Noetherian Ring Research Articles

Totally Acyclic Approximations

Let $$Q \rightarrow R$$ be a surjective homomorphism of Noetherian rings such that Q is Gorenstein and R as a Q-bimodule admits a finite resolution by modules which are projective on both sides. We define an adjoint pair of functors between the homotopy category of totally acyclic R-complexes and that of Q-complexes. This adjoint pair is analogous to the classical adjoint pair of functors between the module categories of R and Q. As a consequence, we obtain a precise notion of approximations of totally acyclic R-complexes by totally acyclic Q-complexes.

Cofiniteness of torsion functors of a pair of cofinite modules

Let I be an ideal of a commutative Noetherian ring R. It is shown that the R-modules are I-cofinite for all if M and N are I-cofinite R-modules and N is of dimension at most 1. Also, it is shown that R-modules are I-weakly cofinite for all if M and N are I-cofinite R-modules, N is of dimension at most 2, and R is a local ring.

𝐾-theory and topological cyclic homology of henselian pairs

Given a henselian pair ( R , I ) (R, I) of commutative rings, we show that the relative K K -theory and relative topological cyclic homology with finite coefficients are identified via the cyclotomic trace K → T C K \to \mathrm {TC} . This yields a generalization of the classical Gabber–Gillet–Thomason–Suslin rigidity theorem (for mod n n coefficients, with n n invertible in R R ) and McCarthy’s theorem on relative K K -theory (when I I is nilpotent). We deduce that the cyclotomic trace is an equivalence in large degrees between p p -adic K K -theory and topological cyclic homology for a large class of p p -adic rings. In addition, we show that K K -theory with finite coefficients satisfies continuity for complete noetherian rings which are F F -finite modulo p p . Our main new ingredient is a basic finiteness property of T C \mathrm {TC} with finite coefficients.

On the Multiplicity and the Cohen–Macaulayness of Fiber Cones of Graded Modules

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

Rado's theorem for rings and modules

We extend classical results of Rado on partition regularity of systems of linear equations with integer coefficients to the case when the coefficient ring is either an arbitrary integral domain or a noetherian ring. In particular, we show that a system of homogeneous linear equations over an infinite integral domain is partition regular if and only if the corresponding matrix satisfies the columns conditions. The crucial idea is to study partition regularity for general modules rather than only for rings. Contrary to previous techniques, our approach is independent of the characteristic of the coefficient ring.

Quasi-cyclic modules and coregular sequences

We develop the theory of coregular sequences and codepth for modules that need not be finitely generated or artinian over a Noetherian ring. We use this theory to give a new version of a theorem of Hellus characterizing set-theoretic complete intersections in terms of local cohomology modules. We also define quasi-cyclic modules as increasing unions of cyclic modules, and show that modules of codepth at least two are quasi-cyclic. We then focus our attention on curves in \({\mathbb {P}}^3\) and give a number of necessary conditions for a curve to be a set-theoretic complete intersection. Thus an example of a curve for which any of these necessary conditions does not hold would provide a negative answer to the still open problem, whether every connected curve in \({\mathbb {P}}^3\) is a set-theoretic complete intersection.

Grothendieck groups, convex cones and maximal Cohen–Macaulay points

Let R be a commutative noetherian ring. Let $$\mathsf {H}(R)$$ be the quotient of the Grothendieck group of finitely generated R-modules by the subgroup generated by pseudo-zero modules. Suppose that the $$\mathbb {R}$$ -vector space $$\mathsf {H}(R)_\mathbb {R}=\mathsf {H}(R)\otimes _\mathbb {Z}\mathbb {R}$$ has finite dimension. Let $$\mathsf {C}(R)$$ (resp. $$\mathsf {C}_r(R)$$ ) be the convex cone in $$\mathsf {H}(R)_\mathbb {R}$$ spanned by maximal Cohen–Macaulay R-modules (resp. maximal Cohen–Macaulay R-modules of rank r). We explore the interior, closure and boundary, and convex polyhedral subcones of $$\mathsf {C}(R)$$ . We provide various equivalent conditions for R to have only finitely many rank r maximal Cohen–Macaulay points in $$\mathsf {C}_r(R)$$ in terms of topological properties of $$\mathsf {C}_r(R)$$ . Finally, we consider maximal Cohen–Macaulay modules of rank one as elements of the divisor class group $${\text {Cl}}(R)$$ .

Local to global principles for generation time over commutative noetherian rings

In the derived category of modules over a commutative noetherian ring a complex $G$ is said to generate a complex $X$ if the latter can be obtained from the former by taking summands and finitely many cones. The number of cones required in this process is the generation time of $X$. In this paper we present some local to global type results for computing this invariant, and discuss applications.

Hearts for commutative Noetherian rings: torsion pairs and derived equivalences

Over a commutative noetherian ring R , the prime spectrum controls, via the assignment of support, the structure of both \mathsf{Mod}(R) and \mathsf{D}(R) . We show that, just like in \mathsf{Mod}(R) , the assignment of support classifies hereditary torsion pairs in the heart of any nondegenerate compactly generated t -structure of \mathsf{D}(R) . Moreover, we investigate whether these t -structures induce derived equivalences, obtaining a new source of Grothendieck categories which are derived equivalent to \mathsf{Mod}(R) .

