Let S be a finitely generated standard multi-graded algebra over a Noetherian local ring A. This paper first expresses mixed multiplicities of S in term of Hilbert-Samuel multiplicity that explained the mixed multiplicities S as the Hilbert-Samuel multiplicities for quotient modules of S. As an application, we get formulas for the mixed multiplicities of ideals that covers the main result of Trung-Verma 2007 [TV].
Let $R$ be a $G$ graded commutative ring and $M$ be a $G$-graded $R$-module. The set of all graded second submodules of $M$ is denoted by $Spec_G^s(M)$ and it is called the graded second spectrum of $M$. In this paper, we discuss graded rings with Noetherian graded prime spectrum and obtain some conclusions. In addition, we introduce the notion of the graded Zariski socle of graded submodules and explore their properties. Using these conclusions and properties, we also investigate $Spec_G^s(M)$ with the Zariski topology from the viewpoint of being a Noetherian space and give some related outco…
The theory of mixed multiplicities of (not necessarily Noetherian) filtrations of $m_R$-primary ideals in a Noetherian local ring $R$, has been developed by Cutkosky, Sarkar and Srinivasan. The objective of this article is to generalise a Minkowski type inequality given in their paper. We also recover a result of Cutkosky, Srinivasan and Verma as a simple consequence of our inequality
The notion of epsilon multiplicity was originally defined by Ulrich and Validashti as a limsup and they used it to detect integral dependence of modules. It is important to know if it can be realized as a limit. In this article we show that the relative epsilon multiplicity of reduced Noetherian graded algebras over an excellent local ring exists as a limit. An important special case of a result of Cutkosky concerning epsilon multiplicity, is obtained as a corollary of our main theorem. We also develop the notion of mixed epsilon multiplicity for monomial ideals.
We show the existence (and define) the mixed multiplicities of arbitrary graded families of ideals under mild assumptions. In particular, our methods and results are valid for the case of arbitrary $\mathfrak{m}$-primary graded families. Furthermore, we provide a far-reaching "Volume = Multiplicity formula" for the mixed multiplicities of graded families of ideals.
In this paper we define and explore properties of mixed multiplicities of (not necessarily Noetherian) filtrations of $m_R$-primary ideals in a Noetherian local ring $R$, generalizing the classical theory for $m_R$-primary ideals. We construct a real polynomial whose coefficients give the mixed multiplicities. This polynomial exists if and only if the dimension of the nilradical of the completion of $R$ is less than the dimension of $R$, which holds for instance if $R$ is excellent and reduced. We show that many of the classical theorems for mixed multiplicities of $m_R$-primary ideals hold fo…
This paper is a review of concepts from graded commutative algebra with specific attention given to length and multiplicity. The author's motivation for this paper comes from the study of equivariant cohomology in algebraic topology where the graded commutative algebra of the module is intimately connected to topological properties of the space (as shown by Quillen). Results (and their proofs) which are often left as exercises, or considered 'folklore' in the commutative algebra community are included in this paper, as are references to relevant applications in topology. As such, this paper is…
We develop new methods to study $\mathfrak{m}$-adic stability in an arbitrary Noetherian local ring. These techniques are used to prove results about the behavior of Hilbert-Samuel and Hilbert-Kunz multiplicities under fine $\mathfrak{m}$-adic perturbations.
Let $G$ be an abelian group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper we will obtain some results concerning the graded generalized 2-absorbing submodules and their homogeneous components. Special attention has been paid, when graded rings are graded gr-Noetherian, to find extra properties of these graded submodules.
We show that the mixed volumes of arbitrary convex bodies are equal to mixed multiplicities of graded families of monomial ideals, and to normalized limits of mixed multiplicities of monomial ideals. This result evinces the close relation between the theories of mixed volumes from convex geometry and mixed multiplicities from commutative algebra.
Let $R$ be a commutative Noetherian ring graded by a torsionfree abelian group $G$. We introduce the notion of $G$-graded irreducibility and prove that $G$-graded irreducibility is equivalent to irreducibility in the usual sense. This is a generalization of Chen and Kim's result in the $\mathbb{Z}$-graded case. We also discuss the concept of the index of reducibility and give an inequality for the indices of reducibility between any radical non-graded ideal and its largest graded subideal.
