Broto, Carlos; Zarati, S. Nil-localization of unstable algebras over the Steenrod algebra. (English) Zbl 0639.55012 Math. Z. 199, No. 4, 525-537 (1988). Let p be an odd prime, \(A\) \(*_ p\) the mod p Steenrod algebra, K the category of unstable \(A\) \(*_ p\)-algebras and K’ the full subcategory of K of those algebras which are concentrated in even degrees. We say that an unstable \(A\) \(*_ p\)-algebra is Nil-closed if the underlying \(A\) \(*_ p\)-module is Nil-closed in the sense of J. Lannes and the second author [Ann. Sci. Éc. Norm. Sup. IV, Sér. 19, 303-333 (1986; Zbl 0608.18006), 6.2.1]. In the first part of the note we introduce the concept of Nil- localization of unstable \(A\) \(*_ p\)-algebras and we prove: Theorem. For any unstable \(A\) \(*_ p\)-algebra K, there exists a unique (up to isomorphism) nil-closed unstable \(A\) \(*_ p\)-algebra, denoted \(N_ K^{-1}(K)\), and a natural map of \(A\) \(*_ p\)-algebras \(\mu _ K: K\to N_ K^{-1}(K)\) which is initial among maps of unstable \(A\) \(*_ p\)- algebras from K to nil-closed unstable \(A\) \(*_ p\)-algebras. In the second part we show that some results of Adams-Wilkerson and of Rector, which occur in K’ can be extended to K. This allows us to determine the Nil-localization of certain unstable \(A\) \(*_ p\)-algebras. Reviewer: C.Broto Cited in 9 Documents MSC: 55S10 Steenrod algebra 55P60 Localization and completion in homotopy theory Keywords:mod p Steenrod algebra; category of unstable algebras over the Steenrod algebra; Nil-closed; Nil-localization of unstable algebras Citations:Zbl 0608.18006 PDFBibTeX XMLCite \textit{C. Broto} and \textit{S. Zarati}, Math. Z. 199, No. 4, 525--537 (1988; Zbl 0639.55012) Full Text: DOI EuDML References: [1] Adams, J.F., Wilkerson, C.W.: Finite H-spaces and algebras over the Steenrod algebra. Ann. Math.111, 95-143 (1980) · Zbl 0417.55018 · doi:10.2307/1971218 [2] Bousfield, A.K., Kan, D.M.: The homotopy spectral sequence of a space with coefficients in a ring. Topology11, 79-106 (1972) · Zbl 0219.55015 · doi:10.1016/0040-9383(72)90024-9 [3] Goerss, P., Smith, L., Zarati, S.: Sur lesA-algèbres instables. To appear in the proceedings of the Barcelona C.A.T. (L.N.M.) · Zbl 0642.55014 [4] Lannes, J., Schwartz, L.: Sur la structure desA-modules instables injectifs. To appear in the proceedings of the Barcelona C.A.T. (L.N.M.) · Zbl 0683.55016 [5] Lannes, J., Zarati, S.: Sur lesU-injectifs. Ann. Sci. Ec. Norm. Super. 4è série19, 303-333 (1986) · Zbl 0608.18006 [6] Li, W.H.: Iterated loop functors and the homology of the Steenrod algebraA(p). Thesis, Fordham Univ., New York, 1980 [7] Miller, H.: The Sullivan conjecture on maps from classifying spaces. Ann. Math.120, 39-87 (1984) · Zbl 0552.55014 · doi:10.2307/2007071 [8] Quillen, D.: The spectrum of an equivariant cohomology ring: I. Ann. Math.94, 549-572 (1971) · Zbl 0247.57013 · doi:10.2307/1970770 [9] Rector, D.L.: Noetherian cohomology rings and finite loop spaces with torsion. J. Pure Appl. Algebra32, 191-217 (1984) · Zbl 0543.57030 · doi:10.1016/0022-4049(84)90051-3 [10] Zarati, S.: Dérivés du foncteur dé destabilization en caractéristique impaire et applications. Thèse de Doctorat d’État, Orsay 1984 [11] Zarati, S.: Quelques propriétés du foncteur HomU p (,H * V). Algebraic Topology, Göttingen 1984, L.N.M. 1172, 204-209 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.