Related papers: Small perturbations in generalized Cohen-Macaulay …

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $I$ an ideal of $R$. We study how local cohomology modules with support in $\mathfrak{m}$ change for small perturbations $J$ of $I$, that is, for ideals $J$ such that $I\equiv J\bmod…

This paper shows that if $R$ is a homomorphic image of a Cohen-Macaulay local ring, then $R$-module $M$ is sequentially generalized Cohen-Macaulay if and only if the difference between Hilbert coefficients and arithmetic degrees for all…

We answer affirmatively a question of Srinivas--Trivedi: in a Noetherian local ring $(R,\mathfrak{m})$, if $I=(f_1,\dots,f_r)$ is an ideal generated by a filter-regular sequence and $J$ is an ideal such that $I+J$ is $\mathfrak{m}$-primary,…

In this paper we show that a large class of one-dimensional Cohen-Macaulay local rings $(A,\mathfrak{m})$ has the property that if $M$ is a maximal Cohen-Macaulay $A$-module then the Hilbert function of $M$ ( with respect to $\mathfrak{m}$)…

Let $(R, {\frak m})$ be a local ring, $I$ a proper ideal of $R$ and $M$ a finitely generated $R$-module of dimension $d$. We discuss the local homology modules of $H^d_I(M)$. When $M$ is Cohen-Macaulay, it is proved that $H^d_{{\frak…

We introduce the notions of sequential sequence and sequential f-sequence in order to characterize sequentially Cohen-Macaulay modules and sequentially generalized Cohen-Macaulay modules. Let R be a Noetherian local ring and M a finitely…

Let $(R,\mathfrak m)$ be a local ring and $I, J$ two arbitrary ideals of $R$. Let $\operatorname{gr}_J(R/I)$ denote the associated ring of $R/I$ with respect to $J$, which corresponds to the normal cone in geometry. The main result of this…

A finitely generated module $M$ over a local ring is called a sequentially generalized Cohen-Macaulay module if there is a filtration of submodules of $M$: $M_0\subset M_1\subset ... \subset M_t=M$ such that $\dim M_0<\dim M_1< >... <\dim…

Let $x_1,\ldots,x_s$ be a filter regular sequence in a local ring $(R,\mathfrak{m})$. Denote by $R_{x_1,\ldots,x_s}$ the Koszul complex of $x_1,\ldots,x_s$ over $R$. In this paper, we give an explicit number $N$ such that the sum of lengths…

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…

In this paper we determine the possible Hilbert functions of a Cohen-Macaulay local ring of dimension $d$, multiplicity $e$ and first Hilbert coefficient $e_1$ in the case $e_1 = e + 2$.

We prove that deformation of F-injectivity holds for local rings $(R,\mathfrak{m})$ that admit secondary representations of $H^i_{\mathfrak{m}}(R)$ which are stable under the natural Frobenius action. As a consequence, F-injectivity deforms…

Let $R$ be a local ring and let ($x_1\biss x_r$) be part of a system of parameters of a finitely generated $R$-module $M,$ where $r < \dim_R M$. We will show that if ($y_1\biss y_r$) is part of a reducing system of parameters of $M$ with…

Let $(R, \mathfrak{m} )$ be a Noetherian local ring, $M$ a finitely generated $R$-module of dimension $d$. Set $\mathfrak{a}(M):=\mathfrak{a}_0(M)\cdots \mathfrak{a}_{d-1}(M)$, where $\mathfrak{a}_i(M):={\rm Ann}_RH^i_{\mathfrak{m}}(M)$ for…

In this paper, we explore the relation between the index of reducibility and the Hilbert coefficients in local rings. Consequently, the main result of this study provides a characterization of a sequentially Cohen-Macaulay ring in terms of…

Let $S$ be an unramified regular local ring of mixed characteristic two and $R$ the integral closure of $S$ in a biquadratic extension of its quotient field obtained by adjoining roots of sufficiently general square free elements $f,g\in…

Let $(R,m, \kappa)$ be a local ring. We give a characterization of $R$-modules $M$ whose local cohomology is finite length up to some index in terms of asymptotic vanishing of Koszul cohomology on parameter ideals up to the same index. In…

In this paper we present characterizations of sequentially Cohen-Macaulay modules in terms of systems of parameters, which are generalizations of well-known results on Cohen-Macaulay and generalized Cohen-Macaulay modules. The sequentially…

We study Hilbert functions of maximal Cohen-Macaulay(=CM) modules over CM local rings. We show that if $A$ is a hypersurface ring with dimension $d > 0$ then the Hilbert function of $M$ \wrt $\m$ is non-decreasing. If $A = Q/(f)$ for some…

Inspired by Jorgensen et. al., it is proved that if a Cohen--Macaulay local ring $R$ with dualizing module admits a suitable chain of semidualizing $R$--modules of length $n$, then $R\cong Q/(I_1+\cdots+I_n)$ for some Gorenstein ring $Q$…