Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let \fa be an ideal of R and \fM denote the intersection of all prime ideals \fp in Supp_RH^d_{\fa}(M). It is shown that H^d_{\fa}(M)\simeq H^d_{\fM}(M)/\displaystyle{\sum_{n\in \mathbb{N}}}<\fM>(0:_{H^d_{\fM}(M)}\fa^n), where for an Artinian R-module A we put <\fM>A=\cap_{n\in \mathbb{N}} \fM^nA. As a consequence, it is proved that for all ideals \fa of R, there are only finitely many non-isomorphic top local cohomology modules H^d_{\fa}(M) having the same support. In addition, we establish an analogue of the Lichtenbaum-Hartshorne Vanishing Theorem over rings that need not be local.
We study rings which have Noetherian cohomology under the action of a ring of cohomology operators. The main result is a criterion for a complex of modules over such a ring to have finite injective dimension. This criterion generalizes, by removing finiteness conditions, and unifies several previous results. In particular we show that for a module over a ring with Noetherian cohomology, if all higher self-extensions of the module vanish then it must have finite injective dimension. Examples of rings with Noetherian cohomology include commutative complete intersection rings and finite dimension…
Let $R$ be a Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. In this article, we examine the question of whether an arbitrary top local cohomology module, $\operatorname{H}^{\operatorname{cd}(I,M)}_I(M)$, is Artinian, or not. Several results related to this question are obtained; in particular, we prove that over a Noetherian local unique factorization domain $R$ of dimension three, for a finitely generated faithful module $M$, a top local cohomology module is Artinian if and only if $\operatorname{cd}(I,M)= 3$.
Let $R$ denote a commutative Noetherian (not necessarily local) ring, $\frak a$ an ideal of $R$ and $M$ a finitely generated $R$-module. The purpose of this paper is to show that $f^n_{\frak a}(M)=\inf \{0\leq i\in\mathbb{Z}|\, \dim H^{i}_{\frak a}(M)/N \geq n\, \, \text{for any finitely generated submodule}\,\, N \subseteq H^{i}_{\frak a}(M)\}$, where $n$ is a non-negative integer and the invariant $f^n_{\frak a}(M):=\inf\{f_{\frak a R_{\frak p}}(M_{\frak p})\,\,|\,\,{\frak p}\in \Supp M/\frak a M\,\,{\rm and}\,\,\dim R/{\frak p}\geq n\}$ is the $n$-th finiteness dimension of $M$ relative to …
We examine local cohomology in the setting of valuation rings. The novelty of this investigation stems from the fact that valuation rings are usually non-Noetherian, whereas local cohomology has been extensively developed mostly in a Noetherian setting. We prove various vanishing results on local cohomology for valuation rings of finite Krull dimension. These vanishing results stem from a uniform bound on the global dimension of such rings. Our investigation reveals differences in the sheaf theoretic definition of local cohomology, and the algebraic definition in terms of a limit of certain Ex…
We show how subintegral extensions of certain local Noetherian domains $S$ can be constructed with specified invariants including reduction number, Hilbert function, multiplicity and local cohomology. The construction behaves analytically like Nagata idealization but rather than a ring extension of $S$, it produces a subring $R$ of $S$ such that $R \subseteq S$ is subintegral.
We define and characterise small support for complexes over non-Noetherian rings and in this context prove a vanishing theorem for modules. Our definition of support makes sense for any rigidly compactly generated tensor triangulated category. Working in this generality, we establish basic properties of support and investigate when it detects vanishing. We use pointless topology to relate support, the topology of the Balmer spectrum, and the structure of the idempotent Bousfield lattice.
Let $\fa$ be an ideal of a $d$-dimensional commutative Noetherian ring $R$. In this paper we give some information on some last non-zero local cohomology modules known as top local cohomology modules in particular, $H^{d-1}_{\fa}(R)$.
