Characterizing local rings via homological dimensions and regular sequences

Let (R,m) be a Noetherian local ring of depth d and C a semidualizing R-complex. Let M be a finite R-module and t an integer between 0 and d. If the G_C-dimension of M/aM is finite for all ideals a generated by an R-regular sequence of length at most d−t then either the G_C-dimension of M is at most t or C is a dualizing complex. Analogous results for other homological dimensions are also given.     

Transfer of Gorenstein dimensions along ring homomorphisms

A central problem in the theory of Gorenstein dimensions over commutative noetherian rings is to find resolution-free characterizations of the modules for which these invariants are finite. Over local rings, this problem was recently solved for the Gorenstein flat and the Gorenstein projective dimensions; here we give a solution for the Gorenstein injective dimension. Moreover, we establish two formulas for the Gorenstein injective dimension of modules in terms of the depth invariant; they extend formulas for the injective dimension due to Bass and Chouinard.     

Stability of Gorenstein categories

We show that an iteration of the procedure used to define theGorenstein projective modules over a commutative ring R yieldsexactly the Gorenstein projective modules. Specifically, givenan exact sequence of Gorenstein projective R-modules

such that the complexes Hom_R(G, H) and Hom_R(H,G) are exact for each Gorenstein projective R-module H, themodule Coker() is Gorensteinprojective. The proof of this result hinges upon our analysisof Gorenstein subcategories of abelian categories.     

Extended Local Cohomology and Local Homology

We present an in-depth exploration of the module structures of local (co)homology modules (moreover, for complexes) over the completion \(\widehat {R}^{\mathcal {a}}\) of a commutative noetherian ring R with respect to a proper ideal \(\mathcal {a}\). In particular, we extend Greenlees-May Duality and MGM Equivalence to track behavior over \(\widehat {R}^{\mathcal {a}}\), not just over R. We apply this to the study of two recent versions of homological finiteness for complexes, and to certain isomorphisms, with a view toward further applications. We also discuss subtleties and simplifications in the computations of these functors.     

Complete intersection dimensions for complexes

We extend the notions of complete intersection dimension and lower complete intersection dimension to the category of complexes with finite homology and verify basic properties analogous to those holding for modules. We also discuss the question of the behavior of complete intersection dimension with respect to short exact sequences.     

Adic finiteness: Bounding homology and applications

We prove the versions of amplitude inequalities of Iversen, Foxby and Iyengar, and Frankild and Sather-Wagstaff that replace finite generation conditions with adic finiteness conditions. As an application, we prove that a local ring R of prime characteristic is regular if and only if for some proper ideal 𝔟 the derived local cohomology complex RΓ_𝔟(R) has finite flat dimension when viewed through some positive power of the Frobenius endomorphism.     

Detecting completeness from Ext-vanishing

Motivated by work of C. U. Jensen, R.-O. Buchweitz, and H. Flenner, we prove the following result. Let be a commutative noetherian ring and an ideal in the Jacobson radical of . Let be the -adic completion of . If is a finitely generated -module such that for all , then is -adically complete.

     

Modules of finite homological dimension with respect to a semidualizing module

We prove versions of results of Foxby and Holm about modules of finite (Gorenstein) injective dimension and finite (Gorenstein) projective dimension with respect to a semidualizing module. We also verify special cases of a question of Takahashi and White. This research was conducted while S.S.-W. visited the IPM in Tehran during July 2008. The research of S.Y. was supported in part by a grant from the IPM (No. 87130211).