Some results on local cohomology modules

Let $R$ be a commutative Noetherian ring, \fa an ideal of $R$, and let $M$ be a finitely generated $R$-module. For a non-negative integer $t$, we prove that H_{\fa}^t(M) is \fa-cofinite whenever H_{\fa}^t(M) is Artinian and H_{\fa}^i(M) is \fa-cofinite for all $i<t$. This result, in particular, characterizes the \fa-cofiniteness property of local cohomology modules of certain regular local rings. Also, we show that for a local ring (R,\fm), f-\depth(\fa,M) is the least integer $i$ such that H_{\fa}^i(M)\ncong H_{\fm}^i(M). This result in conjunction with the first one, yields some interesting consequences. Finally, we extend the non- vanishing Grothendieck's Theorem to \fa-cofinite modules.Comment: 7pages, Archiv der Mathemati

On the associated primes of generalized local cohomology modules

Let \fa be an ideal of a commutative Noetherian ring $R$ with identity and let $M$ and $N$ be two finitely generated $R$-modules. Let $t$ be a positive integer. It is shown that \Ass_R(H_{\fa}^t(M,N)) is contained in the union of the sets \Ass_R(\Ext_R^i(M,H_{\fa}^{t-i}(N))), where $0\leq i\leq t$. As an immediate consequence, it follows that if either H_{\fa}^i(N) is finitely generated for all $i<t$ or \Supp_R(H_{\fa}^i(N)) is finite for all $i<t$, then \Ass_R(H_{\fa}^t(M,N)) is finite. Also, we prove that if d=\pd(M) and $n=\dim(N)$ are finite, then H_{\fa}^{d+n}(M,N) is Artinian. In particular, \Ass_R(H_{\fa}^{d+n}(M,N)) is a finite set consisting of maximal ideals.Comment: 7 pages, to appear in Communications in Algebr

On the computation of the Ratliff-Rush closure, associated graded ring and invariance of a length

Let (R,\fm) be a Cohen-Macaulay local ring of positive dimension $d$ and infinite residue field. Let $I$ be an \fm-primary ideal of $R$ and $J$ be a minimal reduction of $I$. In this paper we show that if $\widetilde{I^k}=I^k$ and $J\cap I^n=JI^{n-1}$ for all $n\geq k+2$, then $\widetilde{I^n}=I^n$ for all $n\geq k$. As a consequence, we can deduce that if $r_J(I)=2$, then $\widetilde{I}=I$ if and only if $\widetilde{I^n}=I^n$ for all $n\geq 1$. Moreover, we recover some main results [\ref{Cpv}] and [\ref{G}]. Finally, we give a counter example for question 3 of [\ref{P1}].Comment: 9 page

On stability properties of powers of polymatroidal ideals

Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ with the maximal ideal $\frak{m}=(x_1,...,x_n)$. Let \astab(I) and \dstab(I) be the smallest integer $n$ for which \Ass(I^n) and \depth(I^n) stabilize, respectively. In this paper we show that \astab(I)=\dstab(I) in the following cases: \begin{itemize} \item[(i)] $I$ is a matroidal ideal and $n\leq 5$. \item[(ii)] $I$ is a polymatroidal ideal, $n=4$ and \frak{m}\notin\Ass^{\infty}(I), where \Ass^{\infty}(I) is the stable set of associated prime ideals of $I$. \item[(iii)] $I$ is a polymatroidal ideal of degree $2$. \end{itemize} Moreover, we give an example of a polymatroidal ideal for which \astab(I)\neq\dstab(I). This is a counterexample to the conjecture of Herzog and Qureshi, according to which these two numbers are the same for polymatroidal ideals.Comment: 10 pages. To appear in Collec. Mat

Stability properties of powers of ideals over regular local rings of small dimension

Let $(R,\mathfrak{m})$ be a regular local ring or a polynomial ring over a field, and let $I$ be an ideal of $R$ which we assume to be graded if $R$ is a polynomial ring. Let astab$(I)$ resp. $\overline{\rm astab}(I)$ be the smallest integer $n$ for which Ass$(I^n)$ resp. Ass$(\overline{I^n})$ stabilize, and dstab$(I)$ be the smallest integer $n$ for which depth$(I^n)$ stabilizes. Here $\overline{I^n}$ denotes the integral closure of $I^n$. We show that astab$(I)=\overline{\rm astab}(I)={\rm dstab}(I)$ if dim$\,R\leq 2$, while already in dimension $3$, astab$(I)$ and $\overline{\rm astab}(I)$ may differ by any amount. Moreover, we show that if dim$\,R=4$, then there exist ideals $I$ and $J$ such that for any positive integer $c$ one has ${\rm astab}(I)-{\rm dstab}(I)\geq c$ and ${\rm dstab}(J)-{\rm astab}(J)\geq c$.Comment: 9 pages, Comments are welcom

