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.
Some criteria for graded local cohomology to commute with coarsening functors are proven, and an example is given where graded local cohomology does not commute with coarsening.
In this paper, we prove some well-known results on local cohomology with respect to a pair of ideals in graded version, such as, Independence Theorem, Lichtenbaum-Harshorne Vanishing Theorem, Basic Finiteness and Vanishing Theorem, among others. Besides, we present a generalized version of Melkersson Theorem about Artinianess of modules and a result concerning Artinianess of local cohomology modules.
In this paper, considering the difference between the finiteness dimension and cohomological dimension for a finitely generated module, we investigate the asymptotic behavior of grades of components of graded local cohomology modules with respect to irrelevant ideal; as long as we study some artinian and tameness property of such modules.
Let Q be an affine semigroup generating Z^d, and fix a finitely generated Z^d-graded module M over the semigroup algebra k[Q] for a field k. We provide an algorithm to compute a minimal Z^d-graded injective resolution of M up to any desired cohomological degree. As an application, we derive an algorithm computing the local cohomology modules H^i_I(M) supported on any monomial (that is, Z^d-graded) ideal I. Since these local cohomology modules are neither finitely generated nor finitely cogenerated, part of this task is defining a finite data structure to encode them.
Let $A$ be a regular ring containing a field of characteristic zero and let $R = A[X_1,\ldots, X_m]$. Consider $R$ as standard graded with $deg \ A = 0$ and $deg \ X_i = 1$ for all $i$. In this paper we present a comprehensive study of graded components of local cohomology modules $H^i_I(R)$ where $I$ is an \emph{arbitrary} homogeneous ideal in $R$. Our study seems to be the first in this regard.
Let $R = \bigoplus_{n \in \mathbb{N}_{0}} R_{n}$ be a standard graded ring, $M$ be a finite graded $R$-module and $J$ be a homogenous ideal of $R$. In this paper we study the graded structure of the $i$-th local cohomology module of $M$ defined by a pair of ideals $(R_{+},J)$, i.e. $H^{i}_{R_{+},J}(M)$. More precisely, we discuss finiteness property and vanishing of the graded components $H^{i}_{R_{+},J}(M)_{n}$. Also, we study the Artinian property and tameness of certain submodules and quotient modules of $H^{i}_{R_{+},J}(M)$.
Let $R$ be a polynomial ring over a field. We describe the extremal rays and the facets of the cone of local cohomology tables of finitely generated graded $R$-modules of dimension at most two. Moreover, we show that any point inside the cone can be written as a finite linear combination, with positive rational coefficients, of points belonging to the extremal rays of the cone. We also provide algorithms to obtain decompositions in terms of extremal points and facets.
Let $K$ be a field of characteristic zero, $R = K[X_1,...,X_n]$. Let $A_n(K) = K<X_1,...,X_n, \partial_1, ..., \partial_n>$ be the $n^{th}$ Weyl algebra over $K$. We consider the case when $R$ and $A_n(K)$ is graded by giving $°X_i = ω_i $ and $°\partial_i = -ω_i$ for $i =1,...,n$ (here $ω_i$ are positive integers). Set $ω= \sum_{k=1}^{n}ω_k$. Let $I$ be a graded ideal in $R$. By a result due to Lyubeznik the local cohomology modules $H^i_I(R)$ are holonomic $A_n(K)$-modules for each $i \geq 0$. In this article we prove that the De Rahm cohomology modules $H^*(\bP ; H^*_I(R))$ is concentrated …
Let $A$ be a regular ring containing a field $K$ of characteristic zero and let $R = A[X_1,\ldots, X_m]$. Consider $R$ as standard graded with $°A = 0$ and $°X_i = 1$ for all $i$. Let $G$ be a finite subgroup of $GL_m(A)$. Let $G$ act linearly on $R$ fixing $A$. Let $S = R^G$. In this paper we present a comprehensive study of graded components of local cohomology modules $H^i_I(S)$ where $I$ is an \emph{arbitrary} homogeneous ideal in $S$. We prove stronger results when $G \subseteq GL_m(K)$. Some of our results are new even in the case when $A$ is a field.
Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ and let $I$ be an $\mathfrak{m}$-primary ideal. Let $G$ be the associated graded ring of $A$ \wrt \ $I$ and let $\R = A[It,t^{-1}]$ be the extended Rees ring of $A$ with respect to $I$. Notice $t^{-1}$ is a non-zero divisor on $\R$ and $\R/t^{-1}\R = G$. So we have \textit{Bockstein operators} $β^i \colon H^i_{G_+}(G)(-1) \rt H^{i+1}_{G_+}(G)$ for $i \geq 0$. Since $β^{i+1}(+1)\circ β^i = 0$ we have \textit{Bockstein cohomology} modules $BH^i(G)$ for $i = 0,\ldots,d$. In this paper we show that certain natural conditions on…
Assume that $R=\oplus_{n\in \mathbb{N}_0}R_n$ is a standard graded algebra over the local ring $(R_0,\mathfrak{m}_0)$, $\mathfrak{a}$ is a homogeneous ideal of $R$, $M$ is a finitely generated graded $R$-module and $R_+:=\oplus_{n\in \mathbb{N}}R_n$ denotes the irrelevant ideal of $R$. In this paper, we study the asymptotic behaviour of the set $\{ \operatorname{grade}(\mathfrak{a} \cap R_0, H^{\operatorname{grade}(R_+,M)}_{R_+}(M)_n) \}_{n \in \mathbb{Z}}$ as $n \rightarrow -\infty$, in the case where $\mathfrak{a}$ and $R_+$ are homogenously linked over $M$.
Let $A$ be a commutative Noetherian ring containing a field of characteristic zero. Let $R= A[X_1, \ldots, X_m]$ be a polynomial ring and $A_m(A) = A \langle X_1, \ldots, X_m, \partial_1, \ldots, \partial_m \rangle$ be the $m^{th}$ Weyl algebra over $A$, where $\partial_i=\partial/\partial X_i$. Consider both $R$ and $A_m(A)$ as standard graded with $\mathrm{deg}~ z=0$ for all $z \in A$, $\mathrm{deg}~ X_i=1$, and $\mathrm{deg}~ \partial_i =-1$ for $i=1, \ldots, m$. We present a few results about the behavior of the graded components of local cohomology modules $H_I^i(R)$, where $I$ is an arbi…
Let $R=k[x_1,..., x_n]$ be a polynomial ring over a field $k$ of characteristic $p>0,$ let $\m=(x_1,..., x_n)$ be the maximal ideal generated by the variables, let $^*E$ be the naturally graded injective hull of $R/\m$ and let $^*E(n)$ be $^*E$ degree shifted downward by $n.$ We introduce the notion of graded $F$-modules (as a refinement of the notion of $F$-modules) and show that if a graded $F$-module $\M$ has zero-dimensional support, then $\M,$ as a graded $R$-module, is isomorphic to a direct sum of a (possibly infinite) number of copies of $^*E(n).$ As a consequence, we show that if the …
Let $A$ be a regular domain containing a field $K$ of characteristic zero, $G$ be a finite subgroup of the group of automorphisms of $A$ and $B=A^G$ be the ring of invariants of $G$. Let $S= A[X_1,\ldots, X_m]$ and $R= B[X_1, \ldots, X_m]$ be standard graded with $\ deg \ A=0$, $\ deg \ B=0$ and $\ deg \ X_i=1$ for all $i$. Extend the action of $G$ on $A$ to $S$ by fixing $X_i$. Note $S^G=R$. Let $I$ be an arbitrary homogeneous ideal in $R$. The main goal of this paper is to establish a comparative study of graded components of local cohomology modules $H_I^i(R)$ that would be analogs to those…
Assume that $R = \bigoplus_{i\in \mathbb{N}_{0}} R_{i}$ is a homogeneous graded Noetherian ring, and that $M$ is a $ \mathbb{Z}$--graded $R$--module, where $ \mathbb{N}_{0}$ (resp. $ \mathbb{Z}$) denote the set all non--negative integers (resp. integers). The set of all homogeneous attached prime ideals of the top non--vanishing local cohomology module of a finitely generated module $M$, $\LH_{R_{+}}^{c}(M)$, with respect to the irrelevant ideal $R_{+}: =\bigoplus_{i\geq 1} R_{i}$ and the set of associated primes of $\LH _{R_{+}}^{i}(M)$ is studied. The asymptotic behavior of $\Hom_{R}(R/R_{+}…
Let $R = \bigoplus_{n \in \mathbb{N}_0} R_n$ be a Noetherian homogeneous ring with local base ring $(R_0, \mathfrak{m}_0)$ and let $M$ and $N$ be finitely generated graded $R$-modules. Let $i,j\in\mathbb{N}_0$. In this paper we will study Artinianess of $Γ_{\mathfrak m_0R}(H_{R_+}^i(M,N)), H_{\mathfrak m_0R}^1(H_{R_+}^i(M,N)), H_{R_+}^i(M,N)/{\mathfrak m_0}H_{R_+}^i(M,N), H_{R_+}^j(M,H_{\mathfrak m_0R}^i(N)), H_{\mathfrak m_0R}^j(M,H_{R_+}^i(N))$, where $R_+$ denotes the irrelevant ideal of $R$.
