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Splitting of composite functors. (Scindements de foncteurs composés.) (French) Zbl 1077.18001
The framework is the category of functors between vector spaces over a prime field. The symmetric power \(S^n\) is the functor obtained as the quotient of the tensor power by the action of the symmetric group (under permutation of factors). A functor \(B\) is said to be \(p\)-Boolean if it takes values in an algebra in which \(x^p= x\). The notion of polynomial and analytic functors leads to the Taylor filtration of a functor. The main result shows that the Taylor filtration of \(S^n\) splits after (right) composition with a \(p\)-Boolean functor \(S^n\circ B\). As applications of such splittings are computations of extensions appearing in the study of the cohomology of Eilenberg-MacLane spaces.
18A25 Functor categories, comma categories
55P20 Eilenberg-Mac Lane spaces
Full Text: DOI
[1] Betley, S., Ext-groups for the composition of functors, (), 31-45 · Zbl 1057.18007
[2] Betley, S., Stable derived functors, the Steenrod algebra and homological algebra in the category of functors, Fund. math., 168, 279-293, (2001) · Zbl 0986.55008
[3] Breen, L., Extensions du groupe additif, Publ. inst. hautes études sci., 48, 39-125, (1978) · Zbl 0404.14018
[4] Carlisle, D.; Kuhn, N., Subalgebras of the Steenrod algebra and the action of matrices on truncated polynomial algebras, J. algebra, 121, 370-387, (1989) · Zbl 0691.55015
[5] Cartan, H.; Eilenberg, S., Homological algebra, (1956), Princeton Univ. Press · Zbl 0075.24305
[6] Franjou, V., Extensions entre puissances extérieures et entre puissances symétriques, J. algebra, 179, 2, 501-522, (1996) · Zbl 0841.55012
[7] Franjou, V.; Friedlander, E.; Scorichenko, A.; Suslin, A., General linear and functor cohomology over finite fields, Ann. of math., 150, 2, 663-728, (1999) · Zbl 0952.20035
[8] Franjou, V.; Lannes, J.; Schwartz, L., Autour de la cohomologie de mac Lane des corps finis, Invent. math., 115, 513-538, (1994) · Zbl 0798.18009
[9] Friedlander, E.; Suslin, A., Cohomology of finite group schemes over a field, Invent. math., 127, 2, 209-270, (1997) · Zbl 0945.14028
[10] Henn, H.-W.; Lannes, J.; Schwartz, L., Analytic functors, unstable algebras and cohomology of classifying spaces, (), 197-220
[11] Latapy, M., Partitions of an integer into powers, (), 215-228 · Zbl 1002.11074
[12] Pirashvili, T., Higher additivizations, Trudy tbiliss. mat. inst. razmadze akad. nauk gruzin. SSR, 91, 44-54, (1988), (in Russian) · Zbl 0705.18008
[13] Pirashvili, T., Dold – kan type theorem for γ-groups, Math. ann., 318, 277-298, (2000) · Zbl 0963.18006
[14] Piriou, L.; Schwartz, L., Extensions de foncteurs simples, K-theory, 15, 3, 269-291, (1998) · Zbl 0918.20036
[15] Richter, B., Taylor towers for γ-modules, Ann. inst. Fourier, 51, 4, 995-1023, (2001) · Zbl 0997.18008
[16] Schwartz, L., Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture, (1994), Univ. of Chicago Press · Zbl 0871.55001
[17] Troesch, A., Cohomologie de compositions de puissances symétriques, C. R. acad. sci. Paris, 333, 509-512, (2001) · Zbl 0990.18009
[18] Troesch, A., Quelques calculs de cohomologie de compositions de puissances symétriques, Comm. algebra, 30, 7, 3351-3382, (2002) · Zbl 1005.18010
[19] A. Troesch, Cohomologie de compositions de puissances symétriques en caractéristique 2, Prépublication du LAGA (Paris 13), no. 2002-12
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