Filtrations in Module Categories, Derived Categories, and Prime Spectra

AbstractLet $R$ be a commutative noetherian ring. The notion of $n$-wide subcategories of ${\operatorname{\mathsf{Mod}}}\ R$ is introduced and studied in Matsui–Nam–Takahashi–Tri–Yen in relation to the cohomological dimension of a specialization-closed subset of ${\operatorname{Spec}}\ R$. In this paper, we introduce the notions of $n$-coherent subsets of ${\operatorname{Spec}}\ R$ and $n$-uniform subcategories of $\mathsf{D}({\operatorname{\mathsf{Mod}}}\ R)$ and explore their interactions with $n$-wide subcategories of ${\operatorname{\mathsf{Mod}}}\ R$. We obtain a commutative diagram that yields filtrations of subcategories of ${\operatorname{\mathsf{Mod}}}\ R$, $\mathsf{D}({\operatorname{\mathsf{Mod}}}\ R)$ and subsets of ${\operatorname{Spec}}\ R$ and complements classification theorems of subcategories due to Gabriel, Krause, Neeman, Takahashi, and Angeleri Hügel–Marks–Šťovíček–Takahashi–Vitória.

Cohomological dimensions of specialization-closed subsets and subcategories of modules

Let R be a commutative noetherian ring. In this paper, we study specialization-closed subsets of Spec R. More precisely, we first characterize the specialization-closed subsets in terms of various closure properties of subcategories of modules. Then, for each nonnegative integer n we introduce the notion of n-wide subcategories of R-modules to consider the question asking when a given specialization-closed subset has cohomological dimension at most n.

A converse to a construction of Eisenbud–Shamash

Let (Q,𝔫,k) be a commutative local Noetherian ring, f1,…,fc a Q-regular sequence in 𝔫, and R=Q∕(f1,…,fc). Given a complex of finitely generated free R-modules, we give a construction of a complex of finitely generated free Q-modules having the same homology. A key application is when the original complex is an R-free resolution of a finitely generated R-module. In this case our construction is a sort of converse to a construction of Eisenbud and Shamash which yields a free resolution of an R-module M over R given one over Q.

Resolutions of length four which are differential graded algebras

Let $P$ be a commutative Noetherian ring and $F$ be a self-dual acyclic complex of finitely generated free $P$-modules. Assume that $F$ has length four and $F_0$ has rank one. We prove that $F$ can be given the structure of a Differential Graded Algebra with Divided Powers; furthermore, the multiplication on $F$ exhibits Poincar\'e duality. This result is already known if $P$ is a local Gorenstein ring and $F$ is a minimal resolution. The purpose of the present paper is to remove the unnecessary hypotheses that $P$ is local, $P$ is Gorenstein, and $F$ is minimal.

On simple-direct modules

Recently, in a series of papers “simple” versions of direct-injective and direct-projective modules have been investigated. These modules are termed as “simple-direct-injective” and “simple-direct-projective,” respectively. In this paper, we give a complete characterization of the aforementioned modules over the ring of integers and over semilocal rings. The ring is semilocal if and only if every right module with zero Jacobson radical is simple-direct-projective. The rings whose simple-direct-injective right modules are simple-direct-projective are fully characterized. These are exactly the left perfect right H-rings. The rings whose simple-direct-projective right modules are simple-direct-injective are right max-rings. For a commutative Noetherian ring, we prove that simple-direct-projective modules are simple-direct-injective if and only if simple-direct-injective modules are simple-direct-projective if and only if the ring is Artinian. Various closure properties and some classes of modules that are simple-direct-injective (resp. projective) are given.

On the Restricted Minimum Condition for Rings

Generalizing Artinian rings, a ring R is said to have right restricted minimum condition ( $${\mathrm{r.RMC}}$$ , for short) if R/A is an Artinian right R-module for any essential right ideal A of R. It is asked in Jain et al. [Cyclic Modules and the Structure of Rings, Oxford University Press, Oxford, 2012, 3.17 Questions (2)] that (i) Is a left self-injective ring with $${\mathrm{r.RMC}}$$ quasi-Frobenius? (ii) Whether a serial ring with $${\mathrm{r.RMC}}$$ must be Noetherian? We carry out a study of rings with $${\mathrm{r.RMC}}$$ and determine when a right extending ring has $${\mathrm{r.RMC}}$$ in terms of rings $${\begin{bmatrix} S&{}\quad M\\ 0&{}\quad R\end{bmatrix}}$$ such that S is right Artinian, $$M_{Q}$$ is semisimple ( $$Q={\mathrm{Q}}(R)$$ ) and R is a semiprime ring with Krull dimension 1. We proved that a left self-injective ring R with $${\mathrm{r.RMC}}$$ is quasi-Frobenius if and only if $$\hbox {Z}_{r}(R) = \hbox {Z}_{l}(R)$$ if and only if $$\hbox {Z}_{r}(R)$$ is a finitely generated left ideal and $${\mathrm{N}}(R)\cap {\mathrm{Soc}}(R_{R})$$ is a finitely generated right ideal. Right serial rings with $${\mathrm{r.RMC}}$$ are studied and proved that a non-singular serial ring has $${\mathrm{r.RMC}}$$ if and only if it is a left Noetherian ring. Examples are presented to describe our results and to show that $$\mathrm{RMC}$$ is not symmetric for a ring.

