The m-adic topologies and, in particular the notions of m-complete ring and m-completion A of a commutative ring A, occur frequently in commutative algebra and are also a useful tool in algebraic geometry. The aim of this work is to collect together some criteria concerning the ascent (from A to A) and the descent (from A to A) of several properties of commutative rings such as, for example: integrity, regularity, factoriality, normality, etc. More precisely, we want to show that many of the above criteria, although not trivial at all, are elementary consequences of some fundamental notions of commutative algebra and local algebra. Sometimes we are able to get only partial results, which probably can be improved by further deeper investigations. No new result has been included in this work. Its only origiยญ nality is the choice of material and the mode of presentation. The comprehension of the most important statements included in this book needs only a very elementary background in algebra, ideal theory and general topology. In order to emphasize the elementary character of our treatment, we have recalled several well known definitions and, sometimes, even the proofs of the first properties which follow directly from them. On the other hand, we did not insert in this work some important results, such as the Cohen structure theorem on complete noetherian local rings, as we did not want to get away too much from the spirit of the book
CONTENT
Content -- ยง 1 Compatibilities of algebraic and topological structures on a set. The ?-adic topology. Artin-Rees lemma. Krullโ{128}{153}s intersection theorem. Zariski rings -- ยง 2 Completions of filtered groups, rings and modules. Applications to ?-adic topologies -- ยง 3 Rings of formal and restricted power series. Preparation theorems. Hensel lemma -- ยง 4 Completions of finitely generated modules. Flatness and faithful flatness -- ยง 5 Noetherian properties of ?-adic completions -- ยง 6 Some general criteria of ascent and descent for ?-completions -- ยง7 Dimension of ?-completions -- ยง 8 Regularity and global dimension of ?-completions -- ยง 9 โ{128}{156}Cohen Macaulayโ{128}{157} and โ{128}{156}Gorensteinโ{128}{157} properties for ?-completions -- ยง 10 Integrity of ?-completions -- ยง 11 Unique factorization of ?-completions -- ยง 12 Fibers of a ring homomorphism and formal fibers of a ring -- ยง 13 Properties Sn and Rn for ?-completions -- ยง 14 Analytic reducedness -- ยง15 Normality of ?-completions -- References