Let $R$ be a Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module with $\operatorname{cd}(I,M)=c$. In this article, we first show that there exists a descending chain of ideals $I=I_c\supsetneq I_{c-1}\supsetneq \cdots \supsetneq I_0$ of $R$ such that for each $0\leq i\leq c-1$, $\operatorname{cd}(I_i,M)=i$ and that the top local cohomology module $\operatorname{H}^i_{I_i}(M)$ is not Artinian. We then give sufficient conditions for a non-negative integer $t$ to be a lower bound for $\operatorname{cd}(I,M)$ and use this to conclude that in non-catenary Noetherian local integral domains, t…
Let $\mathfrak{a}$ be an ideal of Noetherian ring $R$ and let $M$ be an $R$-module such that $\mathrm{Ext}^i_R(R/\mathfrak{a},M)$ is finite $R$-module for every $i$. If $s$ is the first integer such that the local cohomology module $\mathrm{H}^s_\mathfrak{a}(M)$ is non $\mathfrak{a}$-cofinite, then we show that $\mathrm{Hom}_{R}(R/\mathfrak{a}, \mathrm{H}^s_\mathfrak{a}(M))$ is finite. Specially, the set of associated primes of $\mathrm{H}^s_\mathfrak{a}(M)$ is finite. Next assume $(R,\mathfrak{m})$ is a local Noetherian ring and $M$ is a finitely generated module. We study the last integer $n…
For a flat commutative $k$-algebra $A$ such that the enveloping algebra $A\otimes_k A$ is noetherian, given a finitely generated bimodule $M$, we show that the adic completion of the Hochschild cohomology module $HH^n(A/k,M)$ is naturally isomorphic to $HH^n(\widehat{A}/k,\widehat{M})$. To show this, we (1) make a detailed study of derived completion as a functor $D(A) \to D(\widehat{A})$ over a non-noetherian ring $A$; (2) prove a flat base change result for weakly proregular ideals; and (3) Prove that Hochschild cohomology and analytic Hochschild cohomology of complete noetherian local rings…
Let R be a commutative Noetherian ring. We introduce a theory of formal local cohomology for complexes of R-modules. As an application, we establish some relations between formal local cohomology, local homology, local cohomology and local Tate cohomology through some natural isomorphisms. We investigate vanishing of formal local cohomology modules. Also, we give a characterization of Cohen-Macaulay complexes.
Let $R$ be a commutative Noetherian ring and $M$ be a finitely generated $R$-module. Considering the new concept of linkage of ideals over a module, we study associated prime ideals, cofiniteness and Artinianness of local cohomology modules of $M$ with respect to some linked ideals over it.
In this paper, some results on vanishing and non-vanishing of generalized local cohomology modules are presented and some relations between those modules and, Ext and ordinary local cohomology modules are studied. Also, several cofiniteness propositions for generalized local cohomology modules are established which, among other things, provide an alternative answer to a question in [Y2].
Local cohomology modules, even over a Noetherian ring $R$, are typically unwieldly. As such, it is of interest whether or not they have finitely many associated primes. We prove the affirmative in the case where $R$ is a Stanley-Reisner ring over a field, whose associated simplicial complex is a $T$-space.
Let $R$ be a noetherian local ring. We consider the following quastion: Does there exist an integer $n$ such that all idelas generated by a system of parameters contained in the $n$-th power of the maximal ideal have the same Betti numbers? We obtain a positive answer for some rings with finite local cohomology.
For a connected Noetherian unstable algebra $R$ over the mod $p$ Steenrod algebra, we prove versions of theorems of Duflot and Carlson on the depth of $R$, originally proved when $R$ is the mod $p$ cohomology ring of a finite group. This recovers the aforementioned results, and also proves versions of them when $R$ is the mod $p$ cohomology ring of a compact Lie group, a profinite group with Noetherian cohomology, a Kac--Moody group, a discrete group of finite virtual cohomological dimension, as well as for certain other discrete groups. More generally, our results apply to certain finitely ge…
For a finite module $M$ over a local, equicharacteristic ring $(R,m)$, we show that the well-known formula $\cohdim(m,M)=\dim M$ becomes trivial if ones uses Matlis duals of local cohomology modules together with spectral sequences. We also prove a new, ring-theoretic vanishing criterion for local cohomology modules.
