Related papers: A note on the Eisenbud-Mazur Conjecture

Let $I$ be an ideal of height $d$ in a regular local ring $(R,m,k=R/m)$ of dimension $n$ and let $\Omega$ denote the canonical module of $R/I$. In this paper we first prove the equivalence of the following: the non-vanishing of the edge…

Let $(R, \mathfrak m)$ be a commutative noetherian local ring. We investigate under which conditions an $R$-module $M$ is generated by an ideal $I$, i.e. there exists an epimorphism $I^{(\Lambda)} \twoheadrightarrow M$. If $M$ is uniserial,…

Let $S$ be a standard graded polynomial ring over a field, and $I$ be a homogeneous ideal that contains a regular sequence of degrees $d_1,\ldots,d_n$. We prove the Eisenbud-Green-Harris conjecture when the forms of the regular sequence…

This Note provides first a generalization of the stabilization result of Eisenbud and Ulrich for the regularity of powers of a m-primary ideal to the case of ideals that are not generated in a single degree. We then partially extend our…

We study the ring extensions R \subseteq T having the same set of prime ideals provided Nil(R) is a divided prime ideal. Some conditions are given under which no such T exist properly containing R. Using idealization theory, the examples…

In this paper, we introduce a new generalization of weakly prime ideals called $I$-prime. Suppose $R$ is a commutative ring with identity and $I$ a fixed ideal of $R$. A proper ideal $P$ of $R$ is $I$-prime if for $a, b \in R$ with $ab \in…

In this article, we prove the remaining open cases of the Fontaine-Mazur conjecture on two-dimensional regular Galois representations over $\Gal(\overline{\Q}/\Q)$ when $p=3$, hence concluding the conjecture in the regular case for all odd…

Motivated by the work of Chudnovsky and the Eisenbud-Mazur Conjecture on evolutions, Harbourne and Huneke give a series of conjectures that relate symbolic and regular powers of ideals of fat points in $\mathbb P^n$. The conjectures involve…

Mason's Conjecture asserts that for an $m$--element rank $r$ matroid $\M$ the sequence $(I_k/\binom{m}{k}: 0\leq k\leq r)$ is logarithmically concave, in which $I_k$ is the number of independent $k$--sets of $\M$. A related conjecture in…

When M is a finitely generated graded module over a standard graded algebra S and I is an ideal of S, it is known from work of Cutkosky, Herzog, Kodiyalam, R\"omer, Trung and Wang that the Castelnuovo-Mumford regularity of I^mM has the form…

Let (R,m) be a local ring with prime ideals p and q such that p+q is an m-primary ideal. If R is regular and contains a field, and dim(R/p)+dim(R/q)=dim(R), we prove that p^{(r)}\cap q^{(n)}\subseteq m^{m+n} for all positive integers r and…

Let $S$ be a regular local ring or a polynomial ring over a field and $I$ be an ideal of $S$. Motivated by a recent result of Herzog and Huneke, we study the natural question of whether $I^m$ is a Golod ideal for all $m\geq 2$. We observe…

For the ideal $\mathfrak{p}$ in $k[x, y, z]$ defining a space monomial curve, we show that $\mathfrak{p}^{(2 n - 1)} \subseteq \mathfrak{m} \mathfrak{p}^{n}$ for some positive integer $n$, where $\mathfrak{m}$ is the maximal ideal $(x, y,…

We study the Eisenstein ideal for modular forms of even weight $k>2$ and prime level $N$. We pay special attention to the phenomenon of $\mathit{extra \ reducibility}$: the Eisenstein ideal is strictly larger than the ideal cutting out…

We develop tools to study the problem of containment of symbolic powers $I^{(m)}$ in powers $I^r$ for a homogeneous ideal $I$ in a polynomial ring $k[{\bf P}^N]$ in $N+1$ variables over an algebraically closed field $k$. We obtain results…

Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). A theorem of Lipman implies that I has a unique factorization as a *-product of special *-simple complete ideals with possibly negative exponents for…

The Eisenbud-Green-Harris conjecture states that a homogeneous ideal in k[x_1,...,x_n] containing a homogeneous regular sequence f_1,...,f_n with deg(f_i)=a_i has the same Hilbert function as an ideal containing x_i^{a_i} for 1 \leq i \leq…

It has been conjectured by Eisenbud, Green and Harris that if $I$ is a homogeneous ideal in $k[x_1,...,x_n]$ containing a regular sequence $f_1,...,f_n$ of degrees $\deg(f_i)=a_i$, where $2\leq a_1\leq ... \leq a_n$, then there is a…

Extension conjecture states that if a simple module over an artin algebra has nonzero first self-extension group then it has nonzero i-th self-extension group for infinitely many positive integers i. It is shown by recollement of…

A commutative noetherian local ring $(R,\mathfrak{m})$ is Gorenstein if and only if every parameter ideal of $R$ is irreducible. Although irreducible parameter ideals may exist in non-Gorenstein rings, Marley, Rogers, and Sakurai show there…