Andre-Quillen homology for simplicial algebras and ring spectra

QA Mathematics

Reinhard, Philipp Michael ; Baker, Andrew

University:University of Glasgow

Department:School of Mathematics and Statistics

关键词: Topology, Algebraic topology, Stable homotopy theory, Simplicial algebras, Andre-Quillen homology;   

Others  :  http://theses.gla.ac.uk/507/1/2008reinhardphd.pdf

来源: University of Glasgow

【 摘 要 】

We discuss Andre-Quillen homology for simplicial algebras and algebras over simplicial algebras, extending the classical notion for rings. This extension is also discussed by Goerss and Hopkins, however our statements are proven in a more explicit way. We are then further able to construct spectral sequences for Andre-Quillen homology like the spectral sequence for the indecomposables or the Fundamental spectral sequence according to Quillen. The Andre-Quillen homology for algebras constructed as cellular complexes is calculated and we apply this homology theory to obtain notions of atomic and nuclear algebras, thus extending results from Bakerand May. We define the notion of i-stable algebras and are able to give a comparison theorem between Andre-Quillen homology, stabilisation andGamma-homology for i-stable algebras up to degree i.In the second part of the thesis we discuss topologicalAndre-Quillen homology and extend certain results by Gilmour about cellularcomplexes in this setting.