Local Ring Homomorphisms Research Articles

Ring homomorphisms and local rings with quasi-decomposable maximal ideal

The notion of local rings with quasi-decomposable maximal ideal was formally introduced by Nasseh and Takahashi. In separate works, the authors of the present paper showed that such rings have rigid homological properties; for instance, they are both Ext- and Tor-friendly. One point of this paper is to further explore the homological properties of these rings and also introduce new classes of such rings from a combinatorial point of view. Another point is to investigate how far some of these homological properties can be pushed along certain diagrams of local ring homomorphisms.

A Bass equality for Gorenstein injective dimension of modules finite over homomorphisms

Let $$R\rightarrow S$$ be a local ring homomorphism and N a finitely generated S-module. We prove that if the Gorenstein injective dimension of N over R is finite, then it equals the depth of R.

Restricted injective dimensions over local homomorphisms

Abstract In this paper, we study the small restricted injective dimension over local ring homomorphisms. Some of known results are generalized. For example, the Bass formula for the small restricted injective dimension of complexes is extended.

Gorenstein Dimensions over Ring Homomorphisms

In this paper, we study injective dimension and Gorenstein injective dimension over local ring homomorphisms. Some well-known results are generalized. For example, the Bass formula for Gorenstein injective dimension of complexes is extended. As applications, some characterizations of Gorenstein rings are obtained.

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.

G-dimension over local homomorphisms with respect to a semi-dualizing complex

We study the G-dimension over local ring homomorphisms with respect to a semi-dualizing complex. Some results that track the behavior of Gorenstein properties over local ring homomorphisms under composition and decomposition are given. As an application, we characterize a dualizing complex for R in terms of the finiteness of the G-dimension over local ring homomorphisms with respect to a semi-dualizing complex.

Cohen factorizations: Weak functoriality and applications

We investigate Cohen factorizations of local ring homomorphisms from three perspectives. First, we prove a “weak functoriality” result for Cohen factorizations: certain morphisms of local ring homomorphisms induce morphisms of Cohen factorizations. Second, we use Cohen factorizations to study the properties of local ring homomorphisms (Gorenstein, Cohen–Macaulay, etc.) in certain commutative diagrams. Third, we use Cohen factorizations to investigate the structure of quasi-deformations of local rings, with an eye on the question of the behavior of CI-dimension in short exact sequences.

NAK for Ext and ascent of module structures

We investigate the interplay between properties of Ext modules and the ascent of module structures along local ring homomorphisms. Specifically, let φ : ( R , m , k ) → ( S , m S , k ) \varphi \colon (R,\mathfrak {m},k)\to (S,\mathfrak {m} S,k) be a flat local ring homomorphism. We show that if M M is a finitely generated R R -module such that Ext R i ⁡ ( S , M ) \operatorname {Ext}_{R}^{i}(S,M) satisfies NAK (e.g. if Ext R i ⁡ ( S , M ) \operatorname {Ext}_{R}^{i}(S,M) is finitely generated over S S ) for i = 1 , … , dim R ⁡ ( M ) i=1,\ldots ,\dim _{R}(M) , then Ext R i ⁡ ( S , M ) = 0 \operatorname {Ext}_{R}^{i}(S,M)=0 for all i ≠ 0 i\neq 0 and M M has an S S -module structure that is compatible with its R R -module structure via φ \varphi . We provide explicit computations of Ext R i ⁡ ( S , M ) \operatorname {Ext}_{R}^{i}(S,M) to indicate how large it can be when M M does not have a compatible S S -module structure.

Lower bounds for the number of semidualizing complexes over a local ring

We investigate the set $\mathfrak(R)$ of shift-isomorphism classes of semi-dualizing $R$-complexes, ordered via the reflexivity relation, where $R$ is a commutative noetherian local ring. Specifically, we study the question of whether $\mathfrak(R)$ has cardinality $2^n$ for some $n$. We show that, if there is a chain of length $n$ in $\mathfrak(R)$ and if the reflexivity ordering on $\mathfrak (R)$ is transitive, then $\mathfrak(R)$ has cardinality at least $2^n$, and we explicitly describe some of its order-structure. We also show that, given a local ring homomorphism $\varphi\colon R\to S$ of finite flat dimension, if $R$ and $S$ admit dualizing complexes and if $\varphi$ is not Gorenstein, then the cardinality of $\mathfrak (S)$ is at least twice the cardinality of $\mathfrak (R)$.

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 CI - dim R ( M ) –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 G - dim R ( N ) ⩽ CI - dim R ( N ) ⩽ pd R ( N ) . 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 .

Bass Series of Local Ring Homomorphisms of Finite Flat Dimension

Bass Series of Local Ring Homomorphisms of Finite Flat Dimension

Bass series of local ring homomorphisms of finite flat dimension

Nontrivial relations between Bass numbers of local commutative rings are established in case there exists a local homomorphism φ: R → S which makes S into an R-module of finite flat dimension. In particular, it is shown that an inequality μ R i+depth R ≤ μ s i+depth S holds for all i ∈ Z.This is a consequence of an equality involving the Bass series I R M (t) = Σ i ∈ z μ R i (M)t i of a complex M of R-modules which has bounded above and finite type homology and the Bass series of the complex of S-modules M⊗ R S, where ⊗ denotes the derived tensor product. It is prove that there is an equa lity of formal Laurent series I s M⊗RS (t) = I R M (t)I F(φ) (t), where F(φ) is the fiber of φ considered as a homomorphism of commutative differential graded rings