Commutative Noetherian Local Ring Research Articles

Non-commutative Fitting invariants and annihilation of class groups

One can associate to each finitely presented module M over a commutative ring R an R -ideal Fitt R ( M ) which is called the (zeroth) Fitting ideal of M over R and which is an important natural invariant of M . We generalize this notion to o -orders in separable algebras, where o is a complete commutative noetherian local ring. As an application we construct annihilators of class groups assuming the validity of the Equivariant Tamagawa Number Conjecture for a certain motive attached to a Galois CM-extension of number fields.

On the existence of embeddings into modules of finite homological dimensions

Let R R be a commutative Noetherian local ring. We show that R R is Gorenstein if and only if every finitely generated R R -module can be embedded in a finitely generated R R -module of finite projective dimension. This extends a result of Auslander and Bridger to rings of higher Krull dimension, and it also improves a result due to Foxby where the ring is assumed to be Cohen-Macaulay.

On the Matlis duals of local cohomology modules and modules of generalized fractions

Let (R, m) be a commutative Noetherian local ring with non-zero identity, a a proper ideal of R and M a finitely generated R-module with aM ≠ M. Let D(−) ≔ HomR(−, E) be the Matlis dual functor, where E ≔ E(R/m) is the injective hull of the residue field R/m. In this paper, by using a complex which involves modules of generalized fractions, we show that, if x1, …, xn is a regular sequence on M contained in α, then H(x1, …,xnRnD(Han(M))) is a homomorphic image of D(M), where Hbi(−) is the i-th local cohomology functor with respect to an ideal b of R. By applying this result, we study some conditions on a certain module of generalized fractions under which D(H(x1, …,xn)Rn(D(Han(M)))) ⋟ D(D(M)).

DIMENSION FORMULAS FOR MODULES FINITE OVER LOCAL HOMOMORPHISMS

We extend some major theorems in commutative algebra to the class of modules that are not necessarily finitely generated. The novelty of our extension is that the hypothesis of finite generation over $R$ is replaced by one over $S$, where $R$ and $S$ are commutative Noetherian local rings and there is a local homomorphism $\varphi:R \rightarrow S$. Among the results that we extend are: Intersection Theorem and Intersection dimension formula.

A Characterization of Gorenstein Rings and Grothendieck's Conjecture

Given a commutative Noetherian local ring (R, 𝔪), it is shown that R is Gorenstein if and only if there exists a system of parameters x1,…,xd of R which generates an irreducible ideal and [Formula: see text] for all t > 0. Let n be an arbitrary non-negative integer. It is also shown that for an arbitrary ideal 𝔞 of a commutative Noetherian (not necessarily local) ring R and a finitely generated R-module M, [Formula: see text] is finitely generated if and only if there exists an 𝔞-filter regular sequence x1,…,xn∈ 𝔞 such that [Formula: see text] for all t > 0.

Growth in the minimal injective resolution of a local ring

Let R be a commutative noetherian local ring with residue field k and assume that it is not Gorenstein. In the minimal injective resolution of R, the injective envelope E of the residue field appears as a summand in every degree starting from the depth of R. The number of copies of E in degree i equals the k-vector space dimension of the cohomology module ExtiR(k, R). These dimensions, known as Bass numbers, form an infinite sequence of invariants of R about which little is known. We prove that it is non-decreasing and grows exponentially if R is Golod, a non-trivial fiber product, or Teter, or if it has radical cube zero.

A generalization of a theorem of Foxby

In this paper, it is proved that a commutative noetherian local ring admitting a finitely generated module of finite projective and injective dimensions with respect to a semidualizing module is Gorenstein. This result recovers a celebrated theorem of Foxby.

COFINITENESS AND FINITENESS OF GENERALIZED LOCAL COHOMOLOGY MODULES

AbstractLet I be an ideal of a commutative Noetherian local ring R, and M and N two finitely generated modules. Let t be a positive integer. We mainly prove that (i) if HIi(M,N) is Artinian for all i<t, then HIi(M,N) is I-cofinite for all i<t and Hom(R/I,HIt(M,N)) is finitely generated; (ii) if d=pd(M)<∞ and dim N=n<∞, then HId+n(M,N) is I-cofinite. We also prove that if M is a nonzero cyclic R-module, then HIi(N) is finitely generated for all i<t if and only if HIi(M,N) is finitely generated for all i<t.