We give a combinatorial description of local cohomology modules of a graded module over a semigroup ring, with support at the graded maximal ideal. This combinatorial framework yields Hochster-type formulas for the Hilbert series of such local cohomology modules in terms of the homology of finitely many polyhedral cell complexes. A Cohen--Macaulay criterion immediately follows. We also provide an alternative proof of a result of [18] characterizing Cohen--Macaulay affine semigroup rings.
Notions of Castelnuovo-Mumford regularity and of $a^*$ invariant were extended from standard graded algebras to the toric setting. We here focus our attention on the standard multigraded case, which corresponds to a product of $k$ projective spaces. A natural notion for a $\mathbb Z^k$-graded module is its support: degrees in which it is not zero. A stabilized version of it is adding $-\mathbb N^k$, in order for the complement (vanishing region) to be stable by addition of $\mathbb N^k$. Cohomology of twists of a sheaf on a product of projective spaces, provided by a graded module, are given b…
This first part of the paper describes the support of top graded local cohomology modules. As a corrolary one obtains a simple criteria for the vanishing of these modules and also the fact that they have finitely many minimal primes. The second part of this paper constructs examples of cohomological Hilbert functions which are not of polynomial type.
Let $R=\oplus_{n\in \N_0}R_n$ be a standard graded ring, $M$ be a finitely generated graded $R$-module and $R_+:=\oplus_{n\in \N}R_n$ denotes the irrelevant ideal of $R$. In this paper, considering the new concept of linkage of ideals over a module, we study the graded components $H^i_{\fa}(M)_n$ when $\fa$ is an h-linked ideal over $M$. More precisely, we show that $H^i_{\fa}(M)$ is tame in each of the following cases: \begin{itemize} \item [(i)] $i=f_{\fa}^{R_+}(M)$, the first integer $i$ for which $R_+\nsubseteq \sqrt{0:H^i_{\fa}(M)}$; \item [(ii)] $i=\cd(R_+,M)$, the last integer $i$ for w…
We prove that, if $A$ is a positively graded, graded commutative, local, finite Hopf algebra, its cohomology is finitely generated, thus unifying classical results of Wilkerson and Hopkins-Smith, and of Friedlander-Suslin. We do this by showing the existence of conormal elementary quotients.
Stable cohomology is a generalization of Tate cohomology to associative rings, first defined by Pierre Vogel. For a commutative local ring $R$ with residue field $k$, stable cohomology modules $\widehat{\mathrm{Ext}}{\vphantom E}^{n}_R\;(k,k)$, defined for $n\in\mathbb{Z}$, have been studied by Avramov and Veliche. Stable cohomology carries a structure of $\mathbb{Z}$-graded $k$-algebra. One of the main goals of this paper is to prove that, for a class of Gorenstein rings, this algebra is a trivial extension of absolute cohomology $\mathrm{Ext}_R(k,k)$ and a shift of $\mathrm{Hom}_k(\mathrm{Ex…
Let $R=K[X_1,\ldots, X_n]$ where $K$ is a field of characteristic zero, and let $A_n(K)$ be the $n^{th}$ Weyl algebra over $K$. We give standard grading on $R$ and $A_n(K)$. Let $I$, $J$ be homogeneous ideals of $R$. Let $M = H^i_I(R)$ and $N = H^j_J(R)$ for some $i, j$. We show that $\Ext_{A_n(K)}^ν(M,N)$ is concentrated in degree zero for all $ν\geq 0$, i.e., $\Ext_{A_n(K)}^ν(M,N)_l=0$ for $l \neq0$. This proves a conjecture stated in part I of this paper.
