Complete Noetherian Local Ring Research Articles

Cohen–Macaulay Noetherian algebras

Let (R,m) be a d-dimensional commutative complete Noetherian local ring and Λ be a Noetherian R-algebra. Motivated by the notion of Cohen–Macaulay Artin algebras of Auslander and Reiten, we say that Λ is Cohen–Macaulay if there is a finitely generated Λ-bimodule ω that is maximal Cohen–Macaulay over R such that the adjoint pair of functors (ω⊗Λ′−,HomΛ′(ω,−)) induces quasi-inverse equivalences between the full subcategories of finitely generated Λ′-modules consisting of modules of finite projective dimension, P∞(Λ′), and the modules of finite injective dimension, I∞(Λ′), whenever Λ′=Λ,Λop. It is proved that such a module ω is unique, up to isomorphism, as a Λ′-module. It is also shown that Λ is a Cohen–Macaulay algebra if and only if there is a semidualizing Λ-bimodule ω of finite injective dimension and P∞(Λ′) and I∞(Λ′) are contained in the Auslander and Bass classes, respectively. We prove that Cohen–Macaulayness behaves well under reduction modulo system of parameters of R. Indeed, it will be observed that if Λ is a Cohen–Macaulay algebra, then for any system of parameters x=x1,…,xd of R, the Artin algebra Λ∕xΛ is Cohen–Macaulay as well. Assume that ω is a semidualizing Λ-bimodule of finite injective dimension that is maximal Cohen–Macaulay as an R-module. It will turn out that Λ being a Cohen–Macaulay algebra is equivalent to saying that the pair (CM(Λ′),I∞(Λ′)) forms a hereditary complete cotorsion theory and the pair (CM(Λ′op),P∞(Λ′)) forms a Tor-torsion theory, where CM(Λ′) is the class of all finitely generated Λ′-modules admitting a right resolution by modules in addω. Finally, it is shown that Cohen–Macaulayness ascends from R to RΓ and RQ, where Γ is a finite group and Q is a finite acyclic quiver.

A note on cofinite modules over Noetherian complete local rings

Let $(R,{\frak m})$ be a commutative Noetherian complete local ring and let $I$ be a proper ideal of $R$ such that the category of all $I$-cofinite $R$-modules is an Abelian subcategory of the category of all $R$-modules. Then for each $I$-cofinite $R$-module $M$ it is shown that $\dim R/(I+{\rm Ann}_R M)=\dim M$. As a consequence of this result it is shown that if $J$ is an ideal of $R$ such that for each ${\frak p}\in {\rm mAss}_R R$, $\dim R/(J+{\frak p})\leq 1$ or ${\rm cd}(J,R/{\frak p})\leq 1$, then $\dim R/(J+{\rm Ann}_R M)=\dim M$, for each $J$-cofinite $R$-module $M$.

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.

KUMMER COVERINGS AND SPECIALISATION

AbstractWe prove versions of various classical results on specialisation of fundamental groups in the context of log schemes in the sense of Fontaine and Illusie, generalising earlier results of Hoshi, Lepage and Orgogozo. The key technical result relates the category of finite Kummer étale covers of an fs log scheme over a complete Noetherian local ring to the Kummer étale coverings of its reduction.

Local cohomology modules and their properties

UDC 512.5 Let be a complete Noetherian local ring and let be a generalized Cohen-Macaulay -module of dimension We show thatwhere and is the ideal transform functor. Also, assuming that is a proper ideal of a local ring , we obtain some results on the finiteness of Bass numbers, cofinitness, and cominimaxness of local cohomology modules with respect to

Artinian local cohomology modules of cofinie modules

Let [Formula: see text] be a commutative Noetherian complete local ring and [Formula: see text] be a proper ideal of [Formula: see text]. Suppose that [Formula: see text] is a nonzero [Formula: see text]-cofinite [Formula: see text]-module of Krull dimension [Formula: see text]. In this paper, it shown that [Formula: see text] As an application of this result, it is shown that [Formula: see text], for each [Formula: see text] Also it shown that for each [Formula: see text] the submodule [Formula: see text] and [Formula: see text] of [Formula: see text] is [Formula: see text]-cofinite, [Formula: see text] and [Formula: see text] whenever the category of all [Formula: see text]-cofinite [Formula: see text]-modules is an Abelian subcategory of the category of all [Formula: see text]-modules. Also some applications of these results will be included.

