Sur les \({\mathcal U}\)-injectifs. (On \({\mathcal U}\)-injectives). (French) Zbl 0608.18006
The importance of injectives in the category of unstable modules over the mod p Steenrod algebra has recently been emphasized by the work of H. Miller [Ann. Math., II. Ser. 120, 39-87 (1984; Zbl 0552.55014)] and G. Carlsson [Topology 22, 83-103 (1983; Zbl 0504.55011)] relating to the Sullivan conjecture and its solution. In this paper, a study is made of certain types of these injectives. In particular the authors give conditions that imply that if J and K are such injectives, their tensor product \(K\otimes J\) is also injective. For \(p=2\) this result implies the principal result of J. F. Adams, J. Gunawardena and H. Miller [The Segal conjecture for \(({\mathbb{Z}}/p)^ R\), to appear].
{Remark: The first author in a joint paper with L. Schwartz (cited [19] in this note) has recently obtained more detailed results on these injectives.}
{Remark: The first author in a joint paper with L. Schwartz (cited [19] in this note) has recently obtained more detailed results on these injectives.}
Reviewer: T.Porter
References:
| [1] | J. F. ADAMS , J. GUNAWARDENA and H. MILLER , The Segal conjecture for (\Bbb Z/p)k ; à paraître. · Zbl 0611.55010 |
| [2] | J. F. ADAMS and C. W. WILKERSON , Finite H-spaces and algebras over the Streenrod algebra (Ann. of Math., vol. 111, 1980 , p. 95-143). MR 558398 | Zbl 0404.55020 · Zbl 0404.55020 · doi:10.2307/1971218 |
| [3] | G. CARLSSON . G. B. , Segal Burnside ring conjecture for (\Bbb Z/2)k (Topology, vol. 22, n^\circ 1, 1983 , p. 83-103). MR 682060 | Zbl 0504.55011 · Zbl 0504.55011 · doi:10.1016/0040-9383(83)90046-0 |
| [4] | H. CARTAN and S. EILENBERG , Homological algebra ; Princeton Univ. Press, 1956 . MR 77480 | Zbl 0075.24305 · Zbl 0075.24305 |
| [5] | J. R. HARPER , H-spaces with torsion (Memoirs of the A.M.S., n^\circ 223, vol. 22, 1979 ). MR 546361 | Zbl 0421.55006 · Zbl 0421.55006 |
| [6] | J. LANNES , Sur le n-dual du nème spectre de Brown-Gitler ; preprint 1984 , à paraître. MR 954749 | Zbl 0629.55003 · Zbl 0629.55003 · doi:10.1007/BF01160207 |
| [7] | J. LANNES et S. ZARATI , Foncteurs dérivés de la déstabilisation (Note aux C. R. Acad. Sc. Paris, vol. 296, 1983 , p. 573-576). MR 705164 | Zbl 0534.55008 · Zbl 0534.55008 |
| [8] | J. LANNES et S. ZARATI , Foncteurs dérivés de la déstabilisation ; preprint 1983 , à paraître. MR 705164 | Zbl 0534.55008 · Zbl 0534.55008 |
| [9] | J. LANNES et S. ZARATI , Invariants de Hopf d’ordre supérieur et suite spectrale d’Adams (Note aux C. R. Acad. Sc. Paris, vol. 296, 1983 , p. 695-698). MR 705694 | Zbl 0534.55009 · Zbl 0534.55009 |
| [10] | J. LANNES et S. ZARATI , Invariants de Hopf d’ordre supérieur et suite spectrale d’Adams ; à paraître. · Zbl 0534.55009 |
| [11] | W. H. LI , Iterated loop functors and the homology of the Steenrod algebra A(p) ; Thesis, Fordham Univ., New York, 1980 . |
| [12] | M. MAHOWALD , A new infinite family in 2\pi S* (Topology, vol. 16, 1977 , p. 249-254). MR 445498 | Zbl 0357.55020 · Zbl 0357.55020 · doi:10.1016/0040-9383(77)90005-2 |
| [13] | W. S. MASSEY and F. P. PETERSON , The mod. 2 cohomology structure of certain fibre spaces (Memoirs of the A.M.S., n^\circ 74, 1967 ). MR 226637 | Zbl 0168.44002 · Zbl 0168.44002 |
| [14] | H. MILLER , The Sullivan conjecture on maps from classifying spaces (Ann. of Math., vol. 120, 1984 , p. 39-87). MR 750716 | Zbl 0552.55014 · Zbl 0552.55014 · doi:10.2307/2007071 |
| [15] | L. SMITH and R. W. SWITZER , Polynomial algebras over the Steenrod algebra : variations on a theorem of Adams and Wilkerson (Proc. of the Edinburg Math. Soc., vol. 27 (series 11), Part 1, 1984 , p. 11-19). MR 738590 | Zbl 0545.55009 · Zbl 0545.55009 · doi:10.1017/S0013091500022069 |
| [16] | N. E. STEENROD and D. B. A. EPSTEIN , Cohomology operations ; Princeton Univ. Press, 1962 . MR 145525 | Zbl 0102.38104 · Zbl 0102.38104 |
| [17] | S. ZARATI , Dérivés du foncteur de déstabilisation en caractéristique impaire et applications , Thèse de Doctorat d’État, Orsay, 1984 . |
| [18] | D. M. DAVIS , A family of unstable Steenrod-modules wich includes those of G. Carlsson (Jour. of Pure and Appl. Alg., vol. 35, 1985 , p. 253-267). MR 777258 | Zbl 0564.55016 · Zbl 0564.55016 · doi:10.1016/0022-4049(85)90044-1 |
| [19] | J. LANNES et Lionel SCHWARTZ , Sur la structure des A-modules instables injectifs ; à paraître. · Zbl 0683.55016 · doi:10.1016/0040-9383(89)90018-9 |
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.
