André-Quillen (co-)homology for simplicial algebras over simplicial operads
The paper under review is part of a series of three papers, by the same authors, investigating spectra with multiplicative structures and homotopy types of spaces of multiplicative maps between such spectra. In the authors' words ``[t]he other two \dots{} are more homotopy theoretic in nature; this paper is something of an algebraic intermezzo''. The algebraic problems considered in the paper have a prototype in Quillen's work on simplicial modules over simplicial algebras. Part of this work is known as André-Quillen (co)homology of commutative algebras. In this paper the authors, motivated by problems of stable homotopy theory, develop a similar (co)homology theory for simplicial algebras over simplicial operads. This includes simplicial operads not only in the category of \(R\)-modules over a commutative ring \(R\), but also in the category of \(\Gamma\)-comodules over a Hopf algebroid \((R,\Gamma)\). NEWLINENEWLINENEWLINEThe paper consists of six sections. Preliminaries on operads are collected in the first. In the second, the basic tools of André-Quillen (co)homology are transferred to the operadic context. This is done by utilizing square-zero extensions. In section 3, the authors study \(\Gamma\)-comodules over a Hopf algebroid. Here they introduce the notion of an Adams Hopf algebroid. This simply means that \(\Gamma\) is a flat \(R\)-module filtered by left \(\Gamma \)-comodules \(\Gamma_{\alpha}\) such that each term is a finitely generated projective \(R\)-module and \(\underset{\alpha}{\text{colim }}\Gamma_{\alpha}\) is isomorphic to \(\Gamma\) in the (abelian) category \(\mathcal{C}_{\Gamma}\) of left \(\Gamma\)-comodules. Restricting to such algebroids, the authors show that the category \(s\mathcal{C}_{\Gamma}\) of simplicial objects of \(\mathcal{C}_{\Gamma}\) becomes a simplicial model category. Section 4 extends this result to simplicial operads in \(\mathcal{C}_{\Gamma}\). Section 5 studies square-zero extensions. The results are then applied, in the last section, to the study of Postnikov towers.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00041].
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