Ogneva, O. S. 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 Cited in 4 Documents 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 MathOverflow Questions: Is $C^{\infty}(E)$ a projective Frechet $C^{\infty}(M)$-module for a $C^{\infty}$-fiber bundle $E\to M$ with compact fiber? Keywords:cohomological dimensions PDF BibTeX XML Cite \textit{O. S. Ogneva}, Funct. Anal. Appl. 20, 248--250 (1986; Zbl 0626.46057); translation from Funkts. Anal. Prilozh. 20, No. 3, 92--93 (1986) Full Text: DOI 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). 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.