We prove a criterion for the constancy of the Hilbert–Samuel function for locally Noetherian schemes such that the local rings are excellent at every point. More precisely, we show that the Hilbert–Samuel function is locally constant on such a scheme if and only if the scheme is normally flat along its reduction and the reduction itself is regular. Regularity of the underlying reduced scheme is a significant new property.

We give a descent result for formal smoothness having interesting applications: we deduce that quasiexcellence descends along flat local homomorphisms of finite type, we greatly improve Kunz’s characterization of regular local rings by means of the Frobenius homomorphisms as well as André and Radu relativization of this result, etc. In the second part of the paper, we study a similar question for the complete intersection property instead of formal smoothness, giving also some applications.

Let A be a local Noetherian domain of Krull dimension d. Heinzer, Rotthaus, and Sally have shown that if the generic formal fiber of A has dimension d – 1, then A is birationally dominated by a 1-dimensional analytically ramified local Noetherian ring having residue field finite over the residue field of A. We explore further this correspondence between prime ideals in the generic formal fiber and 1-dimensional analytically ramified local rings. Our main focus is on the case where the analytically ramified local rings are stable, and we show that in this case the embedding dimension of the stable ring reflects the embedding dimension of a prime ideal maximal in the generic formal fiber, thus providing a measure of how far the generic formal fiber deviates from regularity. A number of characterizations of analytically ramified local stable domains are also given.

Let $R$ be a commutative Noetherian ring and $M$ a finite $R$-module. In this paper, we consider Zariski-openness of the FID-locus of $M$, namely, the subset of $\mathrm{spec}\,R$ consisting of all prime ideals ${\mathfrak p}$ such that $M_{\mathfrak p}$ has finite injective dimension as an $R_{\mathfrak p}$-module. We prove that the FID-locus of $M$ is an open subset of $\mathrm{spec}\,R$ whenever $R$ is excellent.

This paper is concerned with a finitely generated module M over a (commutative Noetherian) local ring R. In the case when R is a homomorphic image of a Gorenstein local ring, one can use the well-known associativity formula for multiplicities, together with local duality and Matlis duality, to produce analogous associativity formulae for the local cohomology modules of M with respect to the maximal ideal. The main purpose of this paper is to show that these formulae also hold in the case when R is universally catenary and such that all its formal fibres are Cohen–Macaulay.

These formulae involve certain subsets of the spectrum of R called the pseudosupports of M; these pseudo-supports are closed in the Zariski topology when R is universally catenary and has the property that all its formal fibres are Cohen–Macaulay. However, examples are provided to show that, in general, these pseudo-supports need not be closed. We are able to conclude that the above-mentioned associativity formulae for local cohomology modules do not hold over all local rings.