Related papers: The Monomial Conjecture and order ideals II

Let R be a regular local ring of dimension d, I an ideal of R, and M a finitely generated R-module of dimension n. We prove that the set of associated primes of Ext^i_R(R/I,H^j_I(M)) is finite for all i and j in the following cases: (1) dim…

Let $R$ be a regular ring of dimension $d$ containing a field $K$ of characteristic zero. If $E$ is an $R$-module let $Ass^i E = \{ Q \in \ Ass E \mid \ height Q = i \}$. Let $P$ be a prime ideal in $R$ of height $g$. We show that if $R/P$…

For any two integers $d,r \geq 1$, we show that there exists an edge ideal $I(G)$ such that the ${\rm reg}\left(R/I(G)\right)$, the Castelnuovo-Mumford regularity of $R/I(G)$, is $r$, and ${\rm deg} (h_{R/I(G)}(t))$, the degree of the…

We give explicit criteria that imply the resurgence of a self-radical ideal in a regular ring is strictly smaller than its codimension, which in turn implies that the stable version of Harbourne's conjecture holds for such ideals. This…

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$, $N$ two finitely generated $R$-modules. The aim of this paper is to investigate the $I$-cofiniteness of generalized local cohomology modules $\displaystyle…

Let $\fa$ denote an ideal of a $d$-dimensional Gorenstein local ring $R$ and $M$ and $N$ two finitely generated $R$-modules with $\pd M< \infty$. It is shown that $H^d_{\fa}(M,N)=0$ if and only if $\dim \hat{R}\big/ \fa\hat{R}+\fp>0$ for…

Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. Under various hypotheses, it is proved that the center of $\mbox{End}_R(M)$ coincides with the endomorphism ring of the trace ideal of $M$. These results are…

In this paper we consider reduced (non-normal) commutative noetherian rings $R$. With the help of conductor ideals and trace ideals of certain $R$-modules we deduce a criterion for a reflexive $R$-module to be closed under multiplication…

We exhibit an equigenerated monomial ideal $I\subseteq K[x,y,z,w]$ with $\operatorname{reg}(\overline{I})>\operatorname{reg}(I)$. The ideal $I$ is generated in degree 4 and satisfies $\operatorname{reg}(I)=4$, while its integral closure…

Let $\Gamma$ be a cancelation monoid with the neutral element $e$. Consider a $\Gamma$-graded ring $A=\oplus_{\gamma\in\Gamma}A_{\gamma}$, which is not necessarily commutative. It is proved that $A_e$, the degree-$e$ part of $A$, is a local…

We develop a new technique for studying monomial ideals in the standard polynomial rings $A[X_1,\ldots,X_d]$ where $A$ is a commutative ring with identity. The main idea is to consider induced ideals in the semigroup ring…

We prove an index theorem for the quotient module of a monomial ideal. We obtain this result by resolving the monomial ideal by a sequence of Bergman space like essentially normal Hilbert modules.

We exhibit an example of a product of two proper monomial ideals such that the residue class ring is not Golod. We also discuss the strongly Golod property for rational powers of monomial ideals, and introduce some sufficient conditions for…

Let $\L$ be an artin algebra. Iyama conjectures that the endomorphism ring of any two maximal $l$-orthogonal modules, $M_1$ and $M_2$, are derived equivalent. He proves the conjecture for $l=1$, and for $l>1$ he gives some orthogonality…

Let $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be monomial ideal of $R$. In this paper, we show that if $I$ is a generic monomial ideal, then $R/I$ is pretty clean if and only if $R/I$ is…

Our purpose in this work is multifold. First, we provide general criteria for the finiteness of the projective and injective dimensions of a finite module $M$ over a (commutative) Noetherian ring $R$. Second, in the other direction, we…

Let $R$ be a local or positively graded ring with a regular presentation $R \cong Q/I$ where $I$ is a monomial ideal generated by $n$ elements on a regular sequence. In Briggs-Grifo-Pollitz (2025), the authors classify the cohomological…

Given a Cohen-Macaulay local ring, the cohomology annihilator ideal and the annihilator of the stable category of maximal Cohen-Macaulay modules are two ideals closely related both with each other and the singularities of the ring. Kimura…

Let $S$ be a polynomial algebra over a field. We study classes of monomial ideals (as for example lexsegment ideals) of $S$ having minimal depth. In particular, Stanley's conjecture holds for these ideals. Also we show that if Stanley's…

In this article we first compare the set of elements in the socle of an ideal of a polynomial algebra $K[x_1,\ldots,x_d]$ over a field $K$ that are not in the ideal itself and Macaulay's inverse systems of such polynomial algebras in a…