Noetherian Ring Research Articles

Torsion-simple objects in abelian categories

We introduce the notion of torsion-simple objects in an abelian category: these are the objects which are always either torsion or torsion-free with respect to any torsion pair. We present some general results concerning their properties, and then proceed to investigate the notion in various contexts, such as the category of modules over an artin algebra or a commutative noetherian ring, and the category of quasi-coherent sheaves over the projective line.

Relative injective modules, superstability and noetherian categories

We study classes of modules closed under direct sums, [Formula: see text]-submodules and [Formula: see text]-epimorphic images where [Formula: see text] is either the class of embeddings, RD-embeddings or pure embeddings. We show that the [Formula: see text]-injective modules of theses classes satisfy a Baer-like criterion. In particular, injective modules, RD-injective modules, pure injective modules, flat cotorsion modules and [Formula: see text]-torsion pure injective modules satisfy this criterion. The argument presented is a model theoretic one. We use in an essential way stable independence relations which generalize Shelah’s non-forking to abstract elementary classes. We show that the classical model theoretic notion of superstability is equivalent to the algebraic notion of a noetherian category for these classes. We use this equivalence to characterize noetherian rings, pure semisimple rings, perfect rings and finite products of finite rings and artinian valuation rings via superstability.

Product-complete tilting complexes and Cohen–Macaulay hearts

We show that the cotilting heart associated to a tilting complex T is a locally coherent and locally coperfect Grothendieck category (i.e., an Ind-completion of a small artinian abelian category) if and only if T is product-complete. We then apply this to the specific setting of the derived category of a commutative noetherian ring R . If \dim(R)<\infty , we show that there is a derived duality \mathcal{D}^{\textup{b}}_{\textup{fg}}(R) \cong \mathcal{D}^{\textup{b}}(\mathcal{B})^{\textup{op}} between \mod R and a noetherian abelian category \mathcal{B} if and only if R is a homomorphic image of a Cohen–Macaulay ring. Along the way, we obtain new insights about t-structures in \mathcal{D}^{\textup{b}}_{\textup{fg}}(R) . In the final part, we apply our results to obtain a new characterization of the class of those finite-dimensional noetherian rings that admit a Gorenstein complex.

Effective generic freeness and applications to local cohomology

AbstractLet be a Noetherian domain and be a finitely generated ‐algebra. We study several features regarding the generic freeness over of an ‐module. For an ideal , we show that the local cohomology modules are generically free over under certain settings where is a smooth ‐algebra. By utilizing the theory of Gröbner bases over arbitrary Noetherian rings, we provide an effective method to b make explicit the generic freeness over of a finitely generated ‐module.

Noetherian rings of composite generalized power series

Abstract Let A ⊆ B A\subseteq B be an extension of commutative rings with identity, ( S , ≤ ) \left(S,\le ) a nonzero strictly ordered monoid, and S * = S \ { 0 } {S}^{* }\left=S\backslash \left\{0\right\} . Let A + 〚 B S * , ≤ 〛 = { f ∈ 〚 B S , ≤ 〛 ∣ f ( 0 ) ∈ A } A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] =\{f\in [\kern-2pt[ {B}^{S,\le }]\kern-2pt] \hspace{0.33em}| \hspace{0.33em}f\left(0)\in A\} . In this study, we determine when the ring A + 〚 B S * , ≤ 〛 A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring. We prove that when S S is a strict monoid, if A + 〚 B S * , ≤ 〛 A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring, then A A is a Noetherian ring, B B is a finitely generated A A -module, and S S is finitely generated. We also show that if B B is a finitely generated A A -module over a Noetherian ring A A and ( S , ≤ ) \left(S,\le ) is a positive strictly ordered monoid, which is finitely generated, then A + 〚 B S * , ≤ 〛 A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring.

Cofiniteness of generalized local cohomology modules over local rings

Let [Formula: see text] be a local commutative Noetherian ring and [Formula: see text] an ideal of [Formula: see text] Assume that [Formula: see text] is a finitely generated [Formula: see text]-module of finite projective dimension [Formula: see text] and [Formula: see text] is a finitely generated [Formula: see text]-module of dimension [Formula: see text] We prove that [Formula: see text] is Artinian. Moreover, if [Formula: see text] is finitely generated, then [Formula: see text] is an [Formula: see text]-cofinite [Formula: see text]-module.

Lynch’s conjecture and annihilators of local cohomology modules

Let $R$ be a commutative Noetherian ring and $I$ be an ideal of $R$ with $\cd(I,R)=t\geq 1$. In this paper, we consider the Lynch's conjecture and we obtain a partial answer for this conjecture. More precisely, we show that if $M$ is an $R$-module such that $0\neq H^t_I(M)$ is $I$-cofinite, then $\Ann_RH^t_I(R)\subseteq \p$ for some minimal prime ideal $\p$ of $R$.

