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Coincidence of the homological dimensions of the Fréchet algebra of smooth functions on a manifold with the dimension of the manifold. (English. Russian original) Zbl 0626.46057
Funct. Anal. Appl. 20, 248-250 (1986); translation from Funkts. Anal. Prilozh. 20, No. 3, 92-93 (1986).
The principal result of this paper is the following: ds $$C^{\infty}(M)=dg C^{\infty}(M)=db C^{\infty}(M)=m$$; here M is a smooth real m-dimensional manifold, $$C^{\infty}(M)$$ is the topological algebra of $$C^{\infty}$$ functions on M and ds A, dg A, db A denote the cohomological dimensions of a topological algebra A in the sense of A. Ya. Khelemskij [Homology in Banach and topological algebras (in Russian) (1986; Zbl 0608.46046)].
Reviewer: L.Maxim Răileanu

##### MSC:
 46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) 46H05 General theory of topological algebras 46J05 General theory of commutative topological algebras
##### Keywords:
cohomological dimensions
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##### References:
 [1] A. Ya. Khelemskii, Homology in Banach and Topological Algebras [in Russian], Moscow State Univ. (1986). [2] J. L. Taylor, Adv. Math.,9, 183-252 (1972). · Zbl 0271.46041 · doi:10.1016/0001-8708(72)90017-5 [3] S. MacLane, Homology, Springer-Verlag (1975). [4] A. Ya. Khelemskii, Tr. Sem. im. I. G. Petrovsk., No. 3, 223-242 (1978). [5] J. L. Taylor, Adv. Math.,9, 137-182 (1972). · Zbl 0271.46040 · doi:10.1016/0001-8708(72)90016-3 [6] A. Grothendieck, Mem. Am. Math. Soc.,16 (1955).
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