Commutative Noetherian Local Ring Research Articles

Ascent and descent of Artinian module structures under flat base changes

Let φ : R → S be a flat local homomorphism between commutative Noetherian local rings. In this paper, the ascent and descent of Artinian module structures between R and S are investigated. For an Artinian R-module A, the structure of A ⊗ R S is described. As an application, the Artinianess of certain local cohomology modules is clarified.

Homological dimensions of Burch ideals, submodules and quotients

The notion of Burch ideals and Burch submodules were introduced (and studied) by Dao-Kobayashi-Takahashi in 2020 and Dey-Kobayashi in 2022 respectively. The aim of this article is to characterize various local rings in terms of homological invariants of Burch ideals, Burch submodules, or that of the corresponding quotients. Specific applications of our results include the following: Let (R,m) be a commutative Noetherian local ring. Let M=I be an integrally closed ideal of R such that depth(R/I)=0, or M=mN≠0 for some submodule N of a finitely generated R-module L such that either depth(N)⩾1 or L is free. It is shown that: (1) I has maximal projective (resp., injective) complexity and curvature. (2) R is Gorenstein if and only if ExtRn(M,R)=0 for any three consecutive values of n⩾max⁡{depth(R)−1,0}. (3) R is CM (Cohen-Macaulay) if and only if CM-dimR(M) is finite.

On the reducing projective dimension over local rings

Abstract In this paper, we are concerned with certain invariants of modules, called reducing invariants, which have been recently introduced and studied by Araya–Celikbas and Araya–Takahashi. We raise the question whether the residue field of each commutative Noetherian local ring has finite reducing projective dimension and obtain an affirmative answer for the question for a large class of local rings. Furthermore, we construct new examples of modules of infinite projective dimension that have finite reducing projective dimension and study several fundamental properties of reducing dimensions, especially properties under local homomorphisms of local rings.

Coartinianness of extension and torsion functors

Let be a commutative noetherian local ring with -adic topology, I an ideal of R. We investigate coartinianness of and , show that the R-module is I-coartinian if M is a linearly compact I-coartinian R-module and N is an I-cofinite R-module of dimension at most 1; the R-module is I-coartinian in the case M is semi-discrete linearly compact I-coartinian and N is finitely generated with dimension at most 2.

Modules with finite reducing Gorenstein dimension

If M is a nonzero finitely generated module over a commutative Noetherian local ring R such that M has finite injective dimension and finite Gorenstein dimension, then it follows from a result of Holm that M has finite projective dimension, and hence a result of Foxby implies that R is Gorenstein. We prove that the same conclusion holds for certain nonzero finitely generated modules that have finite injective dimension and finite reducing Gorenstein dimension, where the reducing Gorenstein dimension is a finer invariant than the classical Gorenstein dimension, in general. Along the way, we also prove new results, independent of the reducing dimensions, concerning modules of finite Gorenstein dimension.

Exceptional complete intersection maps of local rings

This work concerns surjective maps $\varphi\colon R\to S$ of commutative noetherian local rings with kernel generated by a regular sequence that is part of a minimal generating set for the maximal ideal of $R$. The main result provides criteria for detecting such exceptional complete intersection maps in terms of the lattices of thick subcategories of the derived category of complexes of finite length homology. A key input is a characterization of such maps in terms of the truncated Atiyah class of $\varphi$.

Multiplicities and Betti numbers in local algebra via lim Ulrich points

This work concerns finite free complexes with finite length homology over a commutative noetherian local ring $R$. The focus is on complexes that have length $\mathrm{dim}\, R$, which is the smallest possible value, and in particular on free resolutions of modules of finite length and finite projective dimension. Lower bounds are obtained on the Euler characteristic of such short complexes when $R$ is a strict complete intersection, and also on the Dutta multiplicity, when $R$ is the localization at its maximal ideal of a standard graded algebra over a field of positive prime characteristic. The key idea in the proof is the construction of a suitable Ulrich module, or, in the latter case, a sequence of modules that have the Ulrich property asymptotically, and with good convergence properties in the rational Grothendieck group of $R$. Such a sequence is obtained by constructing an appropriate sequence of sheaves on the associated projective variety.

Approximately Cohen-Macaulay modules

Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. There is a variety of nice results about approximately Cohen-Macaulay rings. These results were done by Goto. In this paper we prove some these results for modules and generalize the concept of approximately Cohen-Macaulay rings to approximately Cohen-Macaulay modules. It is seen that when $M$ is an approximately Cohen-Macaulay module, for any proper ideal $I$ we have $grade(I,M) \geq \dim_R M -\dim_R M/IM -1$. Specially when $M$ is $R$ itself, we obtain an interval for $grade(I,R)$. We also give a definition for these modules in case that $R$ is not necessarily local and show that approximately Cohen-Macaulay modules are in close relationship with perfect modules. Finally we consider the behaviour of these modules under faithful flat extensions.

On the depth and reflexivity of tensor products

In this paper we study the depth of tensor products of homologically finite complexes over commutative Noetherian local rings. As an application of our main result, we determine new conditions under which the factors of a nonzero reflexive tensor product of finitely generated modules over hypersurface rings can be reflexive.A result of Asgharzadeh shows that nonzero symbolic powers of prime ideals in a local ring cannot have finite projective dimension, unless the ring in question is a domain. We make use of this fact in the appendix and consider the reflexivity of tensor products of prime ideals over hypersurface rings.

Dominant Local Rings and Subcategory Classification

AbstractWe introduce a new notion of commutative noetherian local rings, which we call dominant. We explore fundamental properties of dominant local rings and compare them with other local rings. We also provide several methods to get a new dominant local ring from a given one. Finally, we classify resolving subcategories of the module category $\operatorname {\textsf {mod}} R$ and thick subcategories of the derived category $\textsf {D}^{b}(R)$ and the singularity category $\textsf {D}^{sg}(R)$ for a local ring $R$ whose certain localizations are dominant local rings. Our results recover and refine all the known classification theorems described in this context.

