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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.

MSC:
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
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