Generators for the mod-\(p\) cohomology of the Steinberg summand of Thom spectra over \(\mathrm{B}(\mathbb{Z}/p)^n\)-odd primary cases
In this paper, the author studies the important family \[ L_{n,k} := \mathrm{st}_n (\mathfrak{e}^k_n H^*V_n) \] of modules over the mod \(p\) Steenrod algebra \(\mathcal{A}\), at odd primes, for \(k,n \in \mathbb{N}\). Here \(H^*V_n\) is the mod \(p\) cohomology of the rank \(n\) elementary abelian \(p\)-group \(V_n\), \(\mathfrak{e}_n\) is an explicit `Euler class', and \(\mathrm{st}_n\) is the Steinberg idempotent for \(GL_n (\mathbb{F}_p)\). In particular, he determines the indecomposables \(Q (L_{n,k}):= \mathbb{F}_p \otimes _{\mathcal{A}} L_{n,k}\) for all \(n,k\), thus extending work of Inoue [\textit{M. Inoue}, J. Lond. Math. Soc., II. Ser. 75, No. 2, 317--329 (2007; Zbl 1122.55012)] and mirroring his results for \(p=2\) [\textit{N. D. H. Hai}, J. Algebra 381, 164--175 (2013; Zbl 1279.55010)]. This follows inductively from the following decomposition results, in which \(\Phi\) is the Frobenius functor and \(\Sigma\) the suspension functor for \(\mathcal{A}\)-modules. If \(m= m'p\), with \(m' \geq 1\), then \[ Q (\Sigma ^n L_{n, 2m-1} ) \cong \Phi Q (\Sigma^n L_{n,2m' -1 }). \] If \(m = m'p+r\), with \(m' \geq 0\) and \(1 \leq r \leq p-1\), then there is a short exact sequence \[ 0 \rightarrow \Phi Q (\Sigma^n L_{n, 2m'+1} ) \rightarrow Q (\Sigma ^n L_{n,2m-1}) \rightarrow Q (\Sigma^{2m(p-1)+n-1} L_{n-1,2mp-1}) \rightarrow 0. \] For all \(m \geq 0\), there is a short exact sequence \[ 0 \rightarrow Q (L_{n,2m+1}) \rightarrow Q(L_{n,2m}) \rightarrow Q (\Sigma^{2m (p-1)} L_{n-1, 2mp+1} ) \rightarrow 0. \] The proofs use generalizations of fundamental tools from the case \(p=2\) to odd primes, provided here by the author. Namely, he gives odd primary versions of Takayasu's exact sequences in \(\mathcal{A}\)-modules [ \textit{S.-i. Takayasu}, J. Math. Kyoto Univ. 39, No. 2, 377--398 (1999; Zbl 1002.55006)]: for \(m \geq 0\): \[ 0 \rightarrow L_{n,2m+1} \rightarrow L_{n,2m} \rightarrow \Sigma^{2m (p-1)} L_{n-1, 2mp+1} \rightarrow 0; \] and, for \(m \geq 1\): \[ 0 \rightarrow L_{n,2m} \rightarrow L_{n,2m-1} \rightarrow \Sigma^{2m (p-1) -1} L_{n-1 , 2mp-1} \rightarrow 0. \] He also constructs an odd primary version of the Kameko homomorphism [\textit{M. Kameko}, Products of projective spaces as Steenrod modules. (PhD thesis) (1990)]: \[ \psi_n : \Sigma^n H^* V_n \rightarrow \Phi \Sigma^n H^* V_n. \] These results are of independent interest.
scientific article; zbMATH DE number 7438394
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