We consider the (graded) Matlis dual $\DD(M)$ of a graded $\D$-module $M$ over the polynomial ring $R = k[x_1, \ldots, x_n]$ ($k$ is a field of characteristic zero), and show that it can be given a structure of $\D$-module in such a way that, whenever $\dim_kH^i_{dR}(M)$ is finite, then $H^i_{dR}(M)$ is $k$-dual to $H^{n-i}_{dR}(\DD(M))$. As a consequence, we show that if $M$ is a graded $\D$-module such that $H^n_{dR}(M)$ is a finite-dimensional $k$-space, then $\dim_k(H^n_{dR}(M))$ is the maximal integer $s$ for which there exists a surjective $\D$-linear homomorphism $M \rightarrow E^s$, wh…
Let R be a Noetherian standard graded ring, and M and N two finitely generated graded R-modules. We introduce reg_R (M,N) by using the notion of generalized local cohomology instead of local cohomology, in the definition of regularity. We prove that reg_R (M,N)is finite in several cases. In the case that the base ring is a field, we show that reg_R (M,N)=reg (N)-indeg (M). This formula, together with a graded version of duality for generalized local cohomology, gives a formula for the minimum of the initial degrees of some Ext modules(in the case R is Cohen-Macaulay), of which the three usual …
The asymptotic stability of several homological invariants of the graded pieces of a graded module has attracted quite a lot of attention over the last decades. We provide in this text several stability results together with estimates of the degree from which it stabilizes. Before we establish these regularity results, we prove several facts about depth and cohomological dimension with respect to a finitely generated ideal and about Castelnuovo-Mumford regularity of a graded module.
A general method for establishing results over a commutative complete intersection local ring by passing to differential graded modules over a graded exterior algebra is described. It is used to deduce, in a uniform way, results on the growth of resolutions of complexes over such local rings.
The main focus is the generic freeness of local cohomology modules in a graded setting. The present approach takes place in a quite nonrestrictive setting, by solely assuming that the ground coefficient ring is Noetherian. Under additional assumptions, such as when the latter is reduced or a domain, the outcome turns out to be stronger. One important application of these considerations is to the specialization of rational maps and of symmetric and Rees powers of a module.
Given a module M over a ring R which has a grading by a semigroup Q, we present a spectral sequence that computes the local cohomology of M at any Q-graded ideal I in terms of Ext modules. This method is used to obtain finiteness results for the local cohomology of graded modules over semigroup rings; in particular we prove that for a semigroup Q whose saturation is simplicial, the Bass numbers of such local cohomology modules are finite. Conversely, if the saturation of Q is not simplicial, one can find a graded ideal I and a graded R-module M whose local cohomology at I in some degree has an…
Let $A$ be a Dedekind domain of characteristic zero such that its localization at every maximal ideal has mixed characteristic with finite residue field. Let $R=A[X_1,\ldots, X_n]$ be a polynomial ring and $I=(a_1U_1, \ldots, a_c U_c)\subseteq R$ an ideal, where $a_j \in A$ (not necessarily units) and $U_j$'s are monomials in $X_1, \ldots, X_n$. We call such an ideal $I$ as a $\mathfrak{C}$-monomial ideal. Consider the standard multigrading on $R$. We produce a structure theorem for the multigraded components of the local cohomology modules $H^i_I(R)$ for $i \geq 0$. We further analyze the tor…
In this paper, we will consider derived equivalences for differential graded endomorphism algebras by Keller's approaches. First we construct derived equivalences of differential graded algebras which are endomorphism algebras of the objects from a triangle in the homotopy category of differential graded algebras. We also obtain derived equivalences of differential graded endomorphism algebras from a standard derived equivalence of finite dimensional algebras. Moreover, under some conditions, the cohomology rings of these differential graded endomorphism algebras are also derived equivalent. T…
We develop a local cohomology theory for FI$^m$-modules, and show that it in many ways mimics the classical theory for multi-graded modules over a polynomial ring. In particular, we define an invariant of FI$^m$-modules using this local cohomology theory which closely resembles an invariant of multi-graded modules over Cox rings defined by Maclagan and Smith. It is then shown that this invariant behaves almost identically to the invariant of Maclagan and Smith.
The Hilbert functions and the regularity of the graded components of local cohomology of a bigraded algebra are considered. Explicit bounds for these invariants are obtained for bigraded hypersurface rings.
We prove a duality theorem for graded algebras over a field that implies several known duality results : graded local duality, versions of Serre duality for local cohomology and of Suzuki duality for generalized local cohomology, and Herzog-Rahimi bigraded duality.