On the first generalized Hilbert coefficient and depth of associated graded rings

Let $(R,\mathfrak{m})$ be a $d$-dimensional Cohen-Macaulay local ring with infinite residue field. Let $I$ be an ideal of $R$ that has analytic spread $\ell(I)=d$, satisfies the $G_d$ condition, the weak Artin-Nagata property $AN_{d-2}^-$ and depth$(R/I)\geq\min\lbrace 1,\dim R/I \rbrace$. In this paper, we show that if $j_1(I) = \lambda (I/J) +\lambda [R/(J_{d-1} :_{R} I+(J_{d-2} :_{R}I+I) :_R, \mathfrak{m}^\infty)]+1$, then depth$(G(I))\geq d -1$ and $r_J(I)\leq 2$, where $J$ is a general minimal reduction of $I$. In addition, we extend the result by Sally who has studied the depth of associated graded rings and minimal reductions for an $,\mathfrak{m}$-primary ideals.Comment: 9 pages, comments welcome. arXiv admin note: text overlap with arXiv:1312.0651, arXiv:1401.1571 by other author

Results on the Hilbert coefficients and reduction numbers

Let $(R,\frak{m})$ be a $d$-dimensional Cohen-Macaulay local ring, $I$ an $\frak{m}$-primary ideal and $J$ a minimal reduction of $I$. In this paper we study the independence of reduction ideals and the behavior of the higher Hilbert coefficients. In addition, we give some examples in this regards.Comment: accepted for publication in the Proceedings Mathematical Science

On the Hilbert coefficients, depth of associated graded rings and reduction numbers

Let $(R,\mathfrak{m})$ be a $d$-dimensional Cohen-Macaulay local ring, $I$ an $\mathfrak{m}$-primary ideal of $R$ and $J=(x_1,...,x_d)$ a minimal reduction of $I$. We show that if $J_{d-1}=(x_1,...,x_{d-1})$ and $\sum\limits_{n=1}^\infty\lambda{({I^{n+1}\cap J_{d-1}})/({J{I^n} \cap J_{d-1}})=i}$ where i=0,1, then depth $G(I)\geq{d-i-1}$. Moreover, we prove that if $e_2(I) = \sum_{n=2}^\infty (n-1) \lambda (I^n/JI^{n-1})-2;$ or if $I$ is integrally closed and $e_2(I) = \sum_{n=2}^\infty (n-1)\lambda({{I^{n}}}/JI^{n-1})-i$ where $i=3,4$, then $e_1(I) = \sum_{n=1}^\infty \lambda(I^n / JI^{n-1})-1.$ In addition, we show that $r(I)$ is independent. Furthermore, we study the independence of $r(I)$ with some other conditions.Comment: to appear in JC

A note on linear resolution and polymatroidal ideals

Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal generated in degree $d$. Bandari and Herzog conjectured that a monomial ideal $I$ is polymatroidal if and only if all its monomial localizations have a linear resolution. In this paper we give an affirmative answer to the conjecture in the following cases: $(i)$ ${\rm height}(I)=n-1$; $(ii)$ $I$ contains at least $n-3$ pure powers of the variables $x_1^d,...,x_{n-3}^d$; $(iii)$ $I$ is a monomial ideal in at most four variables.Comment: 12 pages. Comments welcom

Associated primes of local cohomology module

Let \fa be an ideal of a commutative Noetherian ring $R$ and $M$ a finitely generated $R$-module. Let $t$ be a natural integer. It is shown that there is a finite subset $X$ of \Spec R, such that \Ass_R(H_{\fa}^t(M)) is contained in $X$ union with the union of the sets \Ass_R(\Ext_R^j(R/\fa,H_{\fa}^i(M))), where $0\leq i<t$ and $0\leq j\leq t^2+1$. As an immediate consequence, we deduce that the first non \fa-cofinite local cohomology module of $M$ with respect to \fa has only finitely many associated prime ideals.Comment: 7 pages, to appear in Proceedings of the American Mathematical Societ