Towards the MacLane cohomology of finite fields. (Autour de la cohomologie de MacLane des corps finis.) (French) Zbl 0798.18009

This paper describes a new method to compute the MacLane cohomology of finite fields.
The present theory is closely related to L. Breen’s “extensions du groupe additif” [Inst. Haut. Étud. Sci., Publ. Math. 48, 39-125 (1978; Zbl 0404.14018)] and M. Bökstedt’s topological Hochschild homology [work to appear; see also T. Pirashvili and F. Waldhausen, MacLane homology and topological Hochschild homology, J. Pure Appl. Algebra 82, No. 1, 81-98 (1992; Zbl 0767.55010)], and hence to stable \(K\)-theory.
The main tool is a cancellation theorem for MacLane cohomology of the field \(\mathbb{F}_ p\) with coefficients in the symmetric algebra where the Frobenius endomorphism is inverted. Then comes the analysis of the Koszul and the De Rham complexes in non-zero characteristic.


18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
55N15 Topological \(K\)-theory
Full Text: DOI EuDML


[1] [A-M] Artin, M., Milne, J.S.: Duality in the flat cohomology of curves. Invent. Math.35, 111-129 (1976) · Zbl 0342.14007 · doi:10.1007/BF01390135
[2] [Bö 1] Bökstedt, M.: Topological Hochschild homology. (à paraître)
[3] [Bö 2] Bökstedt, M.: The topological Hochschild homology of? and?/p. (à paraître) · Zbl 0989.19003
[4] [B 1] Breen, L.: Extensions du groupe additif. Publ. Sci., Inst. Hautes Etud. Sci.48, 39-125 (1978) · Zbl 0404.14018 · doi:10.1007/BF02684313
[5] [B 2] Breen, L.: Extensions du groupe additif sur le site parfait. In: Giraud, J., Illusie, L., Raynaud, M. (éds.), Surfaces algébriques. (Lect. Notes Math., vol. 868, pp. 238-262) Berlin Heidelberg New-York: Springer 1981
[6] [Ct] Cartier, P.: Une nouvelle opération sur les formes différentielles. C.R. Acad. Sci. Paris244, 426-428 (1957) · Zbl 0077.04502
[7] [C-S] Campbell, H.E.A., Selick, P.S.: Polynomial algebras over the Steenrod algebra. Comment. Math. Helv.65, 171-180 (1990) · Zbl 0716.55016 · doi:10.1007/BF02566601
[8] [Cs] Carlsson, G.: G.B. Segal’s Burnside ring conjecture for 537-1. Topology22-1, 83-103 (1983) · Zbl 0504.55011 · doi:10.1016/0040-9383(83)90046-0
[9] [C-E] Cartan, H., Eilenberg, S.: Homological Algebra. Princeton: Princeton University Press 1956
[10] [D-P] Dold, A., Puppe, D.: Homologie nicht-additiver Funktoren. Anwendungen. Ann. Inst. Fourier11, 201-312 (1961) · Zbl 0098.36005
[11] [E-ML 1] Eilenberg, S., MacLane, S.: Cohomology theory of abelian groups and homotopy theory I, II. Proc. Natl. Acad. Sci. USA36, 443-447, 657-663 (1950) · Zbl 0037.39504 · doi:10.1073/pnas.36.8.443
[12] [E-ML 2] Eilenberg, S., MacLane, S.: On the groupsH(?,n) I, II. Ann. Math.58, 55-106 (1953); Ann. Math.60, 49-139 (1954) · Zbl 0050.39304 · doi:10.2307/1969820
[13] [H-L-S] Henn, H.-W., Lannes, J., Schwartz, L.: The categories of unstable modules and unstable algebras modulo nilpotent objects. Am. J. Math. (à paraître 1993) · Zbl 0805.55011
[14] [J-P] Jibladze, M., Pirashvili, T.: Cohomology of algebraic theories. J. Algebra137, 253-296 (1991) · Zbl 0724.18005 · doi:10.1016/0021-8693(91)90093-N
[15] [K] Kuhn, N.J.: Generic representations of the finite general linear groups and the Steenrod algebra, I. Am. J. Math. (à paraître); II (à paraître); III. Prépublications, Charlottesville (1990, 1992, 1993)
[16] [L-S] Lannes, L., Schwartz, L.: Sur la structure desA-modules instables injectifs. Topology28-2, 153-169 (1989) · Zbl 0683.55016 · doi:10.1016/0040-9383(89)90018-9
[17] [Md] Macdonald, I.G.: Symmetric Functions and Hall Polynomials. (Oxf. Math. Monogr.) Oxford: Clarendon 1979 · Zbl 0487.20007
[18] [ML] MacLane, S.: Homologie des anneaux et des modules. In: CBRM, Colloque de topologie algébrique, pp. 55-80. Louvain: Librairie universitaire 1957
[19] [Ml] Miller, H.R.: The Sullivan conjecture on maps from classifying spaces. Ann. Math.120, 39-87 (1984) · Zbl 0552.55014 · doi:10.2307/2007071
[20] [Mn] Milne, J.S.: Duality in the flat cohomology of a surface. Ann. Sci. Ec. Norm. Supér.9, 171-202 (1976) · Zbl 0334.14010
[21] [P] Pirashvili, T.: Higher additivisations. Proc. Math. Inst. Tbilissi91, 44-54 (1988)
[22] [P-W] Pirashvili, T., Waldhausen, F.: MacLane homology and topological Hochschild homology. J. Pure Appl. Algebra82-1, 81-98 (1992) · Zbl 0767.55010 · doi:10.1016/0022-4049(92)90012-5
[23] [S] Schwartz, L.: Unstable Modules over the Steenrod Algebra and Sullivan’s Fixed Point Conjecture. (Chicago Lect. Notes) (à paraître) · Zbl 0871.55001
[24] [Sm] Smith, J.H.: Homotopy additive simplicial functors. (Prépublication 1992)
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.