Description

A complement to standard books on commutative algebra providing complete and relatively easy proofs of important results. This short book gives complete and relatively easy proofs of well-known important results found in standard books on commutative algebra. It also serves as a friendly introduction to the powerful methods used in the André–Quillen homology of commutative algebras. This short book gives complete and relatively easy proofs of well-known important results found in standard books on commutative algebra. It also serves as a friendly introduction to the powerful methods used in the André–Quillen homology of commutative algebras. Written to complement standard texts on commutative algebra, this short book gives complete and relatively easy proofs of important results, including the standard results involving localisation of formal smoothness (M. André) and localisation of complete intersections (L. Avramov), some important results of D. Popescu and André on regular homomorphisms, and some results from A. Grothendieck's EGA on smooth homomorphisms. The authors make extensive use of the André–Quillen homology of commutative algebras, but only up to dimension 2, which is easy to construct, and they deliberately avoid using simplicial methods. The book also serves as an accessible introduction to some advanced topics and techniques. The only prerequisites are a basic course in commutative algebra and the first definitions in homological algebra. Introduction; 1. Definition and first properties of (co-)homology modules; 2. Formally smooth homomorphisms; 3. Structure of complete noetherian local rings; 4. Complete intersections; 5. Regular homomorphisms: Popescu's theorem; 6. Localization of formal smoothness; Appendix: some exact sequences; Bibliography; Index. ' … thisbook makes for a useful addition to the (rather meagre) literature on the uses of André-Quillen homology and cohomology in commutative algebra and algebraic geometry. Mathematical Reviews