The Projective Dimension of a Compact Abelian Group

Eric C. Nummela
1973 Proceedings of the American Mathematical Society  
Let A" be a compact group, FGX the (Graev) free topological group generated by X, and K the kernel of the canonical quotient morphism from FGX to X. Then A" is a (Graev) free topological group. A corollary to the abelian analogue of this theorem is that the projective dimension of a compact abelian group, relative to the class of all continuous epimorphisms admitting sections, is exactly one. Introduction. Let 'S (respectively, s/) denote the category of topological (abelian) groups and
more » ... groups and continuous group homomorphisms. We take as known the definitions and elementary properties of topological groups as presented in, for example, [2, pp. 219-250]. We adopt the convention that all groups are topological, using the modifier "algebraic" for those groups without topologies. Moreover, all our topologies satisfy the Hausdorff separation axiom. We assume also some familiarity with the notions of free (topological) groups due to A. A. Markov [8] and M. I. Graev [5] . Given a completely regular space X, we denote by FX (ZX) the Markov free (free abelian) group over X, and by FGX (ZGX) the Graev free (free abelian) group over X. Recall that if A' is a compact space, then FGX is the colimit of the expanding sequence {Fm} of compact spaces, where Fm is the set of all words in FGX whose reduced length relative to X\{e} is less than or equal to m (e is the identity of FGX). An analogous result holds for FX, ZGX, and ZX. Recall that a group P in 'S (s/) is projective relative to a class S of epimorphisms in 'S (stf) if, given any diagram P I
doi:10.2307/2038930 fatcat:i7bdkjg525cfveiogm5fk22cvm