [y_1, \ldots , y_n]\big)^{GL_n}\). The algebra \(DM_n\) contains the Dickson algebra \( D_n:= \mathbb{F}_p [y_1, \ldots , y_n]^{GL_n}\) and is stable under the action of the Steenrod algebra. The purpose of this paper is to outline at odd primes how to determine the Margolis homology groups \(H_*(DM_n; Q_j)\) for the Milnor operations \(Q_j\), \(j \geq 0\). This is more involved than the case \(p=2\), which was treated by the author in [\textit{N. H. V. Hung}, C. R., Math., Acad. Sci. Paris 358, No. 4, 505--510 (2020; Zbl 1448.55019)]. Since \(Q_j\) is a derivation, \(H_*(DM_n; Q_j)\) is a graded commutative algebra; it acts trivially on \(D_n\), hence \(H_*(DM_n; Q_j)\) is a \(D_n\)-module. The author distinguishes the following cases: \(n=1\) (which can be treated directly); the unstable cases \(1<n\) and \(0 \leq j <n\), for which he gives a presentation of \(H_*(DM_n; Q_j)\) as an algebra; the stable cases \(1 <n \) and \(j \geq n\). The main results of the paper concern the stable case. In particular, the author gives explicit generators for \(H_*(DM_n; Q_j)\) as a \(D_n\)-module, which he terms \textit{critical elements}. Since the Margolis homology is \(c_0\)-torsion, where \(c_0 \in D_n\) denotes the top Dickson invariant, these critical elements are constructed by considering elements of the image of \(Q_j\) and dividing by \(c_0\) as far as possible.