Let $\mathfrak{a}$ be an ideal of a commutative Noetherian ring $R$ and $M$ a finitely generated $R$-module. In this paper we proved that if $\operatorname{Supp}\mathfrak{F}_\mathfrak{a}^i(M)$ is finite for all $i<t$, then so is $\operatorname{Ass}(\mathfrak{F}_{\mathfrak{a}}^t(M))$.
The concept of Faltings' local-global principle for the in dimension $< n$ of local cohomology modules over a Noetherian ring $R$ is introduced, and it is shown that this principle holds at levels 1, 2. We also establish the same principle at all levels over an arbitrary Noetherian ring of dimension not exceeding 3. These generalize the main results of Brodmann et al. in \cite{BRS}. Moreover, as a generalization of Raghavan's result, we show that the Faltings' local-global principle for the in dimension $<n$ of local cohomology modules holds at all levels $r\in \mathbb{N}$ whenever the ring $R…
It is proved that if one of the finite modules M and N, over a local ring R, has reducible complexity and has finite Gorenstein dimension then the depth formula holds, provided TorR_i(M,N) = 0 for i>>0. We also study the vanishing of cohomology of a module of finite complete intersection dimension.
Let $(R,\mathfrak{m})$ be a Noetherian regular local ring containing a field of characteristic $p>0$ and $I$ a nonzero ideal of $R$. In this short note, we prove that if $\operatorname{H}^i_I(R)\neq 0$, then $ \operatorname{Supp}_R(D(\operatorname{H}^i_{I}(R)))=\operatorname{Spec}(R)$.
We provide new results on the vanishing of local cohomology modules supported at ideals of minors of matrices over arbitrary commutative Noetherian rings. In the process, we compute the local cohomology of rings of polynomials with integer coefficients---supported at generic determinantal ideals---and also obtain results on F-modules and D-modules that are likely to be of independent interest.
Let R be a complete intersection ring and let M and N be R-modules. It is shown that the vanishing of Ext^i_R(M,N) for a certain number of consecutive values of i starting at n forces the complete intersection dimension of M to be at most n-1. We also estimate the complete intersection dimension of the dual of M, in terms of vanishing of the cohomology modules, Ext^i_R(M,N).
Let $A$ be a commutative noetherian ring, let $\mathfrak a$ be an ideal of $A$ and let $n$ be a non-negative integer. In this paper, we study $\mathcal{S}_{n}(\mathfrak{a})$, a certain class of $A$-modules and we find some sufficient conditions so that a module belongs to $\mathcal{S}_{n}(\mathfrak{a})$. Moreover, we study the cofiniteness of local cohomology modules when $\dim A/\frak a\geq 3$.
We investigate symmetry in the vanishing of Tate cohomology for finitely generated modules over local Gorenstein rings. For finitely generated R-modules M and N over Gorenstein local ring R, it is shown that $\widehat{Ext}^i_R(M,N)=0$ for all $i\in\mathbb{Z}$ if and only if $\widehat{Ext}^i_R(N,M)=0$ for all $i\in\mathbb{Z}$.
Let $J$ be an ideal of a noetherian local ring $R$. We show new results on the set of attached primes $\Att_R(\LCMo ^l_J(R))$ of a local cohomology module $\LCMo ^l_J(R)$. To prove our results we establish and use new relations between the set $\Att_R(\LCMo ^l_J(R))$ of attached primes of a local cohomology module and the set $\Ass_R(D(\LCMo ^l_J(R)))$ of associated primes of the Matlis dual of the same local cohomology module.
We generalize some known results on the relation between the cohomological and projective dimension. Then we examine the set-theoretically Cohen-Macaulay ideals to find some cohomological characterization of these kind of ideals.
We prove that if M, N are finite modules over a Gorenstein local ring R of codimension at most 4, then the vanishing of Ext^n_R(M,N) for n\gg 0 is equivalent to the vanishing of Ext^n_R(N,M) for n\gg 0. Furthermore, if the completion of $R$ has no embedded deformation, then such vanishing occurs if and only if M or N has finite projective dimension.