Big pure projective modules over commutative noetherian rings: Comparison with the completion

Abstract A module over a ring R is pure projective provided it is isomorphic to a direct summand of a direct sum of finitely presented modules. We develop tools for the classification of pure projective modules over commutative noetherian rings. In particular, for a fixed finitely presented module M, we consider Add ⁡ ( M ) {\operatorname{Add}(M)} , which consists of direct summands of direct sums of copies of M. We are primarily interested in the case where R is a one-dimensional, local domain, and in torsion-free (or Cohen–Macaulay) modules. We show that, even in this case, Add ⁡ ( M ) {\operatorname{Add}(M)} can have an abundance of modules that are not direct sums of finitely generated ones. Our work is based on the fact that such infinitely generated direct summands are all determined by finitely generated data. Namely, idempotent/trace ideals of the endomorphism ring of M and finitely generated projective modules modulo such idempotent ideals. This allows us to extend the classical theory developed to study the behaviour of direct sum decomposition of finitely generated modules comparing with their completion to the infinitely generated case. We study the structure of the monoid V * ⁢ ( M ) {V^{*}(M)} , of isomorphism classes of countably generated modules in Add ⁡ ( M ) {\operatorname{Add}(M)} with the addition induced by the direct sum. We show that V * ⁢ ( M ) {V^{*}(M)} is a submonoid of V * ⁢ ( M ⊗ R R ^ ) {V^{*}(M\otimes_{R}\widehat{R})} , this allows us to make computations with examples and to prove some realization results.

Proxy small thick subcategories of derived categories

Let R be a commutative noetherian ring. Denote by Db(R) the bounded derived category of finitely generated R-modules. Extending the notion of a proxy small object of Db(R) in the sense of Dwyer, Greenlees, Iyengar and Pollitz, we introduce the notion of a proxy small thick subcategory of Db(R). When R is a locally dominant ring, we give a complete classification of the proxy small thick subcategories of Db(R) in terms of pairs of specialization-closed subsets of Spec R and Sing R.

On the Auslander–Bridger–Yoshino theory for complexes of finitely generated projective modules

Let R be a two-sided noetherian ring. Auslander and Bridger developed a theory of projective stabilization of the category of finitely generated R-modules, which is called the stable module theory. Recently, Yoshino established a stable “complex” theory, i.e., a theory of a certain stabilization of the homotopy category of complexes of finitely generated projective R-modules. We introduce higher versions of several notions introduced by Yoshino, such as ⁎torsionfreeness and ⁎reflexivity. Also, we prove the Auslander–Bridger approximation theorem for complexes of finitely generated projective R-modules.

On Noetherian algebras, Schur functors and Hemmer–Nakano dimensions

Important connections in representation theory arise from resolving a finite-dimensional algebra by an endomorphism algebra of a generator-cogenerator with finite global dimension; for instance, Auslander’s correspondence, classical Schur–Weyl duality and Soergel’s Struktursatz. Here, the module category of the resolution and the module category of the algebra being resolved are linked via an exact functor known as the Schur functor. In this paper, we investigate how to measure the quality of the connection between module categories of (projective) Noetherian algebras, B B , and module categories of endomorphism algebras of generator-relative cogenerators over B B which are split quasi-hereditary Noetherian algebras. In particular, we are interested in finding, if it exists, the highest degree n n so that the endomorphism algebra of a generator-cogenerator provides an n n -faithful cover, in the sense of Rouquier, of B B . The degree n n is known as the Hemmer–Nakano dimension of the standard modules. We prove that, in general, the Hemmer–Nakano dimension of standard modules with respect to a Schur functor from a split highest weight category over a field to the module category of a finite-dimensional algebra B B is bounded above by the number of non-isomorphic simple modules of B B . We establish methods for reducing computations of Hemmer–Nakano dimensions in the integral setup to computations of Hemmer–Nakano dimensions over finite-dimensional algebras, and vice-versa. In addition, we extend the framework to study Hemmer–Nakano dimensions of arbitrary resolving subcategories. In this setup, we find that the relative dominant dimension over (projective) Noetherian algebras is an important tool in the computation of these degrees, extending the previous work of Fang and Koenig. In particular, this theory allows us to derive results for Schur algebras and the BGG category O \mathcal {O} in the integral setup from the finite-dimensional case. More precisely, we use the relative dominant dimension of Schur algebras to completely determine the Hemmer–Nakano dimension of standard modules with respect to Schur functors between module categories of Schur algebras over regular Noetherian rings and module categories of group algebras of symmetric groups over regular Noetherian rings. We exhibit several structural properties of deformations of the blocks of the Bernstein-Gelfand-Gelfand category O \mathcal {O} establishing an integral version of Soergel’s Struktursatz. We show that deformations of the combinatorial Soergel’s functor have better homological properties than the classical one.

