Related papers: Uniform Symbolic Topologies via Multinomial Expans…

Given a symbolic power of a homogeneous ideal in a polynomial ring, we study the problem of determining which powers of the ideal contain it. For ideals defining 0-dimensional subschemes of projective space, as an immediate corollary of our…

We prove that over a commutative noetherian ring the three approaches to introducing depth for complexes: via Koszul homology, via Ext modules, and via local cohomology, all yield the same invariant. Using this result, we establish a far…

For a Noetherian $R$-algebra $\Lambda$, there is a canonical inclusion $\mathsf{tors}\Lambda\to\prod_{\mathfrak{p}\in \mathrm{Spec} R}\mathsf{tors}(\kappa(\mathfrak{p})\Lambda)$, and each element in the image satisfies a certain…

This paper is concerned with tight closure in a commutative Noetherian ring $R$ of prime characteristic $p$, and is motivated by an argument of K. E. Smith and I. Swanson that shows that, if the sequence of Frobenius powers of a proper…

Let $D$ be an integrally closed local Noetherian domain of Krull dimension 2, and let $f$ be a nonzero element of $D$ such that $fD$ has prime radical. We consider when an integrally closed ring $H$ between $D$ and $D_f$ is determined…

In this paper we prove the following theorem. Let L/\Q_p be a finite extension with ring of integers O_L and maximal ideal lambda. Theorem 1. Suppose that p >= 5. Suppose also that \rho:G_\Q -> GL_2(O_L) is a continuous representation…

We study the symbolic powers of determinantal ideals of generic, generic symmetric, and Hankel matrices of variables, and of Pfaffians of generic skew-symmetric matrices, in prime characteristic. Specifically, we show that the limit…

Let $R$ be a commutative Noetherian ring of prime characteristic $p$. In this paper we give a short proof using filter regular sequences that the set of associated prime ideals of $H^t_I(R)$ is finite for any ideal $I$ and for any $t \ge 0$…

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal in a polynomial ring over a field $\mathbb{K}$. In this paper, we obtain upper bounds for the Castelnuovo-Mumford regularity of symbolic powers of certain…

We prove conditions ensuring that a Lie ideal or an invariant additive subgroup in a ring contains all additive commutators. A crucial assumption is that the subgroup is fully noncentral, that is, its image in every quotient is noncentral.…

We give an algebraic and self-contained proof of the existence of the so-called Noetherian operators for primary submodules over general classes of Noetherian commutative rings. The existence of Noetherian operators accounts to provide an…

We study the symbolic powers of square-free monomial ideals via symbolic Rees algebras and methods in prime characteristic. In particular, we prove that the symbolic Rees algebra and the symbolic associated graded algebra are split with…

Let $I$ be an ideal whose symbolic Rees algebra is Noetherian. For $m \geq 1$, the $m$-th symbolic defect, sdefect$(I,m)$, of $I$ is defined to be the minimal number of generators of the module $\frac{I^{(m)}}{I^m}$. We prove that…

We study the asymptotic behaviour of $v$-number and local $v$-numbers of Noetherian generalized symbolic power filtrations $\mathcal I=\{I_n\}$ in a Noetherian $\mathbb N$-graded domain and show that they are quasi-linear type. We provide…

For finitely generated modules $N \subsetneq M$ over a Noetherian ring $R$, we study the following properties about primary decomposition: (1) The Compatibility property, which says that if $\ass (M/N)=\{P_1, P_2, ..., P_s\}$ and $Q_i$ is a…

Let R be a commutative ring with identity and N(R) be the set of all nilpotent elements of R. The aim of this paper is to introduce and study the notion of nil-prime ideals as a generalization of prime ideals. We say that a proper ideal P…

Let $p$ be a prime, let $d \geq 1$ be an integer and $A$ be the algebra of square matrices of size $d$ over the field of order $p$. Let $P, Q \in A[x_1, \dots x_n]$ be polynomials in $n$ indeterminates with coefficients in $A$, such that…

The so-called Dao numbers are a sort of measure of the asymptotic behaviour of full properties of certain product ideals in a Noetherian local ring $R$ with infinite residue field and positive depth. In this paper, we answer a question of…

Let $R$ be a commutative Noetherian ring with non-zero identity, $\fa$ an ideal of $R$, and $X$ an $R$--module. In this paper, for fixed integers $s, t$ and a finite $\fa$--torsion $R$--module $N$, we first study the membership of…

In 2007, G.E. Andrews introduced the $(n+1)$-variable combinatorial generating function $R_n(x_1,x_2,\cdots,x_n;q)$ for ranks of $n$-marked Durfee symbols, an $(n+1)$-dimensional multisum, as a vast generalization to the ordinary…