Flat ring epimorphisms and universal localizations of commutative rings

Abstract We study different types of localizations of a commutative noetherian ring. More precisely, we provide criteria to decide: (a) if a given flat ring epimorphism is a universal localization in the sense of Cohn and Schofield; and (b) when such universal localizations are classical rings of fractions. In order to find such criteria, we use the theory of support and we analyse the specialization closed subset associated to a flat ring epimorphism. In case the underlying ring is locally factorial or of Krull dimension one, we show that all flat ring epimorphisms are universal localizations. Moreover, it turns out that an answer to the question of when universal localizations are classical depends on the structure of the Picard group. We furthermore discuss the case of normal rings, for which the divisor class group plays an essential role to decide if a given flat ring epimorphism is a universal localization. Finally, we explore several (counter)examples which highlight the necessity of our assumptions.

Residual coordinates over one-dimensional rings

Given a noetherian ring R and n ≥ 2 , it is well-known that residual coordinates of the polynomial algebra R [ n ] are m -stable coordinates for some m ≥ 1 , that is they become coordinates in the larger polynomial algebra R [ n + m ] . In this paper we prove that, over a large class of noetherian one-dimensional rings, m = 1 is enough. This includes affine algebras over an algebraically closed field as well as noetherian complete local rings containing a field.

Superstability, noetherian rings and pure-semisimple rings

We uncover a connection between the model-theoretic notion of superstability and that of noetherian rings and pure-semisimple rings.We characterize noetherian rings via superstability of the class of left modules with embeddings. Theorem 0.1For a ring R the following are equivalent.(1)R is left noetherian.(2)The class of left R-modules with embeddings is superstable.(3)For everyλ≥|R|+ℵ0, there isχ≥λsuch that the class of left R-modules with embeddings has uniqueness of limit models of cardinality χ.(4)Every limit model in the class of left R-modules with embeddings is Σ-injective.We characterize left pure-semisimple rings via superstability of the class of left modules with pure embeddings. Theorem 0.2For a ring R the following are equivalent.(1)R is left pure-semisimple.(2)The class of left R-modules with pure embeddings is superstable.(3)There existsλ≥(|R|+ℵ0)+such that the class of left R-modules with pure embeddings has uniqueness of limit models of cardinality λ.(4)Every limit model in the class of left R-modules with pure embeddings is Σ-pure-injective.Both equivalences provide evidence that the notion of superstability could shed light in the understanding of algebraic concepts.As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.

On the cofiniteness of generalized local cohomology modules

Let $R$ be a commutative Noetherian ring with non-zero identity and $I$ be an ideal of $R$. Let $M$ and $N$ be two finitely generated $R$-modules. In this paper it is shown that, if ${\rm cd}(I,R)\leq 1$ and ${\rm pd_{R}}(M)<\infty$, then $H_{I}^{i}(M,N)$ are $I$-cofinite for each $i\geq 0$. Moreover, it is shown that if $\mbox{q}(I,R) \leq 1$ and ${\rm pd_{R}}(M)<\infty$, then the Bass numbers of generalized local cohomology modules $H_{I}^{i}(M,N)$ are finite for each $i\geq 0$. We also characterize the greatest integer $i$ such that $H_{I}^{i}(M,N)$ is not Artinian and $I$-cofinite.

English

Let R be a Noetherian ring, and I and J be two ideals of R. Let S be a Serre subcategory of the category of R-modules satisfying the condition CI and M be a ZD-module. As a generalization of the S-depth(I, M) and depth(I, J, M), the S-depth of (I, J) on M is defined as $$S{\rm{ - depth}}\left( {I,J,M} \right) = \inf \left\{ {S{\rm{ - depth}}\left( {\mathfrak{a},M} \right): \mathfrak{a}\in \widetilde{W}\left( {I,J} \right)} \right\}$$ , and some properties of this concept are investigated. The relations between S-depth(I, J, M) and H (M) are studied, and it is proved that S-depth(I, J, M) = inf{i: H (M) ∉ S}, where S is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology modules with respect to a pair of ideals coincide with ordinary local cohomology modules under these conditions. Let SuppRH (M) be a finite subset of Max(R) for all i < t, where M is an arbitrary R-module and t is an integer. It is shown that there are distinct maximal ideals $$\mathfrak{m}_1,\mathfrak{m}_2,...\mathfrak{m}_k\in{W(I,J)}$$ such that $$H_{I,J}^i\left( M \right) \cong H_{\mathfrak{m}{_1}}^i\left( M \right) \oplus H_{\mathfrak{m}{_2}}^i\left( M \right) \oplus \ldots \oplus H_{\mathfrak{m}_{_k}}^i\left( M \right)$$ for all i < t.