Descent and ascent of local properties along homomorphisms of finite flat dimension (Q1082381)

Let \(f: R\to S\) be a local homomorphsm of Noetherian local rings. It is well known that if f is flat then the property of being (1) regular, (2) complete intersection, (3) Gorenstein, (4) Cohen-Macaulay, respectively, descends from S to R. In fact, properties (1), (2) and (4) are known to descend even under the weaker condition that f makes S an R-module of finite flat dimension. One of the results proved in this paper is that (3) also descends under this weaker condition. Moreover, a quantitative expression is provided to these results by proving that if f makes S an R-module of finite flat dimension then appropriate inequalities hold between certain numerical functions of S measuring its deviation from one of the above properties and the corresponding functions of R. The authors also consider the ascent of the above properties. Assuming that f makes S an R-module of finite flat dimension, conditions are found on the homotopy fibre of f which ensure, respectively, the ascent of properties (1), (2) and (3) from R to S.