Revistes Catalanes amb Accés Obert (RACO)

$P$-nilpotent completion is not idempotent

G. C. Tan


Let $P$ be an arbitrary set of primes. The $P$-nilpotent completion of a group $G$ is defined by the group homomorphism $\eta: G \to G_{\widehat{P}}$ where $G_{\widehat{P}} = \operatorname{invlim} (G/\Gamma_iG)_P$. Here $\Gamma_2G$ is the commutator subgroup $[G,G]$ and $\Gamma_iG$ the subgroup $[G,\Gamma_{i - 1}G]$ when $i > 2$. In this paper, we prove that $P$-nilpotent completion of an infinitely generated free group $F$ does not induce an isomorphism on the first homology group with ${\bold Z}_P$ coefficients. Hence, $P$-nilpotent completion is not idempotent. Another important consequence of the result in homotopy theory (as in \cite{4}) is that any infinite wedge of circles is $R$-bad, where $R$ is any subring of rationals.

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