Division of the Dickson algebra by the Steinberg unstable module. (English. French summary) Zbl 1278.55029
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 \(M_{n}\).
In 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.
In 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.
Reviewer: Saïd Zarati (Tunis)
MSC:
| 55S10 | Steenrod algebra |
| 18G15 | Ext and Tor, generalizations, Künneth formula (category-theoretic aspects) |
| 55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |
| 55T15 | Adams spectral sequences |
References:
| [1] | Dickson, Leonard Eugene, A fundamental system of invariants of the general modular linear group with a solution of the form problem, Trans. Amer. Math. Soc., 12, 1, 75-98 (1911) · JFM 42.0136.01 |
| [2] | Hai, Nguyen Dang Ho; Schwartz, Lionel; Nam, Tran Ngoc, La fonction génératrice de Minc et une « conjecture de Segal » pour certains spectres de Thom, Adv. Math., 225, 3, 1431-1460 (2010) · Zbl 1200.55026 |
| [3] | Kuhn, Nicholas J., Chevalley group theory and the transfer in the homology of symmetric groups, Topology, 24, 3, 247-264 (1985) · Zbl 0606.20044 |
| [4] | Kuhn, Nicholas J., The rigidity of \(L(n)\), (Algebraic Topology. Algebraic Topology, Seattle, Wash., 1985. Algebraic Topology. Algebraic Topology, Seattle, Wash., 1985, Lecture Notes in Math., vol. 1286 (1987), Springer: Springer Berlin), 286-292 · Zbl 0632.55004 |
| [5] | Lannes, Jean, Sur les espaces fonctionnels dont la source est le classifiant dʼun \(p\)-groupe abélien élémentaire, Inst. Hautes Études Sci. Publ. Math., 75, 135-244 (1992), With an appendix by Michel Zisman · Zbl 0857.55011 |
| [6] | Lannes, Jean; Zarati, Saïd, Sur les foncteurs dérivés de la déstabilisation, Math. Z., 194, 1, 25-59 (1987) · Zbl 0627.55014 |
| [7] | Mitchell, Stephen A.; Priddy, Stewart B., Stable splittings derived from the Steinberg module, Topology, 22, 3, 285-298 (1983) · Zbl 0526.55010 |
| [8] | Schwartz, Lionel, Unstable Modules over the Steenrod Algebra and Sullivanʼs Fixed Point Set Conjecture, Chicago Lectures in Mathematics (1994), University of Chicago Press: University of Chicago Press Chicago, IL · Zbl 0871.55001 |
| [9] | Steinberg, Robert, Prime power representations of finite linear groups, Canad. J. Math., 8, 580-591 (1956) · Zbl 0073.01502 |
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