Division of the Dickson algebra by the Steinberg unstable module (Q367177)

Let \(A\) be the mod. \(2\) Steenrod algebra, \(\mathcal{U}\) be the category of unstable \(A\)-modules, \(D_{n}=(H^{*}((\mathbb{Z}/2\mathbb{Z})^{n}; \; \mathbb{F}_{2}))^{GL_{n}(\mathbb{F}_{2})},\; n \geq 1\) be the Dickson algebra, \(M_{n}\) be the direct summand of \(H^{*}((\mathbb{Z}/2\mathbb{Z})^{n}; \; \mathbb{F}_{2})\) associated with the Steinberg idempotent of \(\mathbb{F}_{2}[GL_{n}(\mathbb{F}_{2})]\) and \(L_{n}\) be the indecomposable summand of \(M_{n}\).NEWLINENEWLINEIn this paper the author studies the functor ``Division'' \((- , L_{n})_{\mathcal{U}}: \mathcal{U} \rightsquigarrow \mathcal{U}\) adjoint to the functor \(- \otimes L_{n}: \mathcal{U} \rightsquigarrow \mathcal{U}\). He computes the unstable \(A\)-module \((D_{k}, L_{n}), n \geq 1, k \geq 1\) and, as an application, he gives, via the Adams spectral sequence in mod. \(2\) cohomology, some informations on the cohomotopy groups of a particular spectra.