Related papers: An Upper Bound for the First Hilbert Coefficient o…

In this paper we propose a general method for computing a minimal free right resolution of a finitely presented graded right module over a finitely presented graded noncommutative algebra. In particular, if such module is the base field of…

Polynomial representations of Boolean functions over various rings such as $\mathbb{Z}$ and $\mathbb{Z}_m$ have been studied since Minsky and Papert (1969). From then on, they have been employed in a large variety of fields including…

We construct a class of Gorenstein local rings $R$ which admit minimal complete $R$-free resolutions $\bd C$ such that the sequence $\{\rank_R C_i\}$ is constant for $i< 0$, and grows exponentially for all $i>0$. Over these rings we show…

Consider a Noetherian domain $R$ and a finite group $G \subseteq Gl_n(R)$. We prove that if the ring of invariants $R[x_1, \ldots, x_n]^G$ is a Cohen-Macaulay ring, then it is generated as an $R$-algebra by elements of degree at most…

We obtain various characterizations of commutative Noetherian local rings $(R, \fm)$ in terms of homological dimensions of certain finitely generated modules. For example, we establish that $R$ is Gorenstein if the Gorenstein injective…

We describe the canonical module of a simplicial affine semigroup ring $\mathbb{K}[S]$ and its trace ideal. As a consequence, we characterize when $\mathbb{K}[S]$ is nearly Gorenstein in terms of arithmetic properties of the semigroup $S$.…

In this paper, for the development of the study of almost Gorenstein graded rings, we discuss some relations between almost Gorensteinness of Cohen--Macaulay homogeneous rings and their $h$-vectors. Concretely, for a Cohen--Macaulay…

Let $R=\bigoplus_{n\geq 0}R_n$, $\fa\supseteq \bigoplus_{n> 0}R_n$ and $M$ and $N$ be a standard graded ring, an ideal of $R$ and two finitely generated graded $R$-modules, respectively. This paper studies the homogeneous components of…

For any finitely generated module $M$ with non-zero rank over a commutative one dimensional Noetherian local domain, the numerical invariant $h(M)$ was introduced and studied in the author's previous work "Partial Trace Ideals and Berger's…

Let G be a simple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. Let \sigma:G->G be a strict endomorphism (i. e., the subgroup G(\sigma) of \sigma-fixed points is finite). Also,…

We study the depth properties of the associated graded ring of an m-primary ideal I in terms of numerical data attached to the ideal I. We also find bounds on the Hilbert coefficients of I by means of the Sally module S_J(I) of I with…

Let $A$ be an Artin algebra, $M$ be a Gorenstein projective $A$-module and $B =$ End$_A M$, then $M$ is a $A$-$B$-bimodule. We use the restricted flat dimension of $M_B$ to give a characterization of the homological dimensions of $A$ and…

The Hilbert function of standard graded algebras are well understood by Macaulay's theorem and very little is known in the local case, even if we assume that the local ring is a complete intersection. An extension to the power series ring…

The complexity of computing the solutions of a system of multivariate polynomial equations by means of Groebner bases computations is upper bounded by a function of the solving degree. In this paper, we discuss how to rigorously estimate…

Let $R$ be a noetherian algebra over a Cohen--Macaulay ring admitting a canonical module, and assume that $R$ is maximal Cohen--Macaulay over the base ring. We provide a characterization of when $R$ is left weakly Gorenstein. We further…

To every Gorenstein algebra $A$ of finite dimension greater than 1 over a field ${\Bbb F}$ of characteristic zero, and a projection $\pi$ on its maximal ideal ${\mathfrak m}$ with range equal to the annihilator $\hbox{Ann}({\mathfrak m})$…

We study the first Hilbert coefficient (after the multiplicity) $e_1$ of a local ring $(A,\m). $ Under various circumstances, it is also called the {\bf Chern number} of the local ring $A.$ Starting from the work of D.G. Northcott in the…

Let S be a polynomial ring and R=S/I where I is a graded ideal of S. The Multiplicity Conjecture of Herzog, Huneke, and Srinivasan which was recently proved using the Boij-Soederberg theory states that the multiplicity of R is bounded above…

We provide a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace K inside the second exterior product of a vector space, as well as a sharp upper bound for its Hilbert function.…

Let $ R $ be a Cohen-Macaulay local ring. We prove that the $ n $th syzygy module of a maximal Cohen-Macaulay $ R $-module cannot have a semidualizing direct summand for every $ n \ge 1 $. In particular, it follows that $ R $ is Gorenstein…