The first Brauer–Thrall conjecture over one-dimensional local rings

Let [Formula: see text] be a one-dimensional commutative Noetherian complete local ring. Assume that the cohomology annihilator [Formula: see text] is [Formula: see text]-primary. We use the notion of Gabriel–Roiter (co)measure in the category of maximal Cohen–Macaulay [Formula: see text]-modules and prove that, if there is an infinite set [Formula: see text] of indecomposable maximal Cohen–Macaulay [Formula: see text]-modules of bounded multiplicity, then there are indecomposable maximal Cohen–Macaulay modules of arbitrary large multiplicity which are cogenerated by modules in [Formula: see text]. This, in particular, guarantees the validity of the first Brauer–Thrall-type theorem for the category of maximal Cohen–Macaulay [Formula: see text]-modules.

Attached primes and annihilators of top local cohomology modules defined by a pair of ideals

Assume that \(R\) is a complete Noetherian local ring and \(M\) is a non-zero finitely generated \(R\)-module of dimension \(n=\dim(M)\geq 1\). It is shown that any non-empty subset \(T\) of \(\mathrm{Assh}(M)\) can be expressed as the set of attached primes of the top local cohomology modules \(H_{I,J}^n(M)\) for some proper ideals \(I,J\) of \(R\). Moreover, for ideals \(I, J=\bigcap_ {\mathfrak p\in \mathrm{Att}_R(H_{I}^n(M))}\mathfrak p\) and \(J'\) of \(R\) it is proved that \(T=\mathrm{Att}_R(H_{I,J}^n(M))=\mathrm{Att}_R(H_{I,J'}^n(M))\) if and only if \(J'\subseteq J\). Let \(H_{I,J}^n(M)\neq 0\). It is shown that there exists \(Q\in \mathrm{Supp}(M)\) such that \(\dim(R/Q)=1\) and \(H_Q^n(R/{\mathfrak p})\neq 0\), for each \(\mathfrak p \in \mathrm{Att}_R(H_{I,J}^n(M))\). In addition, we prove that if \(I\) and \(J\) are two proper ideals of a Noetherian local ring \(R\), then \(\mathrm{Ann}_R(H_{I,J}^{n}(M))=\mathrm{Ann}_R(M/{T_R(I,J,M)})\), where \(T_R(I,J,M)\) is the largest submodule of \(M\) with \(\mathrm{cd}(I,J,T_R(I,J,M))<\mathrm{cd}(I,J,M)\), here \(\mathrm{cd}(I,J,M)\) is the cohomological dimension of \(M\) with respect to \(I\) and \(J\). This result is a generalization of [1, Theorem 2.3] and [2, Theorem 2.6].

On the existence of maximal Cohen-Macaulay modules over Noetherian complete local rings

Let be a Noetherian complete local ring of dimension d and be a system of parameters for R. Assume that M is a big Cohen-Macaulay module with respect to In this paper it is shown that if the R-module is of finite length then is a maximal Cohen-Macaulay R-module, where D denotes the Matlis dual functor.

Cofiniteness over Noetherian complete local rings

In this article, we prove the following generalization of a result of Hartshorne: Let be a regular local ring of dimension 4. Assume that is a regular system of parameters for S and . Then for each finitely generated S-module N with the socle of is infinite dimensional. Also, using this result, for any commutative Noetherian complete local ring , we characterize the class of all ideals I of R with the property that, for every finitely generated R-module M, the local cohomology modules are I-cofinite for all .

On a question of D. Rees on classical integral closure and integral closure relative to an Artinian module

Let [Formula: see text] be a commutative Noetherian complete local ring and [Formula: see text] and [Formula: see text] ideals of [Formula: see text]. Motivated by a question of Rees, we study the relationship between [Formula: see text], the classical Northcott–Rees integral closure of [Formula: see text], and [Formula: see text], the integral closure of [Formula: see text] relative to an Artinian [Formula: see text]-module [Formula: see text] (also called here ST-closure of [Formula: see text] on [Formula: see text]), in order to study a relation between [Formula: see text], the multiplicity of [Formula: see text], and [Formula: see text], the multiplicity of [Formula: see text] relative to an Artinian [Formula: see text]-module [Formula: see text]. We conclude [Formula: see text] when every minimal prime ideal of [Formula: see text] belongs to the set of attached primes of [Formula: see text]. As an application, we show what happens when [Formula: see text] is a generalized local cohomology module.

