Related papers: The second Hilbert coefficient of modules with alm…

Intuitively, the filter dimension of an algebra or a module measures how `close' standard filtrations of the algebra or the module are. In particular, for a simple algebra it also measures the growth of how `fast' one can prove that the…

Let G be GL_N or SL_N as reductive linear algebraic group over a field k of positive characteristic p. We prove several results that were previously established only when N < 6 or p > 2^N. Let G act rationally on a finitely generated…

Hilbert algebras are the implicative subreducts of Heyting algebras. It is shown that having depth at most n is an equational condition in Hilbert algebras. This generalizes an analogous well-known result in the setting of Heyting algebras.

Let $(R,\mathfrak{m})$ be a Noetherian local ring such that $\widehat{R}$ is reduced. We prove that, when $\widehat{R}$ is $S_2$, if there exists a parameter ideal $Q\subseteq R$ such that $\bar{e}_1(Q)=0$, then $R$ is regular and…

Numerical invariants of a minimal free resolution of a module $M$ over a regular local ring $(R,\n)$ can be studied by taking advantage of the rich literature on the graded case. The key is to fix suitable $\n$-stable filtrations ${\mathbb…

We introduce a general notion of depth two for ring homomorphism N --> M, and derive Morita equivalence of the step one and three centralizers, R = C_M(N) and C = End_{N-M}(M \o_N M), via dual bimodules and step two centralizers A =…

In this paper we introduce a new technique to study associated graded modules. Let $(A,\m)$ be a Noetherian local ring with $\depth A \geq 2$. Our techniques gives a necessary and sufficient condition for $\depth G_{\m^n}(A) \geq 2$ for all…

Let $(R, \frak m)$ be a local ring and $M$ a finitely generated $R$-module. It is shown that if $M$ is relative Cohen-Macaulay with respect to an ideal $\frak a$ of $R$, then $\text{Ann}_R(H_{\mathfrak{a}}^{\text{cd}(\mathfrak{a},…

Let $M$ be a complex $n$-dimensional projective manifold in $\mathbb{P}^{n+r}$ endowed with the Fubini-Study metric of constant holomorphic sectional curvature $1$, $\sigma$ its second fundamental form, and $\underline{|\sigma|}^2$ the mean…

Let $R$ be a commutative Noetherian ring, $\mathfrak a$ and $\mathfrak b$ ideals of $R$. In this paper, we study the finiteness dimension $f_{\mathfrak a}(M)$ of $M$ relative to $\mathfrak a$ and the $\mathfrak b$-minimum $\mathfrak…

Stanley decompositions of multigraded modules $M$ over polynomials rings have been discussed intensively in recent years. There is a natural notion of depth that goes with a Stanley decomposition, called the Stanley depth. Stanley…

In 2006, Kaneko and Koike defined extremal quasimodular forms and proved their existence in depth $1$ and $2$. After normalizing and restricting to the case of depth at most $4$, they conjectured a certain bound on the Fourier coefficients…

This paper gives new bounds on the first Hilbert coefficient of an ideal of finite colength in a Cohen-Macaulay local ring. The bound given is quadratic in the multiplicity of the ideal. We compare our bound to previously known bounds, and…

Let $(A,\mathfrak{m})$ be a \CM\ local ring, then notion of $T$-split sequence was introduced in part-1 of this paper for $\mathfrak{m}$-adic filtration with the help of numerical function $e^T_A$. We have explored the relation between…

The goal of this paper is to derive in the two-dimensional case necessary and sufficient minimality conditions in terms of the second variation for the functional \[ v\mapsto\int_{\Omega}\big(|\nabla v|^{2}+\chi_{\{v>0\}}Q^{2} \big)\,dx, \]…

We generalize Iarrobino's symmetric decomposition for the associated graded algebra of an Artinian Gorenstein algebra to a symmetric decomposition of finite-length self-dual modules over a local algebra, and we deduce consequences for the…

Let $G$ be the simple algebraic group $\mathrm{SL}_2$ defined over an algebraically closed field $k$ of characteristic $p > 0$. Using results of A. Parker, we develop a method which gives, for any $q \in \mathbb{N}$, a closed form…

We initiate a study of Hilbert modules over the polynomial algebra A=C[z_1,...,z_d] that are obtained by completing A with respect to an inner product having certain natural properties. A standard Hilbert module is a finite multiplicity…

Let $h:\mathbb Z \to \mathbb Z_{\geq 0}$ be a nonzero function with $h(k)=0$ for $k\ll 0$. We define the Hilbert depth of $h$ by $\operatorname{hdepth}(h)=\max\{d\;:\; \sum_{j\leq k} (-1)^{k-j}\binom{d-j}{k-j}h(j)\geq 0\text{ for all }k\leq…

The aim of this paper is to introduce a method for computing Hilbert decompositions (and consequently the Hilbert depth) of a finitely generated multigraded module $M$ over the polynomial ring $K[X_1,..., X_n]$ by reducing the problem to…