An Extension of a Depth Inequality of Auslander

In this paper, we consider a depth inequality of Auslander which holds for finitely generated Tor-rigid modules over commutative Noetherian local rings. We raise the question of whether such a depth inequality can be extended for $n$-Tor-rigid modules, and obtain an affirmative answer for $2$-Tor-rigid modules that are generically free. Furthermore, in the appendix, we use Dao's eta function and determine new classes of Tor-rigid modules over hypersurfaces that are quotient of unramified regular local rings.

A FREENESS CRITERION WITHOUT PATCHING FOR MODULES OVER LOCAL RINGS

AbstractIt is proved that if $\varphi \colon A\to B$ is a local homomorphism of commutative noetherian local rings, a nonzero finitely generated B-module N whose flat dimension over A is at most $\operatorname {edim} A - \operatorname {edim} B$ is free over B and $\varphi $ is a special type of complete intersection. This result is motivated by a ‘patching method’ developed by Taylor and Wiles and a conjecture of de Smit, proved by the first author, dealing with the special case when N is flat over A.

Vasconcelos’ conjecture on the conormal module

For any ideal I of finite projective dimension in a commutative noetherian local ring R, we prove that if the conormal module $$I/I^2$$ has finite projective dimension over R/I, then I must be generated by a regular sequence. This resolves a conjecture of Vasconcelos. We prove a similar result for the first Koszul homology module of I. When R is a localisation of a polynomial ring over a field K of characteristic zero, Vasconcelos conjectured that R/I is a reduced complete intersection if the module $$\Omega _{(R/I)/K}$$ of Kähler differentials has finite projective dimension; we prove this contingent on the Eisenbud-Mazur conjecture. The arguments exploit the structure of the homotopy Lie algebra associated to I in an essential way. By work of Avramov and Halperin, if every degree 2 element of the homotopy Lie algebra is radical, then I is generated by a regular sequence. Iyengar has shown that free summands of $$I/I^2$$ give rise to central elements of the homotopy Lie algebra, and we establish an analogous criterion for constructing radical elements, from which we deduce our main result.

$C$-CANONICAL MODULES

Let $C$ be a semidualizing module over a commutative Noetherian local ring $R$. In this paper we introduce a new class of modules, namely $C$-canonical modules which are a generalization of canonical modules. It is shown that if the canonical module exists then the $C$-canonical module exists and the converse holds under special conditions. Also, a new characterization of Gorenstein local rings is given via $C$-canonical modules.

Finiteness properties of extension functors of cominimax modules

Let be a complete commutative Noetherian local ring, I an ideal of R, M an R-module (not necessarily I-torsion) and N a finitely generated R-module with . It is shown that if M is I-ETH-cominimax (i.e. is minimax (or Matlis reflexive), for all ) and or more generally , then the R-module is finitely generated, for all . As an application to local cohomology, let be a system of ideals of R and , if (e.g., ) for all , then the R-modules are finitely generated, for all and . Similar results are true for local cohomology defined by a pair of ideals and ordinary local cohomology modules.

Cohomological supports over derived complete intersections and local rings

A theory of cohomological support for pairs of DG modules over a Koszul complex is investigated. These specialize to the support varieties of Avramov and Buchweitz defined over a complete intersection ring, as well as support varieties over an exterior algebra. The main objects of study are certain DG modules over a polynomial ring; these determine the aforementioned cohomological supports and are shown to encode (co)homological information about pairs of DG modules over a Koszul complex. The perspective in this article leads to new proofs of well-known results for pairs of complexes over a complete intersection. Furthermore, these cohomological supports are used to define a support theory for pairs of objects in the derived category of an arbitrary commutative noetherian local ring. Finally, we calculate several examples; one of which answers a question of D. Jorgensen in the negative.

Resolving Subcategories Closed Under Certain Operations and a Conjecture of Dao and Takahashi

Let R be a commutative Noetherian local ring with residue field k. Let X be a resolving subcategory of finitely generated R-modules. This paper mainly studies when X contains k or consists of totally reflexive modules. It is proved that X does so if X is closed under cosyzygies. A conjecture of Dao and Takahashi is also shown to hold in several cases.

Little dimension and the improved new intersection theorem

Let $R$ be a commutative noetherian local ring. We define a new invariant for $R$-modules which we call the little dimension. Using it, we extend the improved new intersection theorem.

POWERS OF THE MAXIMAL IDEAL AND VANISHING OF (CO)HOMOLOGY

AbstractWe prove that each positive power of the maximal ideal of a commutative Noetherian local ring is Tor-rigid and strongly rigid. This gives new characterizations of regularity and, in particular, shows that such ideals satisfy the torsion condition of a long-standing conjecture of Huneke and Wiegand.

Specifying the Auslander transpose in submodule category and its applications

Let (R,m) be a d-dimensional commutative Noetherian local ring. Let M denote the morphism category of finitely generated R-modules, and let S be the full subcategory of M consisting of monomorphisms, known as the submodule category. This article reveals that the Auslander transpose in the category S can be described explicitly within modR, the category of finitely generated R-modules. This result is exploited to study the linkage theory as well as the Auslander–Reiten theory in S. In addition, motivated by a result of Ringel and Schmidmeier, we show that the Auslander–Reiten translations in the subcategories H and G, consisting of all morphisms which are maximal Cohen–Macaulay R-modules and Gorenstein projective morphisms, respectively, may be computed within modR via G-covers. The corresponding result for the subcategory of epimorphisms in H is also obtained.