Andre-Quillen Homology and Special Classes of Ring Homomorphisms

This thesis is comprised of three chapters. The first chapter deals with a purely algebraic proof of a deep result of Quillen stating that the category of simplicial commutative algebras over a commutative ring is a model category. The central focus of our approach is on the study of shuffle product of connective chain complexes that provides a bridge to translate the constructions in the simplicial algebra world to the chain complex world. The second chapter delves into Quillen's fundamental spectral sequences that relate Andre-Quillen homology and cohomology to Tor and Ext functors. Our comprehensive treatment develops and streamlines the requisite background constructions to provide a suitable framework in order to establish complete proofs. The third chapter studies certain special classes of ring homomorphisms, the so-called complete intersection maps and exceptional complete intersection maps. Some research has been conducted to characterize or detect the exceptional property. We investigate this property further and discover new conditions to detect it.