Applications and homological properties of local rings with decomposable maximal ideals

Abstract

We construct a local Cohen-Macaulay ring R with a prime ideal p ∈ Spec(R) such that R satisfies the uniform Auslander condition (UAC), but the localization Rp does not satisfy Auslander's condition (AC). Given any positive integer n, we also construct a local Cohen-Macaulay ring R with a prime ideal p ∈ Spec(R) such that R has exactly two non-isomorphic semidualizing modules, but the localization Rp has 2 n non-isomorphic semidualizing modules. Each of these examples is constructed as a fiber product of two local rings over their common residue field. Additionally, we characterize the non-trivial Cohen-Macaulay fiber products of finite Cohen-Macaulay type.

Applications of Differential Graded Algebra Techniques in Commutative Algebra

“…Friendliness and persistence have been studied in numerous works; see for instance [17,18,29,70,71,73,74,76,77,92,93,94,95,106,107]. The main motivation for this section is the following result in which the proofs of parts (a), (b), (c), (e), and (f) use DG algebra techniques.…”

Generic Constructions and Semidualizing Modules

“…However, we do not know whether this map is bijective. (Note that one must have some assumptions on A and U in order for this to be so because of [16,Theorem B].) Several of our proofs in [21] would be simplified if this map were bijective.…”

Minimal Free Resolutions of Fiber Products

“…Both S and T come equipped with a natural surjection S πS − − → k πT ← − − T which are used to build the fiber product of S and T over k given by S × k T := {(s, t) ∈ S × T : π S (s) = π T (t)}. Much research has been conducted comparing and contrasting the homological properties of S and T with those of S × k T , e.g., the Cohen-Macaulay, Gorenstein, Golod, finite representation type, and Arf properties (see [1,2,4,5,[7][8][9][10][11][12][13]).…”