Kahler differentials of a ring homomorphism form a right-exact functor of algebras, and deriving it requires resolving an algebra by simplicial rather than module-theoretic means. Students learn to build free simplicial resolutions by killing cycles, to form the cotangent complex as the differentials of such a resolution, and to read off the Andre-Quillen homology and cohomology modules D_n(S|R;N) and D^n(S|R;N) together with their base-change, localization and Jacobi-Zariski long exact sequence properties. The payoff is that two central classes of ring homomorphisms become homological conditions: a homomorphism essentially of finite type is locally complete intersection exactly when D_2 vanishes and regular exactly when D_1 vanishes.
Expanded lecture notes from the Chicago 2004 summer school 'Interactions between homotopy theory and algebra' (32 pages, to appear in Contemporary Mathematics) developing Andre-Quillen homology for c…