The Hochschild cohomology of a differential graded algebra, or a differential graded category, admits a natural map to the graded center of its homology category: the characteristic homomorphism. We interpret it as an edge homomorphism in a spectral sequence. This gives a conceptual explanation of the failure of the characteristic homomorphism to be injective or surjective, in general. To illustrate this, we discuss modules over the dual numbers, coherent sheaves over algebraic curves, as well as examples related to free loop spaces and string topology.
Let $R$ be a polynomial ring over a field $K$ of arbitrary characteristic and $D$ be the ring of differential operators over $R$. Inspired by Euler formula for homogeneous polynomials, we introduce a class of graded $D$-modules, called Eulerian graded $D$-modules. It is proved that a vast class of $D$-modules, including all composite of local cohomology modules, $H_{J_0}^{i_0}(H_{J_1}^{i_1}...(H_{J_n}^{i_n}(R)))$ where $J_1,...,J_n$ are homogeneous ideals of $R$, are Eulerian graded. As an application of our theory, we prove that in all characteristic, these composite of local cohomology modul…
Let $A$ be a differential graded algebra with cohomology ring $H^*A$. A graded module over $H^*A$ is called \emph{realisable} if it is (up to direct summands) of the form $H^*M$ for some differential graded $A$-module $M$. Benson, Krause and Schwede have stated a local and a global obstruction for realisability. The global obstruction is given by the Hochschild class determined by the secondary multiplication of the $A_{\infty}$-algebra structure of $H^*A$. In this thesis we mainly consider differential graded algebras $A$ with graded-commutative cohomology ring. We show that a finitely presen…
Let $R=\bigoplus_{n\geq 0}R_n$, $\fa\supseteq \bigoplus_{n> 0}R_n$ and $M$ and $N$ be a standard graded ring, an ideal of $R$ and two finitely generated graded $R$-modules, respectively. This paper studies the homogeneous components of graded generalized local cohomology modules. First of all, we show that for all $i\geq 0$, $H^i_{\fa}(M, N)_n$, the $n$-th graded component of the $i$-th generalized local cohomology module of $M$ and $N$ with respect to $\fa$, vanishes for all $n\gg 0$. Furthermore, some sufficient conditions are proposed to satisfy the equality $\sup\{\en(H^i_{\fa}(M, N))| i\g…
We introduce the notion of E-depth of graded modules over polynomial rings to measure the depth of certain Ext modules. First, we characterize graded modules over polynomial rings with (sufficiently) large E-depth as those modules whose (sufficiently) partial general initial submodules preserve the Hilbert function of local cohomology modules supported at the irrelevant maximal ideal, extending a result of Herzog and Sbarra on sequentially Cohen-Macaulay modules. Second, we describe the cone of local cohomology tables of modules with sufficiently high E-depth, building on previous work of the …
Let $S=k[x_1,..., x_n]$ be a polynomial ring over a field $k$ with $n$ variables $x_1$, ..., $x_n$, $\mmmm$ the irrelevant maximal ideal of $S$, $I$ a monomial ideal in $S$ and $I'$ the polarization of $I$ in the polynomial ring $S'$ with $ρ$ variables. We show that each graded piece $H_\mmmm^i(S/I)_\aaa$, $\aaa\in\ZZZ^n$, of the local cohomology module $H_\mmmm^i(S/I)$ is isomorphic to a specific graded piece $H_{\mmmm'}^{i+ρ-n}(S'/I')_\alphaaa$, $\alphaaa\in\ZZZ^ρ$, of the local cohomology module $H_{\mmmm'}^{i+ρ-n}(S'/I')$, where $\mmmm'$ is the irrelevant maximal ideal of $S'$.
Mats Boij and Jonas Soederberg (math.AC/0611081) have conjectured that the Betti table of a Cohen-Macaulay module over a polynomial ring can be decomposed in a certain way as a positive linear combination of Betti tables of modules with pure resolutions. We prove, over any field, a strengthened form of their conjecture. Applications include a proof of the Multiplicity Conjecture of Huneke and Srinivasan and a proof of the convexity of a fan naturally associated to the Young lattice. We also characterize the rational cone of all cohomology tables of vector bundles on projective spaces in terms …
The main result of the paper states that for a graded ideal I in a polynomial ring R over a field of characteristic 0, the Hilbert functions of the local cohomology modules of R/I and of R/Gin(I) coincide if and only if R/I is sequentially Cohen-Macaulay.
