Related papers: Relative Hilbert co-efficients

Let $(R,m)$ be a Cohen-Macaulay local ring of positive dimension $d$ and infinite residue field. Let $I$ be an m-primary ideal and $J$ a minimal reduction of $I$. In this paper, we show that $\widetilde{r_J(I)}\leq r_J(I)$. This answer to a…

Let $I$ be an ideal of a local ring $(R,\mathfrak m)$ with $d = \dim R.$ For the local cohomology module $H^i_I(R)$ it is a well-known fact that it vanishes for $i > d$ and is an Artinian $R$-module for $i = d.$ In the case that the…

We explore the classical Lech's inequality relating the Hilbert--Samuel multiplicity and colength of an $\mathfrak{m}$-primary ideal in a Noetherian local ring $(R,\mathfrak{m})$. We prove optimal versions of Lech's inequality for…

Let $d_1,...,d_r$ be positive integers and let $I = (F_1,...,F_r)$ be an ideal generated by general forms of degrees $d_1,...,d_r$, respectively, in a polynomial ring $R$ with $n$ variables. When all the degrees are the same we give a…

Let $R$ be a commutative Noetherian ring, and let $\mathfrak a$ be a proper ideal of $R$. Let $M$ be a non-zero finitely generated $R$-module with the finite projective dimension $p$. Also, let $N$ be a non-zero finitely generated…

The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal $I$ in the polynomial ring $S=K[x_1,...,x_n]$ and a finitely generated graded $S$-module, the Hilbert coefficients $e_i(M/I^kM)$ are polynomial…

Let $(R,\mathfrak{m})$ be a Noetherian local ring of prime characteristic $p$ and $Q$ be an $\mathfrak{m}$-primary parameter ideal. We give criteria for F-rationality of $R$ using the tight Hilbert function $H^*_Q(n)=\ell(R/(Q^n)^*$ and the…

Let $(R, \mathfrak m)$ be a Cohen-Macaulay local ring of dimension $d \geq 2,$ and $I$ an $\mathfrak m$-primary ideal of $R.$ Denote $r_{J}(I)$ as the reduction number of $I$ with respect to a minimal reduction $J$ of $I,$ and $\rho(I)$ as…

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…

Let F denote a binary form of order d over the complex numbers. If r is a divisor of d, then the Hilbert covariant H_{r,d}(F) vanishes exactly when F is the perfect power of an order r form. In geometric terms, the coefficients of H give…

Let $(R,\mathfrak{M})$ be a two-dimensional regular local ring with algebraically closed residue field. Let $I$ be a simple complete $\mathfrak{M}$-primary ideal of $R$ and let $w$ denote its unique Rees valuation. Then the degree function…

We prove that if M is a finitely-generated module of dimension d with finite local cohomologies over a Noetherian local ring, and if the ith local cohomology module of M is zero unless i = d, i = 0, and i = r for some r strictly between 0…

Let (R,m) be a Noetherian local ring of depth d and C a semidualizing R-complex. Let M be a finite R-module and t an integer between 0 and d. If G_C-dimension of M/IM is finite for all ideals I generated by an R-regular sequence of length…

Let $I$ denote an ideal of a local Gorenstein ring $(R, \mathfrak m)$. Then we show that the local cohomology module $H^c_I(R), c = \height I,$ is indecomposable if and only if $V(I_d)$ is connected in codimension one. Here $I_d$ denotes…

Let R be a regular local ring of dimension d, I an ideal of R, and M a finitely generated R-module of dimension n. We prove that the set of associated primes of Ext^i_R(R/I,H^j_I(M)) is finite for all i and j in the following cases: (1) dim…

We prove that if the first tight Hilbert coefficient vanishes then ring is $F$-rational provided it is a Buchsbaum local ring satisfying the $(S_2)$ condition.

Let $(S, m)$ be an $n$-dimensional regular local ring essentially of finite type over a field and let $I$ be an ideal of $S$. We prove that if $\text{depth} S/I \ge 3$, then the cohomological dimension $\mathrm{cd}(S, I)$ of $I$ is less…

Let $I$ and $J$ be two ideals of a commutative Noetherian ring $R$ and $M$ be an $R$-module. For a non-negative integer $n$ it is shown that, if the sets $\Ass_R(\Ext^{n} _{R}(R/I,M))$ and $\Supp_R(\Ext^{i}_{R}(R/I,H^{j}_{I,J} (M)))$ are…

Let $R$ be a commutative ring with $1\neq0$. In this article, we introduce the concept of weakly $(m,n)-$closed $\delta-$primary ideals of $R$ and explore its basic properties. We show that $I\bowtie^{f}J$ is a weakly $(m,n)-$closed…

The j-multiplicity is an invariant that can be defined for any ideal in a Noetherian local ring $(R, m)$. It is equal to the Hilbert-Samuel multiplicity if the ideal is $m$-primary. In this paper we explore the computability of the…