Modules in resolving subcategories which are free on the punctured spectrum

Let R be a commutative noetherian local ring, and let X be a resolving subcategory of the category of finitely generated R-modules. In this paper, we study modules in X by relating them to modules in X which are free on the punctured spectrum of R. We do this by investigating nonfree loci and establishing an analogue of the notion of a level in a triangulated category which has been introduced by Avramov, Buchweitz, Iyengar and Miller. As an application, we prove a result on the dimension of the nonfree locus of a resolving subcategory having only countably many nonisomorphic indecomposable modules in it, which is a generalization of a theorem of Huneke and Leuschke.

On the Finiteness Results of the Generalized Local Cohomology Modules

Let 𝔞 be an ideal of a commutative Noetherian local ring R, and let M and N be two finitely generated R-modules. Let t be a positive integer. It is shown that if the support of the generalized local cohomology module [Formula: see text] is finite for all i < t, then the set of associated prime ideals of the generalized local cohomology module [Formula: see text] is finite. Also, if the support of the local cohomology module [Formula: see text] is finite for all i < t, then the set [Formula: see text] is finite. Moreover, we prove that gdepth (𝔞+ Ann (M),N) is the least integer t such that the support of the generalized local cohomology module [Formula: see text] is an infinite set.

ATTACHED PRIMES OF THE TOP GENERALIZED LOCAL COHOMOLOGY MODULES

AbstractLet (R,𝔪) be a commutative Noetherian local ring, letIbe an ideal ofRand letMandNbe finitely generatedR-modules. Assume that$\mathrm {pd} (M)=d\lt \infty $,$\dim N=n\lt \infty $. First, we give the formula for the attached primes of the top generalized local cohomology moduleHId+n(M,N); later, we prove that if Att(HId+n(M,N))=Att(HJd+n(M,N)), thenHId+n(M,N)=HJd+n(M,N).

Generalized Regular Sequence and Finiteness of Local Cohomology Modules

Let R be a commutative Noetherian local ring, 𝔞 an ideal of R, and M a finitely generated generalized f-module. Let t be a positive integer such that [Formula: see text] and t > dim M - dim M/𝔞M. In this paper, we prove that there exists an ideal 𝔟 ⊇ 𝔞 such that (1) dim M - dim M/𝔟M = t; and (2) the natural homomorphism [Formula: see text] is an isomorphism for all i > t and it is surjective for i = t. Also, we show that if [Formula: see text] is a finite set for all i < t, then there exists an ideal 𝔟 of R such that dim R/𝔟 ≤ 1 and [Formula: see text] for all i < t.

Graded annihilators and tight closure test ideals

Let R be a commutative Noetherian local ring of prime characteristic p, with maximal ideal m . The main purposes of this paper are to show that if the injective envelope E of R / m has a structure as an x-torsion-free left module over the Frobenius skew polynomial ring over R (in the indeterminate x), then R has a tight closure test element (for modules) and is F-pure, and to relate the test ideal of R to the smallest ‘ E-special’ ideal of R of positive height. A byproduct is an analogue of a result of Janet Cowden Vassilev: she showed, in the case where R is an F-pure homomorphic image of an F-finite regular local ring, that there exists a strictly ascending chain 0 = τ 0 ⊂ τ 1 ⊂ ⋯ ⊂ τ t = R of radical ideals of R such that, for each i = 0 , … , t − 1 , the reduced local ring R / τ i is F-pure and its test ideal (has positive height and) is exactly τ i + 1 / τ i . This paper presents an analogous result in the case where R is complete (but not necessarily F-finite) and E has a structure as an x-torsion-free left module over the Frobenius skew polynomial ring. Whereas Cowden Vassilev's results were based on R. Fedder's criterion for F-purity, the arguments in this paper are based on the author's work on graded annihilators of left modules over the Frobenius skew polynomial ring.

