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Register a new userFaithfulness of Top Local Cohomology Modules in Domains
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We study the conditions under which the highest nonvanishing local cohomology module of a domain $R$ with support in an ideal $I$ is faithful over $R$, i.e., which guarantee that $H^c_I(R)$ is faithful, where $c$ is the cohomological dimension of $I$. In particular, we prove that this is true for the case of positive prime characteristic when $c$ is the number of generators of $I$.
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Let $A$ be a commutative Noetherian ring containing a field $K$ 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 $i$. We present a few results about the behavior of the graded components of local cohomology modules $H_I^i(R)$ where $I$ is an arbitrary homogeneous ideal in $R$. We mostly restrict our attention to the Vanishing, Tameness and Rigidity problems.
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 proven in a previous paper of the first author for $H_J^i(S)$ where $J$ is an arbitrary homogeneous ideal in $S$.
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 $Ext_{A_n(K)}^{ u}(M,N)$ is concentrated in degree zero for all $ u geq 0$, i.e., $Ext_{A_n(K)}^{ u}(M,N)_l=0$ for $l eq0$. This proves a conjecture stated in part I of this paper.
Let fa be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let cd_{fa}(M,N) denote the supremum of the is such that H^i_{fa}(M,N) eq 0. First, by using the theory of Gorenstein homological dimensions, we obtain several upper bounds for cd_{fa}(M,N). Next, over a Cohen-Macaulay local ring (R,fm), we show that cd_{fm}(M,N)=dim R-grade(Ann_RN,M), provided that either projective dimension of M or injective dimension of N is finite. Finally, over such rings, we establish an analogue of the Hartshorne-Lichtenbaum Vanishing Theorem in the context of generalized local cohomology modules.
Let $fa$ be an ideal of a local ring $(R,fm)$ and $M$ a finitely generated $R$-module. We investigate the structure of the formal local cohomology modules ${vpl}_nH^i_{fm}(M/fa^n M)$, $igeq 0$. We prove several results concerning finiteness properties of formal local cohomology modules which indicate that these modules behave very similar to local cohomology modules. Among other things, we prove that if $dim Rleq 2$ or either $fa$ is principal or $dim R/faleq 1$, then $Tor_j^R(R/fa,{vpl}_nH^i_{fm}(M/fa^n M))$ is Artinian for all $i$ and $j$. Also, we examine the notion $fgrade(fa,M)$, the formal grade of $M$ with respect to $fa$ (i.e. the least integer $i$ such that ${vpl}_nH^i_{fm}(M/fa^n M) eq 0$). As applications, we establish a criterion for Cohen-Macaulayness of $M$, and also we provide an upper bound for cohomological dimension of $M$ with respect to $fa$.
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