This paper investigates when local cohomology modules have an annihilator that does not depend on the choice of an ideal. Takahashi classified the dominant resolving subcategories of the category of finitely generated modules over a commutative Noetherian ring. We show that his classification theorem describes annihilation results of local cohomology modules over a finite-dimensional ring with certain assumptions or a Cohen-Macaulay ring.
In this paper we prove the following generalization of a result of Hartshorne: Let $(S,\n)$ be a regular local ring of dimension $4$. Assume that $x,y,u,v$ is a regular system of parameters for $S$ and $a:=xu+yv$. Then for each finitely generated $S$-module $N$ with $\Supp N=V(aS)$ the socle of $H^2_{(u,v)S}(N)$ is infinite dimensional. Also, using this result, for any commutative Noetherian complete local ring $(R,\m)$, 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 $H^i_I(M)$ are $I$-cofinit…
Considering modules of finite complete intersection dimension over commutative Noetherian local rings, we prove (co)homology vanishing results in which we assume the vanishing of nonconsecutive (co)homology groups. In fact, the (co)homology groups assumed to vanish may be arbitrarily far apart from each other.
The concept of Faltings' local-global principle for the minimaxness of local cohomology modules over a commutative Noetherian ring $R$ is introduced, and it is shown that this principle holds at level 2. We also establish the same principle at all levels over an arbitrary commutative Noetherian ring of dimension not exceeding 3. These generalize the main results of Brodmann et al. in \cite{BRS}. Moreover, it is shown that if $M$ is a finitely generated $R$-module, $\frak a$ an ideal of $R$ and $r$ a non-negative integer such that $\frak a^tH^i_{\frak a}(M)$ is skinny for all $i<r$ and for some…
Let R be a commutative Noetherian ring. We introduce the notion of colocalization functors with supports in arbitrary subsets of Spec R, which is a natural generalization of right derived functors of section functors with supports in specialization-closed subsets. We prove that the local duality theorem and the vanishing theorem of Grothendieck type hold for colocalization functors.
For any non-zero finite module M of finite projective dimension over a noetherian local ring R with maximal ideal m and residue field k, it is proved that the natural map Ext_R(k,M)-->Ext_R(k,M/mM) is non-zero when R is regular and is zero otherwise. A noteworthy aspect of the proof is the use of stable cohomology. Applications include computations of Bass series over certain local rings.
Let $R=k[x_1,\dots,x_n]$ be a ring of polynomials over a field $k$ of characteristic $p>0$. There is an algorithm due to Lyubeznik for deciding the vanishing of local cohomology modules $H^i_I(R)$ where $I\subset R$ is an ideal. This algorithm has not been implemented because its complexity grows very rapidly with the growth of $p$ which makes it impractical. In this paper we produce a modification of this algorithm that consumes a modest amount of memory.
In this article, we study the behaviour of smooth algebra $R$ over local Noetherian local ring $A$. At first, we observe that for every $f\in R$, $R_f$ has finite length in the category of $D(R,A)$-module if dimension of $A$ is zero. This extends the result of Theorem 2 of \cite{Ly3}. We use this fact to generalize the result of Theorem 4.1 of \cite{BBLSZ}, from the finiteness of the set of associated primes of local cohomology module to that of Lyubeznik functor. Finally, we introduce the definition of $Σ$-finite $D$-modulue for smooth algebra and we extend the result of Theorem 1.3 of \cite{…
Let $X \overset{f}\longrightarrow S$ be a morphism of Noetherian schemes, with $S$ reduced. For any closed subscheme $Z$ of $X$ finite over $S$, let $j$ denote the open immersion $X\setminus Z \hookrightarrow X$. Kollár asked whether for any coherent sheaf $\mathcal F$ on $X\setminus Z$ and any index $r\geq 1$, the sheaf $f_*(R^rj_*\mathcal F)$ is generically free on $S$ and commutes with base change. We answer this affirmatively, by proving a related statement about local cohomology: Let $ R$ be Noetherian algebra over a Noetherian domain $A$, and let $I \subset R$ be an ideal such that $ R/I…
Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $M$ an $R$-module. We intend to establish the dual of Grothendieck's Vanishing Theorem for local homology modules. We conjecture that $H^{\fa}_i(M)=0$ for all $i>\Mag_RM$. We prove this in several cases.