Let $R$ be a standard graded Noetherian algebra over an infinite field $K$ and $M$ a finitely generated $\mathbb{Z}$-graded $R$-module. Then for any graded ideal $I\subseteq R_+$ of $R$, we show that there exist integers $e_1\geq e_2$ such that $r(I^nM)=ρ_I(M)n+e_1$ and $D(I^nM)=ρ_I(M)n+e_2$ for $n\gg0$. Here $r(M)$ and $D(M)$ denote the reduction number of $M$ and the maximal degree of minimal generators of $M$ respectively, and $ρ_I(M)$ is an integer determined by both $M$ and $I$. We introduce the notion of generalized regularity function $Γ$ for a standard graded algebra over a Noetherian …
Let $(A,\mathfrak{m},\Bbbk)$ denote a local Noetherian ring and $\mathfrak{q}$ an ideal such that $\ell_A(M/\mathfrak{q}M) < \infty$ for a finitely generated $A$-module $M$. Let $\au = a_1,\ldots,a_d$ denote a system of parameters of $M$ such that $a_i \in \mathfrak{q}^{c_i} \setminus \mathfrak{q}^{c_i+1}$ for $i=1,\ldots,d$. It follows that $ χ:= e_0(\au;M) - c \cdot e_0(\mathfrak{q};M) \geq 0$, where $c = c_1\cdot \ldots \cdot c_d$. The main results of the report are a discussion when $χ= 0$ resp. to describe the value of $χ$ in some particular cases. Applications concern results on the mult…
We develop a theory of multiplicities and mixed multiplicities of filtrations, extending the theory for filtrations of $m$-primary ideals to arbitrary (not necessarily Noetherian) filtrations. The mixed multiplicities of $r$ filtrations on an analytically unramified local ring $R$ come from the coefficients of a suitable homogeneous polynomial in $r$ variables of degree equal to the dimension of the ring, analogously to the classical case of the mixed multiplicities of $m$-primary ideals in a local ring. We prove that the Minkowski inequalities hold for arbitrary filtrations. The characterizat…
Let $(A,\mathfrak{m})$ be a Noetherian local ring of dimension $d>0$ and $I$ an $\mathcal{I}$-primary ideal of $A$. In this paper, we discuss a sufficient condition, for the Buchsbaumness of the local ring $A$ to be passed onto the associated graded ring of filtration. Let $\mathcal{I}$ denote an $I$-good filtration. We prove that if $A$ is Buchsbaum and the I-invariant, $I(A)$ and $I(G(\mathcal{I}))$, coincide then the associated graded ring $G(\mathcal{I})$ is Buchsbaum. As an application of our result, we indicate an alternative proof of a conjecture, of Corso on certain boundary conditions…
Let $P$ be a commutative Noetherian ring and $F$ be a self-dual acyclic complex of finitely generated free $P$-modules. Assume that $F$ has length four and $F_0$ has rank one. We prove that $F$ can be given the structure of a Differential Graded Algebra with Divided Powers; furthermore, the multiplication on $F$ exhibits Poincaré duality. This result is already known if $P$ is a local Gorenstein ring and $F$ is a minimal resolution. The purpose of the present paper is to remove the unnecessary hypotheses that $P$ is local, $P$ is Gorenstein, and $F$ is minimal.
We introduce two new invariants of a Noetherian (standard graded) local ring $(R, \mathfrak m)$ that measure the number of generators of certain kinds of reductions of $\mathfrak m,$ and we study their properties. Explicitly, we consider the minimum among the number of generators of ideals $I$ such that either $I^2 = \mathfrak m^2$ or $I \supseteq \mathfrak m^2$ holds. We investigate subsequently the case that $R$ is the quotient of a polynomial ring $k[x_1, \dots, x_n]$ by an ideal $I$ generated by homogeneous quadratic forms, and we compute these invariants. We devote specific attention to t…