Ascent of module structures, vanishing of Ext, and extended modules

Let (R;m) and (S;n) be commutative Noetherian local rings, and let ' : R ! S be a ∞at local homomorphism such that mS = n and the induced map on residue flelds R=m ! S=n is an isomorphism. Given a flnitely generated R-module M, we show that M has an S-module structure compatible with the given R-module structure if and only if Ext i (S;M) is flnitely generated as an R-module for each i ‚ 1. We say that an S-module N is extended if there is a flnitely generated R-module M such that N » S ›R M. Given a short exact sequence 0 ! N1 ! N ! N2 ! 0 of flnitely generated S-modules, with two of the three modules N1;N;N2 extended, we obtain conditions forcing the third module to be extended. We show that every flnitely generated module over the Henselization of R is a direct summand of an extended module, but that the analogous result fails for the m-adic completion.

Associated primes of local cohomology modules and Matlis duality

Let ( R , m ) be a commutative Noetherian local ring of dimension d and I an ideal of R. We show that the set of associated primes of the local cohomology module H I 2 ( R ) is finite whenever R is regular. Also, it is shown that if x 1 , … , x d is a system of parameters for R, then D ( H ( x 1 , … , x i ) i ( R ) ) has infinitely many associated prime ideals for all i ⩽ d − 1 , where D ( − ) : = Hom R ( − , E ) denotes the Matlis dual functor and E : = E R ( R / m ) is the injective hull of the residue field R / m . Finally, we explore a counterexample of Grothendieck's conjecture by showing that, if d ⩾ 3 , then the R-module Hom R ( R / I , H I d − 1 ( R ) ) is not finitely generated, where I = ( x 1 ) ∩ ( x 2 , … , x d ) .

Descent via Koszul extensions

Let R be a commutative noetherian local ring with completion R ˆ . We apply differential graded (DG) algebra techniques to study descent of modules and complexes from R ˆ to R ′ where R ′ is either the henselization of R or a pointed étale neighborhood of R: We extend a given R ˆ -complex to a DG module over a Koszul complex; we describe this DG module equationally and apply Artin approximation to descend it to R ′ . This descent result for Koszul extensions has several applications. When R is excellent, we use it to descend the dualizing complex from R ˆ to a pointed étale neighborhood of R; this yields a new version of P. Roberts' theorem on uniform annihilation of homology modules of perfect complexes. As another application we prove that the Auslander Condition on uniform vanishing of cohomology ascends to R ˆ when R is excellent, henselian, and Cohen–Macaulay.

Finite Gorenstein representation type implies simple singularity

Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive R-modules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R is complete, then it is even a simple hypersurface singularity). The crux of our proof is to argue that if the residue field has a totally reflexive cover, then R is Gorenstein or every totally reflexive R-module is free.

Direct Summands of Syzygy Modules of the Residue Class Field

AbstractLetRbe a commutative Noetherian local ring. This paper deals with the problem asking whetherRis Gorenstein if thenth syzygy module of the residue class field ofRhas a non-trivial direct summand of finite G-dimension for somen. It is proved that ifnis at most two then it is true, and moreover, the structure of the ringRis determined essentially uniquely.

On the Artinianness of Generalized Local Cohomology

Let R be a commutative Noetherian local ring, I a proper ideal of R, M, and N finitely generated R-modules. It is proved that f-depth(I + Ann(M), N) is the least integer r such that the generalized local cohomology module is not Artinian. Let r ≥ 0 be an integer. We also discuss the property that is Artinian for all i ≥ r.

Indecomposable modules of large rank over Cohen-Macaulay local rings

A commutative Noetherian local ring (R, m, k) is called Dedekind-like provided R is one-dimensional and reduced, the integral closure R is generated by at most 2 elements as an R-module, and m is the Jacobson radical of R. If M is an indecomposable finitely generated module over a Dedekind-like ring R, and if P is a minimal prime ideal of R, it follows from a classification theorem due to L. Klingler and L. Levy that M p must be free of rank 0, 1 or 2. Now suppose (R, m, k) is a one-dimensional Cohen-Macaulay local ring that is not Dedekind-like, and let P 1 ,... P t be the minimal prime ideals of R. The main theorem in the paper asserts that, for each non-zero t-tuple (n 1 ,... n t ) of non-negative integers, there is an infinite family of pairwise non-isomorphic indecomposable finitely generated R-modules M satisfying MP i ≅ (Rp i ) (n i ) for each i.