Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRatliff-Rush Filtration, regularity and depth of Higher Associated graded modules: Part II
144
0
0.0
(
0
)
Authors
Tony J. Puthenpurakal
Ask ChatGPT about the research
No Arabic abstract
Let $(A,m)$ be a Noetherian local ring, let $M$ be a finitely generated CM $A$-module of dimension $r geq 2$ and let $I$ be an ideal of definition for $M$. Set $L^I(M) = bigoplus_{ngeq 0}M/I^{n+1}M$. In part one of this paper we showed that $L^I(M)$ is a module over $R$, the Rees algebra of $I$ and we gave many applications of $L^I(M)$ to study the associated graded module, $G_I(M)$. In this paper we give many further applications of our technique; most notable is a reformulation of a classical result due to Narita in terms of the Ratliff-Rush filtration. This reformulation can be extended to all dimensions $geq 2$.
rate research
Read More
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 $(A,mathfrak{m})$ be a hypersurface ring with dimension $d$, and $M$ a MCM $A-$module with reduction no.2 and $mu(M)=2$ or $3$ then we have proved that depth$G(M)geq d-mu(M)+1$. If $e(A)=3$ and $mu(M)=4$ then in this case we have proved that depth$G(M)geq d-3$. When $A = Q/(f)$ where $Q = k[[X_1,cdots, X_{d+1}]]$ then we give estimates for depth $G(M)$ in terms of minimal presentation of $M$. Our paper is the first systematic study of depth of associated graded modules of MCM modules over hypersurface rings.
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 $A$ 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 $deg A = 0$ and $deg X_i = 1$ 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 Henselian Cohen-Macaulay local ring and let CM(A) be the category of maximal Cohen-Macaulay $A$-modules. We construct $T colon CM(A)times CM(A) rightarrow mod(A)$, a subfunctor of $Ext^1_A(-, -)$ and use it to study properties of associated graded modules over $G(A) = bigoplus_{ngeq 0} mathfrak{m}^n/mathfrak{m}^{n+1}$, the associated graded ring of $A$. As an application we give several examples of complete Cohen-Macaulay local rings $A$ with $G(A)$ Cohen-Macaulay and having distinct indecomposable maximal Cohen-Macaulay modules $M_n$ with $G(M_n)$ Cohen-Macaulay and the set ${e(M_n)}$ bounded (here $e(M)$ denotes multiplicity of $M$).
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