Resolution of singularities of arithmetical threefolds

We prove Grothendieck's conjecture on Resolution of Singularities for quasi-excellent schemes X of dimension three and of arbitrary characteristic. This applies in particular to X=SpecA, A a reduced complete Noetherian local ring of dimension three and to algebraic or arithmetical varieties of dimension three. Similarly, if F is a number field, a complete discretely valued field or more generally the quotient field of any excellent Dedekind domain O, any regular projective surface X/F has a proper and flat model X over O which is everywhere regular.

Classifications of exact structures and Cohen–Macaulay-finite algebras

We give a classification of all exact structures on a given idempotent complete additive category. Using this, we investigate the structure of an exact category with finitely many indecomposables. We show that the relation of the Grothendieck group of such a category is generated by AR conflations. Moreover, we obtain an explicit classification of (1) Gorenstein-projective-finite Iwanaga–Gorenstein algebras, (2) Cohen–Macaulay-finite orders, and more generally, (3) cotilting modules U with U⊥ of finite type. In the appendix, we develop the AR theory of exact categories over a noetherian complete local ring, and relate the existence of AR conflations to the AR duality and dualizing varieties.

DISPLAYED EQUATIONS FOR GALOIS REPRESENTATIONS

The Galois representation associated to a $p$ -divisible group over a normal complete noetherian local ring with perfect residue field is described in terms of its Dieudonné display. As a consequence, the Kisin module associated to a commutative finite flat $p$ -group scheme via Dieudonné displays is related to its Galois representation in the expected way.

Cohomological dimension, cofiniteness and Abelian categories of cofinite modules

Let R be a commutative Noetherian ring, I⊆J be ideals of R and M be a finitely generated R-module. In this paper it is shown that q(J,M)≤q(I,M)+cd(J,M/IM). Furthermore, it is shown that, for any ideal I of R and any finitely generated R-module M with q(I,M)≤1, the local cohomology modules HIi(M) are I-cofinite for all integers i≥0. As a consequence of this result it is shown that, if q(I,R)≤1, then for any finitely generated R-module M, the local cohomology modules HIi(M) are I-cofinite for all integers i≥0. Finally, it is shown that the category of all I-cofinite R-modules C(R,I)cof is an Abelian subcategory of the category of all R-modules, whenever (R,m) is a complete Noetherian local ring and I is an ideal of R with q(I,R)≤1. These assertions answer affirmatively two questions raised by R. Hartshorne in [16], in the some special cases.

Some results on top local cohomology and top formal local cohomology modules

ABSTRACTLet (R,𝔪) be a complete noetherian local ring and M a finitely generated R-module of dimension n and 𝔞 an ideal of R.At first, we find a relation between top local cohomology and top formal local cohomology modules. In fact, we show that there exists an ideal 𝔟 such that and there exists an ideal 𝔠 such that . Several applications of this result are given. Among other things, it is shown that and .

Annihilators of Local Cohomology Modules

Let (R, 𝔪) be a commutative Noetherian complete local ring, M a nonzero finitely generated R-module of dimension n, and I be an ideal of R. In this paper we calculate the annihilator of the top local cohomology module . Also, if (R, 𝔪) is a Noetherian local Cohen–Macaulay ring of dimension d and I is a nonzero proper ideal of R, then we calculate the annihilator of the first nonzero local cohomology module . Finally, we show that if R is an arbitrary Noetherian ring, I an ideal of R, and M is a nonzero finitely generated R-module with cd(I, M) = t ≥ 0, then there exists a submodule N of M such that . This is a generalization of the main result of Bahmanpour, A'zami, and Ghasemi [1] for all ideals of an arbitrary Noetherian ring R.