Let $R_0$ be any domain, let $R=R_0[U_1, ..., U_s]/I$, where $U_1, ..., U_s$ are indeterminates of some positive degrees, and $I\subset R_0[U_1, ..., U_s]$ is a homogeneous ideal. The main theorem in this paper is states that all the associated primes of $H:=H^s_{R_+}(R)$ contain a certain non-zero ideal $c(I)$ of $R_0$ called the ``content'' of $I$. It follows that the support of $H$ is simply $V(\content(I)R + R_+)$ (Corollary 1.8) and, in particular, $H$ vanishes if and only if $c(I)$ is the unit ideal. These results raise the question of whether local cohomology modules have finitely many …
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…
The $i$-th local cohomology module of a finitely generated graded module $M$ over a standard positively graded commutative Noetherian ring $R$, with respect to the irrelevant ideal $R_+$, is itself graded; all its graded components are finitely generated modules over $R_0$, the component of $R$ of degree 0. This paper is concerned with the asymptotic behaviour of $\Ass_{R_0}(H^i_{R_+}(M)_n)$ as $n \to -\infty$. The smallest $i$ for which such study is interesting is the finiteness dimension $f$ of $M$ relative to $R_+$, defined as the least integer $j$ for which $H^j_{R_+}(M)$ is not finitely …
In this article we extend a previous definition of Castelnuovo-Mumford regularity for modules over an algebra graded by a finitely generated abelian group. Our notion of regularity is based on Maclagan and Smith's definition, and is extended first by working over any commutative base ring, and second by considering local cohomology with support in an arbitrary finitely generated graded ideal $B$, obtaining, for each $B$, a $B$-regularity region. The first extension provides a natural approach for working with families of sheaves or of graded modules, while the second opens new applications. We…
Graded skew-commutative rings occur often in practice. Here are two examples: 1) The cohomology ring of a compact three-dimensional manifold. 2) The cohomology ring of the complement of a hyperplane arrangement (the Orlik-Solomon algebra). We present some applications of the homological theory of these graded skew-commutative rings. In particular we find compact oriented 3-manifolds without boundary for which the Hilbert series of the Yoneda Ext-algebra of the cohomology ring of the fundamental group is an explicit transcendental function. This is only possible for large first Betti numbers of…
We prove an inequality between Hilbert functions of local cohomology modules supported in the homogeneous maximal ideal of standard graded algebras over a field, within the framework of embeddings of posets of Hilbert functions. As a main application, we prove an analogue for local cohomology of Evans' Lex-Plus-Power Conjecture for Betti numbers. This results implies some cases of the classical Lex-Plus-Power Conjecture, namely an inequality between extremal Betti numbers. In particular, for the classes of ideals for which the Eisenbud-Green-Harris Conjecture is currently known, the projective…
We first provide here a very short proof of a refinement of a theorem of Kodiyalam and Cutkosky, Herzog and Trung on the regularity of powers of ideals. This result implies a conjecture of Hà and generalizes a result of Eisenbud and Harris concerning the case of ideals primary for the graded maximal ideal in a standard graded algebra over a field. It also implies a new result on the regularities of powers of ideal sheaves. We then compare the cohomology of the stalks and the cohomology of the fibers of a projective morphism to the effect of comparing the maximum over fibers and over stalks of …
Let $A$ be a noetherian ring, $\fa$ an ideal of $A$, and $M$ an $A$--module. Some uniform theorems on the artinianness of certain local cohomology modules are proven in a general situation. They generalize and imply previous results about the artinianness of some special local cohomology modules in the graded case.
Let $M$ be a finitely generated graded module over a Noetherian homogeneous ring $R = \bigoplus_{n \in \mathbb{N}_0}R_n$. For each $i \in \mathbb{N}_0$ let $H^i_{R_{+}}(M)$ denote the $i$-th local cohomology module of $M$ with respect to the irrelevant ideal $R_+ = \bigoplus_{n > 0} R_n$ of $R$, furnished with its natural grading. We study the tame loci $\ft^i(M)^{\leq 3}$ at level $i \in \mathbb{N}_0$ in codimension $\leq 3$ of $M$, that is the sets of all primes $\fp_0 \subset R_0$ of height $\leq 3$ such that the graded $R_{\fp_0}$-modules $H^i_{R_{+}}(M)_{\fp_0}$ are tame.