For a Noetherian regular ring $S$ and for a fixed ideal $J\subset S$, assume that the associated primes of local cohomology module $H^i_J(S)$ does not contain $p$ for some $i\geq 0$, and we call this as a property $\textit{\textbf{P}}^{i,p}_{J,S}$ or $\textit{\textbf{P}}$ for brevity. Recently, in Theorem 1.2 of \cite{Nu1}, it is proved that in a Noetherian regular local ring $S$ and for a fixed ideal $J\subset S$, associated primes of local cohomology module $H^i_J(S)$ for $i\geq 0$ is finite, if it does not contain $p$. In this paper, we study how the property $\textit{\textbf{P}}$ (as menti…
Let $(R,m)$ be a Noetherian local ring of characteristic $p>0$. We introduce and study $F$-full and $F$-anti-nilpotent singularities, both are defined in terms of the Frobenius actions on the local cohomology modules of $R$ supported at the maximal ideal. We prove that if $R/(x)$ is $F$-full or $F$-anti-nilpotent for a nonzerodivisor $x\in R$, then so is $R$. We use these results to obtain new cases on the deformation of $F$-injectivity.
In the present paper, we define a notion of numerical equivalence on Chow groups or Grothendieck groups of Noetherian local rings, which is an analogue of that on smooth projective varieties. Under a mild condition, it is proved that the Chow group modulo numerical equivalence is a finite dimensional ${\Bbb Q}$-vector space, as in the case of smooth projective varieties. Numerical equivalence on local rings is deeply related to that on smooth projective varieties. For example, if Grothendieck's standard conjectures are true, then a vanishing of Chow group (of local rings) modulo numerical equi…
We present an Avoidance Principle for certain graded rings. As an application we fill a gap in the proof of a result by Brodmann, Rohrer and Sazeedeh about the antipolynomiality of the Hilbert-Samuel multiplicity of the graded components of the local cohomology modules of a finitely generated module over a Noetherian homogeneous ring with two-dimensional local base ring.
Let $\fa$ denote an ideal of a $d$-dimensional Gorenstein local ring $R$ and $M$ and $N$ two finitely generated $R$-modules with $\pd M< \infty$. It is shown that $H^d_{\fa}(M,N)=0$ if and only if $\dim \hat{R}\big/ \fa\hat{R}+\fp>0$ for all $\fp\in\Ass_{\hat{R}}\hat{M}\cap\Supp_{\hat{R}}\hat{N}$.
We use a result of Hellus about generalized local duality to describe some generalized Matlis duals for certain quasi-\F-modules. Furthermore, we apply this description to obtain examples for non-artinian local cohomology modules by the theory of \F-modules. In particular, we get a new view on Hartshorne's counterexample for a conjecture by Grothendieck about the finiteness of $Hom_R(R/I,H^i_I(R))$ for a noetherian local Ring $R$ and an ideal $I \subseteq R$.
Let $\fa$ denote an ideal of a commutative Noetherian ring $R$ and $M$ and $N$ two finitely generated $R$-modules with $\pd M< \infty$. It is shown that if $\fa$ is principal or $R$ is complete local and $\fa$ a prime ideal with $\dim R/\fa=1$, then the generalized local cohomology module $H^i_{\fa}(M,N)$ is $\fa$-cofinite for all $i \geq 0$. This provides an affirmative answer for the above ideal $\fa$ to a question proposed in [{\bf 13}].
A 2003 counterexample to a conjecture of Auslander brought attention to a family of rings - colloquially called AC rings - that satisfy a natural condition on vanishing of cohomology. Several results attest to the remarkable homological properties of AC rings, but their definition is barely operational, and it remains unknown if they form a class that is closed under typical constructions in ring theory. In this paper, we study transfer of the AC property along local homomorphisms of Cohen--Macaulay rings. In particular, we show that the AC property is preserved by standard procedures in local…
We 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.
