Krull dimension for differential graded algebras

We introduce a naive notion of a system of parameters for a homologically finite complex over a commutative noetherian local ring and compare it to the system of parameters defined by Christensen. We show that these notions differ in general but that they agree when the complex in question is a DG R-algebra. In this case we also show that the Krull dimension defined in terms of the lengths of such systems of parameters agrees with Krull dimensions defined in terms of certain chains of prime ideals.     

Relative Tor Functors with Respect to a Semidualizing Module

Let C be a semidualizing module over a commutative noetherian ring R. We exhibit an isomorphism $\operatorname{Tor}^{{\mathcal{F}_C}\mathcal{M}}_{i}(-,-) \cong \operatorname{Tor}^{{\mathcal{P}_C}\mathcal{M}}_{i}(-,-)$ between the bifunctors defined via C-flat and C-projective resolutions. We show how the vanishing of these functors characterizes the finiteness of ${{\mathcal{F}_C}\text{-}\operatorname{pd}}$ , and use this to give a relation between the ${{\mathcal{F}_C}\text{-}\operatorname{pd}}$ of a module and of a pure submodule. On the other hand, we show that other isomorphisms force C to be trivial.     

A dimension inequality for Cohen-Macaulay rings

The recent work of Kurano and Roberts on Serre's positivity conjecture suggests the following dimension inequality: for prime ideals and in a local, Cohen-Macaulay ring such that we have . We establish this dimension inequality for excellent, local, Cohen-Macaulay rings which contain a field, for certain low-dimensional cases and when is regular.

     

Adic Foxby Classes

Algebras and Representation Theory - We continue our work on adic semidualizing complexes over a commutative noetherian ring R by investigating the associated Auslander and Bass classes...     

Ascent properties for pairs of modules

Given a flat local ring homomorphism \({R \rightarrow S}\) and two finitely generated R-modules M and N, we describe conditions under which the modules \({{\rm Tor}^{R}_{i}(M,N)}\) and \({{\rm Ext}^{i}_{R}(M,N)}\) have S-module structures that are compatible with their R-module structures.     

Generic Constructions and Semidualizing Modules

Algebras and Representation Theory - We investigate some general machinery for describing semidualizing modules over generic constructions like ladder determinantal rings with coefficients in a...     

Complete intersection dimensions and Foxby classes

Let R be a local ring and M a finitely generated R-module. The complete intersection dimension of M-defined by Avramov, Gasharov and Peeva, and denoted -is a homological invariant whose finiteness implies that M is similar to a module over a complete intersection. It is related to the classical projective dimension and to Auslander and Bridger’s Gorenstein dimension by the inequalities .Using Blanco and Majadas’ version of complete intersection dimension for local ring homomorphisms, we prove the following generalization of a theorem of Avramov and Foxby: Given local ring homomorphisms φ:R→S and ψ:S→T such that φ has finite Gorenstein dimension, if ψ has finite complete intersection dimension, then the composition ψ°φ has finite Gorenstein dimension. This follows from our result stating that, if M has finite complete intersection dimension, then M is C-reflexive and is in the Auslander class A_C(R) for each semidualizing R-complex C.     

Applications and homological properties of local rings with decomposable maximal ideals

We construct a local Cohen–Macaulay ring R with a prime ideal $\mathfrak{p}\in \mathrm{Spec}(R)$ such that R satisfies the uniform Auslander condition (UAC), but the localization $R_{\mathfrak{p}}$ 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 $\mathfrak{p}\in \mathrm{Spec}(R)$ such that R has exactly two non-isomorphic semidualizing modules, but the localization $R_{\mathfrak{p}}$ 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.     

Intersections of Symbolic Powers of Prime Ideals

Let (R,m) be a local ring with prime ideals p and q such that. If R is regular and containsa field, and dim(R/p)+dim(R/q)=dim(R), then it is proved thatp^(m) q⁽ⁿ⁾ m^m+n for all positive integers m and n. This isproved using a generalization of Serre's Intersection Theoremwhich is applied to a hypersurface R/fR. The generalizationgives conditions that guarantee that Serre's bound on the intersectiondimension (R/p)+(R/q)dim(R) holds when R is nonregular.