EXTENSION FUNCTORS OF LOCAL COHOMOLOGY MODULES AND SERRE CATEGORIES OF MODULES

Let $(R,m)$ be a complete Noetherian local ring, $I$ a proper ideal of R and $M$, $N$ two finitely generated R-modules such that Supp $(N)\subseteq V(I)$. Let $t\geq 0$ be an integer such that for each $0\leq i\leq t$, the $R$-module $H^i_I(M)$ is in dimension $<n$. Then we show that each element $L$ of the set $\mathfrak{J}$, which is defined as: $$ \{{Ext}^j_R(N,H^i_I(M)):j\ge0 \ and \ 0\le i \le t\} \\ \cup\{{Hom}_R(N,H^{t+1}_I(M)),{Ext}^1_R(N,H^{t+1}_I(M))\} $$ is in dimension $<n-2$ and so as a consequence, it follows that the set $$ {Ass}_R(L)\cap\{{\frak{p}} \in Spec(R): :\,\dim (R/{\frak{p}})\ge n − 2\} $$ is finite. In particular, the set $$ {Ass}_R(\oplus_{i=0}^{t+1}H^i_I(R))\cap \{{\frak{p}}\in \,{Spec}(R)\,:\,\dim(R/{\frak{p}})\geq n-2\} $$ is finite. Also, as an immediately consequence of this result it follows that the $R$-modules ${Ext}^j_R(N,H^i_I(M))$ are in dimension $<n-1$, for all integers $i,j\geq 0$, whenever $\dim(M/IM)=n$. These results generalizes the main results of Huneke-Koh [17], Delfino [10], Chiriacescu [9], Asadollahi-Naghipour [1], Quy [18], Brodmann-Lashgari [7], Bahmanpour-Naghipour [5] and Bahmanpour et al. [6] in thecase of complete local rings.

EXTENSION FUNCTORS OF LOCAL COHOMOLOGY MODULES AND SERRE CATEGORIES OF MODULES

Let $(R,m)$ be a complete Noetherian local ring, $I$ a proper ideal of R and $M$, $N$ two finitely generated R-modules such that Supp $(N)\subseteq V(I)$. Let $t\geq 0$ be an integer such that for each $0\leq i\leq t$, the $R$-module $H^i_I(M)$ is in dimension $<n$. Then we show that each element $L$ of the set $\mathfrak{J}$, which is defined as: $$ \{{Ext}^j_R(N,H^i_I(M)):j\ge0 and 0\le i \le t\} \cup\{{Hom}_R(N,H^{t+1}_I(M)),{Ext}^1_R(N,H^{t+1}_I(M))\} $$ is in dimension $<n-2$ and so as a consequence, it follows that the set $$ {Ass}_R(L)\cap\{{\frak{p}} \in Spec(R): :\,\dim (R/{\frak{p}})\ge n − 2\} $$ is finite. In particular, the set $$ {Ass}_R(\oplus_{i=0}^{t+1}H^i_I(R))\cap \{{\frak{p}}\in \,{Spec}(R)\,:\,\dim(R/{\frak{p}})\geq n-2\} $$ is finite. Also, as an immediately consequence of this result it follows that the $R$-modules ${Ext}^j_R(N,H^i_I(M))$ are in dimension $<n-1$, for all integers $i,j\geq 0$, whenever $\dim(M/IM)=n$. These results generalizes the main results of Huneke-Koh [17], Delfino [10], Chiriacescu [9], Asadollahi-Naghipour [1], Quy [18], Brodmann-Lashgari [7], Bahmanpour-Naghipour [5] and Bahmanpour et al. [6] in thecase of complete local rings.

Poincaré series for curve singularities and its behaviour under projections

Let O be an equicharacteristic reduced complete noetherian local ring of Krull dimension one, and let S be the value semigroup associated with O. The aim of the paper is to investigate the behaviour of the multi-variable Poincaré series associated to S with respect to the property of “forgetting variables”. We prove that, for O Gorenstein, the Poincaré series with one less variable can be explicitly computed in terms of the original series; this provides also a shorter and pure arithmetical way to show that the Poincaré series is a complete invariant of the equisingularity. Moreover we express (without the Gorenstein assumption) the Hilbert series of S in terms of the Poincaré series of the unions of irreducible components of the singularity.