Locally cohomologically complete intersection ideals and cofiniteness

Let [Formula: see text] be a unitary commutative local Noetherian ring and [Formula: see text] be an ideal. The aim of this paper is twofold: In the first part of this paper, we consider locally cohomologically complete intersection ideals of pure height [Formula: see text]. Next, we deal with cofinite modules. In particular, we investigate conditions for cofiniteness of [Formula: see text] and conditions under which, the modules [Formula: see text] are of finite length for all [Formula: see text] and all finitely generated [Formula: see text]-module [Formula: see text].

Cofiniteness of generalized local cohomology modules

Let [Formula: see text] be an ideal of a commutative noetherian ring [Formula: see text] and [Formula: see text] two [Formula: see text]-modules with [Formula: see text] finitely generated. It is shown that if either [Formula: see text] is an [Formula: see text]-cofinite module of dimension [Formula: see text] for all [Formula: see text], or [Formula: see text] is a principal ideal and [Formula: see text] is finitely generated for all [Formula: see text], or [Formula: see text] is finitely generated and [Formula: see text] for all [Formula: see text], then the [Formula: see text]-module [Formula: see text] is [Formula: see text]-cofinite for all [Formula: see text].

Factorization in the monoid of integrally closed ideals

Given a Noetherian ring A, the collection of integrally closed ideals in A which contain a nonzerodivisor forms a cancellative monoid under the operation I * J = IJ ¯ , the integral closure of the product. The monoid is torsion-free and atomic. Restricting to monomial ideals in a polynomial ring, there is a surjective homomorphism from the Integral Polytope Group onto the Grothendieck group of integrally closed monomial ideals under translation invariance of their Newton Polyhedra. The Integral Polytope Group, the Grothendieck group of polytopes with integer vertices under Minkowski addition and translation invariance, has an explicit basis, allowing for explicit factoring in the monoid.

Simple-separable modules

A module $M$ over a ring is called simple-separable if every simple submodule of $M$ is contained in a finitely generated direct summand of $M$. While a direct sum of any family of simple-separable modules is shown to be always simple-separable, we prove that a direct summand of a simple-separable module does not inherit the property, in general. It is also shown that an injective module $M$ over a right noetherian ring is simple-separable if and only if $M=M_1 \oplus M_2$ such that $M_1$ is separable and $M_2$ has zero socle. The structure of simple-separable abelian groups is completely described.

Dimension of complexes related to special Gorenstein projective precovers (III)

We show that a commutative Noetherian ring R with finite Krull dimension is a Gorenstein ring if and only if Gppd ( X ) = Gpd ( X ) for any complex X of R-modules, where Gpd ( X ) is the Gorenstein projective dimension of X and Gppd ( X ) is the dimension of the complex X related to special Gorenstein projective precovers. In order to do this, the notion of DG-Gorenstein projective resolutions of complexes is introduced and a characterization of Gorenstein projective dimension of complexes is given via DG-Gorenstein projective resolutions.

The local-global principle for the artinianness dimensions

Let R be a commutative noetherian ring and a an ideal of R. The goal of this paper is to establish the local-global principle for the artinianness dimension r a ( M ) , where r a ( M ) is the smallest integer such that the local homology module of M is not artinian. For an artinian R-module M with the set Coass R H r a ( M ) a ( M ) finite, we show that r a ( M ) = inf { r a R p ( Hom R ( R p , M ) ) | p ∈ Spec R } . And the class of all modules N such that Coass R N is finite is studied.

A scheme associated to modules over commutative rings

For any module M over a commutative ring R with identity, we consider Sch ( M ) as a certain class of prime submodules modulo the equivalence relation that two such prime submodules are the same if their colon ideals are equal. We topologize Sch ( M ) and introduce a sheaf O Sch ( M ) of commutative rings on it, which makes ( Sch ( M ) , O Sch ( M ) ) into a scheme. In particular, if M is a module over a Noetherian ring R, then ( Sch ( M ) , O Sch ( M ) ) is a locally Noetherian scheme. Among others, we give sufficient conditions for Sch ( M ) to be an affine scheme.

Hilbert polynomial of length functions

AbstractLet $$\lambda $$ λ be a general length function for modules over a Noetherian ring R. We use $$\lambda $$ λ to introduce Hilbert series and polynomials for R[X]-modules, measuring the growth rate of $$\lambda $$ λ . We show that the leading term $$\mu $$ μ of the Hilbert polynomial is an invariant of the module, which refines both the algebraic entropy and the receptive algebraic entropy; its degree is a suitable notion of dimension for R[X]-modules. Similar to algebraic entropy, $$\mu $$ μ in general is not additive for exact sequences of R[X]-modules: we demonstrate how to adapt certain entropy constructions to this new invariant. We also consider multi-variate versions of the Hilbert polynomial.

Asymptotic behavior of homological invariants of localizations of modules

Let R be a commutative noetherian ring, I an ideal of R, and M a finitely generated R-module. We consider the asymptotic injective dimensions, projective dimensions, Bass numbers, and Betti numbers of localizations of M/InM at prime ideals of R and prove that these invariants are stable or have polynomial growth for large integers n that do not depend on the prime ideals.