Homological aspects of semidualizing modules (original) (raw)

Homological invariants associated to semi-dualizing bimodules

Arxiv preprint math/0505466, 2005

Cohen-Macaulay dimension for modules over a commutative noetherian local ring has been defined by AA Gerko [10]. That is to be a homological invariant of a module which shares a lot of properties with projective dimension and Gorenstein dimension. The aim of this paper ...

On the Cohomological Dimension of Finitely Generated Modules

2018

Let (R,m) be a commutative Noetherian local ring and I be an ideal of R. In this paper it is shown that if cd(I, R) = t > 0 and the R-module HomR(R/I,H t I(R)) is finitely generated, then t = sup {dim̂̂ RP/Q : P ∈ V (IR̂), Q ∈ mAsŝ̂ RP ̂̂ RP and P ̂̂ RP = Rad(I ̂̂ RP + Q)}. Moreover, some other results concerning the cohomological dimension of ideals with respect to the rings extension R ⊂ R[X] will be included.

On the Finiteness Dimension of Local Cohomology Modules

Algebra Colloquium, 2014

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M a non-zero finitely generated R-module. Let t be a non-negative integer. In this paper, it is shown that [Formula: see text] for all i < t if and only if there exists an ideal 𝔟 of R such that dim R/𝔟 ≤ 1 and [Formula: see text] for all i < t. Moreover, we prove that [Formula: see text] for all i.

On Injective and Gorenstein Injective Dimensions of Local Cohomology Modules

Algebra Colloquium, 2015

Let (R, 𝔪) be a commutative Noetherian local ring and M an R-module which is relative Cohen-Macaulay with respect to a proper ideal 𝔞 of R, and set n := ht M𝔞. We prove that injdim M < ∞ if and only if [Formula: see text] and that [Formula: see text]. We also prove that if R has a dualizing complex and Gid RM < ∞, then [Formula: see text]. Moreover if R and M are Cohen-Macaulay, then Gid RM < ∞ whenever [Formula: see text]. Next, for a finitely generated R-module M of dimension d, it is proved that if [Formula: see text] is Cohen-Macaulay and [Formula: see text], then [Formula: see text]. The above results have consequences which improve some known results and provide characterizations of Gorenstein rings.

On the Projective Dimension of Artinian Modules

2021

Let (R,mathfrakm)(R, mathfrak{m})(R,mathfrakm) be a Noetherian local ring and MMM, NNN be two finitely generated RRR-modules. In this paper it is shown that RRR is a Cohen-Macaulay ring if and only if RRR admits a non-zero Artinian RRR-module AAA of finite projective dimension; in addition, for all such Artinian RRR-modules AAA, it is shown that mathrmpdR,A=dimRmathrm{pd}_R, A=dim RmathrmpdR,A=dimR. Furthermore, as an application of these results it is shown that$$pdd H^i_{{frak p}R_{frak p}}(M_{frak p}, N_{frak p})leq pd H^{i+dim R/{frak p}}_{frak m}(M,N)$$for each frakpinmathrmSpecR{frak p}in mathrm{Spec} RfrakpinmathrmSpecR and each integer igeq0igeq 0igeq0. This result answers affirmatively a question raised by the present authors in [13].