In this paper, we continue the study of cominimaxness modules with respect to an ideal of a commutative Noetherian ring (cf. \cite{ANV}), and Bass numbers of local cohomology modules. Let $R$ denote a commutative Noetherian local ring and $I$ an ideal of $R$. We first show that the Bass numbers $μ^0(\frak p, H^2_I(R))$ and $μ^1(\frak p, H^2_I(R))$ are finite for all $\frak p\in \Spec R$, whenever $R$ is regular. As a consequence, it follows that the Goldie dimension of $H^2_I(R)$ is finite. Also, for a finitely generated $R$-module $M$ of dimension $d$, it is shown that the Bass numbers of $H^…
Let $\mathfrak{a}$ denote an ideal of a commutative Noetherian ring $R$. Let $M$ and $N$ be two $R$-modules. In this paper, we give partial answers on the extension of Hartshorne's conjecture about the cofiniteness of torsion and extension functors. For this purpose, we study the cofiniteness of the generalized local cohomology module $H^{i}_{\mathfrak{a}}(M, N)$ for a new class of modules, called $\mathfrak{a}$-weakly finite modules, in the local and non-local case. Furthermore, we derive some results on attached primes of top generalized local cohomology modules.
For a noetherian ring R we call an R-module M cofinite if there exists an ideal I of R such that M is I-cofinite; we show that every cofinite module M satisfies dim_R(M)<=injdimR(M). As an application we study the question which local cohomology modules H^i_I(R) satisfy injdim_R(H^i_I(R)) = dim_R(H^i_I(R)). There are two situations where the answer is positive. On the other hand we present two counter-examples, the failure in these two examples coming from different reasons.
Let $\mathfrak a$ be an ideal of a commutative Noetherian ring $R$ and $t$ be a non-negative integer. Let $M$ and $N$ be two finitely generated $R$-modules. In certain cases, we give some bounds under inclusion for the annihilators of $\operatorname{Ext}^t_R(M, N)$ and $\operatorname{H}^t_{\mathfrak a}(M)$ in terms of minimal primary decomposition of the zero submodule of $M$ which are independent of the choice of minimal primary decomposition. Then, by using those bounds, we compute the annihilators of local cohomology and Ext modules in certain cases.
Let $R$ be a commutative noetherian ring, $I,J$ be two ideals of $R$, $M$ be an $R$-module, and $\mathcal{S}$ be a Serre class of $R$-modules. A positive answer to the Huneke$^,$s conjecture is given for a noetherian ring $R$ and minimax $R$-module $M$ of krull dimension less than 3, with respect to $\mathcal{S}$. There are some results on cofiniteness and artinianness of local cohomology modules with respect to a pair of ideals. For a ZD-module $M$ of finite krull dimension and an integer $n\in\mathbb{N}$, if $\lc^{i}_{I,J}(M)\in\mathcal{S}$ for all $i>n$, then $\lc^{i}_{I,J}(M)/\fa^{j}\lc^{i…
This paper deals with the notion of grade of ideals with respect to torsion theories defined via some homological tools such as Ext-modules, Koszul cohomology modules, Čech and local cohomology modules over commutative rings which are not necessarily Noetherian. We also compare these approaches of grade.
Let $R$ be a finitely generated positively graded algebra over a Noetherian local ring $B$, and $\mathfrak{m} = [R]_+$ be the graded irrelevant ideal of $R$. We provide a local criterion characterizing the $B$-freeness of all the local cohomology modules $\text{H}_\mathfrak{m}^i(M)$ of a finitely generated graded $R$-module $M$. We show that fiber-full modules are exactly the ones that satisfy this criterion. When we change $B$ by an arbitrary Noetherian ring $A$, we study the fiber-full locus of a module in $\text{Spec}(A)$: we show that the fiber-full locus is always an open subset of $\text…
