The plus-construction, Bousfield localization, and derived completion
Quillen's plus-construction produces, for a given pointed space \(X\) and a perfect normal subgroup \(P\) of the fundamental group, an integral homology equivalence \(X \rightarrow X^+\) inducing the quotient map \(\pi_1 X \rightarrow \pi_1 X/P\) on fundamental groups. For algebras over operads, such a construction was established in a simplicial setting by \textit{T. I. Pirashvili} [Tr. Tbilis. Mat. Inst. Razmadze 78, 44--78 (1986; Zbl 0616.55010)], and in a differential graded setting by \textit{M. Livernet} [K-Theory 18, No.4, 317--337 (1999; Zbl 0981.18012)]. Classically, the plus-construction is performed by attaching cells, but it is also possible to provide a functorial construction as a Bousfield localization. Such an interpretation of Livernet's construction has been given by \textit{D. Chataur, J. L. Rodríguez} and this reviewer [\(K\)-Theory 33, No.~1, 1--21 (2004; Zbl 1103.19002)]. The subject of this article is to develop both points of view for connective augmented algebras \(A\) over an operad \(\mathcal O\) in symmetric spectra. Integral homology is replaced by Quillen homology and perfect subgroups of the fundamental group by perfect \(\pi_0(\mathcal O)\)-submodules of the kernel of \(\pi_n (A) \rightarrow \pi_n (\mathcal O(0))\). The techniques rely on results of \textit{J. E. Harper} [J. Pure Appl. Algebra 214, No.~8, 1407--1434 (2010; Zbl 1231.55011) and Algebr. Geom. Topol. 9, No.~3, 1637--1680 (2009; Zbl 1235.55004)]. When the operad is either the commutative or the associative operad, and when the augmentation ideal of the algebra is perfect, the plus-construction can be seen as Bousfield localization with respect to the homology theory represented by the base symmetric ring spectrum \(R\). There is in fact a third interpretation of this plus-construction in terms of \textit{G. Carlsson}'s derived completion [J. Pure Appl. Algebra 212, No.~3, 550--577 (2008; Zbl 1146.55006)].
scientific article; zbMATH DE number 5669127
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