Related papers: Deformation of Residual Intersections

Cohen-Macaulayness, unmixedness, the structure of the canonical module and the stability of the Hilbert function of algebraic residual intersections are studied in this paper. Some conjectures about these properties are established for…

Every quotient R/I of a semigroup ring R by a radical monomial ideal I has a unique minimal injective-like resolution by direct sums of quotients of R modulo prime monomial ideals. The quotient R/I is Cohen-Macaulay if and only if every…

Let $(R,m)$ be a Noetherian local ring and $I$ an ideal with finite projective dimension. If $R/I$ satisfies some property $\mathcal{P}$, it is natural to ask whether $R$ would also satisfy this property $\mathcal{P}$. This is called the…

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 focus in this paper is in effective computation of the core core(I) of an ideal I which is defined to be the intersection of all minimal reductions of I. The first main result is a closed formula for the graded core(m) of the maximal…

Let $K$ be a field and $I$ a monomial ideal of the polynomial ring $S=K[x_1,..., x_n]$ generated by monomials $u_1,u_2,..., u_t$. We show that $S/I$ is pretty clean if either: 1) $u_1,u_2,..., u_t$ is a filter-regular sequence, 2)…

Let $I$ be a monomial ideal of the polynomial ring $S=K[x_1,...,x_4]$ over a field $K$. Then $S/I$ is sequentially Cohen-Macaulay if and only if $S/I$ is pretty clean. In particular, if $S/I$ is sequentially Cohen-Macaulay then $I$ is a…

The arithmetic rank of an ideal in a polynomial ring over an algebraically closed field is the smallest number of equations needed to define its vanishing locus set-theoretically. We determine the arithmetic rank of the generic $m$-residual…

A classical result of Micali asserts that a Noetherian local ring is regular if and only if the Rees algebra of its maximal ideal is defined by an ideal of linear forms. In this case, this defining ideal may be realized as a determinantal…

It is shown that a module is sequentially Cohen-Macaulay if and only if the index of reducibility for distinguished parameter ideals are eventually constant with special value. As corollaries to the main theorem we given to characterize the…

We construct a local Cohen-Macaulay ring $R$ with a prime ideal $\mathfrak{p}\in\spec(R)$ such that $R$ satisfies the uniform Auslander condition (UAC), but the localization $R_{\mathfrak{p}}$ does not satisfy Auslander's condition (AC).…

For a Cohen-Macaulay local ring $(R,\mathfrak{m})$ with canonical module, we study how relations between $\text{index}(R)$ and $\text{g}\ell\ell(R)$ and between $\text{index}(R)$ and $e(R)$ are preserved when factoring out regular sequences…

The main result of the paper states that for a graded ideal I in a polynomial ring R over a field of characteristic 0, the Hilbert functions of the local cohomology modules of R/I and of R/Gin(I) coincide if and only if R/I is sequentially…

This paper studies Ulrich ideals in one-dimensional Cohen-Macaulay local rings. A correspondence between Ulrich ideals and overrings is given. Using the correspondence, chains of Ulrich ideals are closely explored. The specific cases where…

Let $(R,\fm)$ be commutative Noetherian local ring. It is shown that $R$ is Cohen--Macaulay ring if there exists a Cohen--Macaulay finite (i.e. finitely generated) $R$--module with finite upper Gorenstein dimension. In addition, we show…

Generic Bourbaki ideals were introduced by Simis, Ulrich and Vasconcelos to study the Cohen-Macaulay property of Rees algebras of modules. In this article we prove that the same technique can sometimes be used to investigate the…

Let $(R, {\mathfrak m})$ be a Noetherian local ring and let $I$ be an $R$-ideal. Inspired by the work of H\"ubl and Huneke, we look for conditions that guarantee the Cohen-Macaulayness of the special fiber ring ${\mathcal F}={\mathcal…

Let $S = K[x_1,..., x_n]$ be a polynomial ring over a field $K$. Let $I(G) \subseteq S$ denote the edge ideal of a graph $G$. We show that the $\ell$th symbolic power $I(G)^{(\ell)}$ is a Cohen-Macaulay ideal (i.e., $S/I(G)^{(\ell)}$ is…

Let $(R, {\frak m})$ be a local ring, $I$ a proper ideal of $R$ and $M$ a finitely generated $R$-module of dimension $d$. We discuss the local homology modules of $H^d_I(M)$. When $M$ is Cohen-Macaulay, it is proved that $H^d_{{\frak…

This paper shows that if $R$ is a homomorphic image of a Cohen-Macaulay local ring, then $R$-module $M$ is sequentially generalized Cohen-Macaulay if and only if the difference between Hilbert coefficients and arithmetic degrees for all…