Let $𝔞$, $I$, $J$ be ideals of a Noetherian local ring $(R,𝔪,k)$. Let $M$ and $N$ be finitely generated $R$-modules. We give a generalized version of the Duality Theorem for Cohen-Macaulay rings using local cohomology defined by a pair of ideals. We study the behavior of the endomorphism rings of $H_{I,J}^t\left(M\right)$ and $D\left(H_{I,J}^t\left(M\right)\right)$, where $t$ is the smallest integer such that the local cohomology with respect to a pair of ideals is nonzero and $D(-):={\mathrm{Hom}}_R(-,E_R\left(k\right))$ is the Matlis dual functor. We show that if $R$ is a $d$-dimensional complete Cohen-Macaulay ring and $H_{I,J}^i\left(R\right)=0$...

In this paper, we will present several necessary and sufficient conditions on a commutative ring such that the algebraic and geometric local cohomologies are equivalent.

Let $R$ be a local ring and $C$ a semidualizing module of $R$. We investigate the behavior of certain classes of generalized Cohen-Macaulay $R$-modules under the Foxby equivalence between the Auslander and Bass classes with respect to $C$. In particular, we show that generalized Cohen-Macaulay $R$-modules are invariant under this equivalence and if $M$ is a finitely generated $R$-module in the Auslander class with respect to $C$ such that $C{\otimes }_{R}M$ is surjective Buchsbaum, then $M$ is also surjective Buchsbaum.

Let Δ denote the discriminant of the generic binary d-ic. We show that for d ≥ 3, the Jacobian ideal of Δ is perfect of height 2. Moreover we describe its SL2-equivariant minimal resolution and the associated differential equations satisfied by Δ. A similar result is proved for the resultant of two forms of orders d, e whenever d ≥ e-1. If Φn denotes the locus of binary forms with total root multiplicity ≥ d-n, then we show that the ideal of Φn is also perfect, and we construct a covariant which...

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and $𝒞$-minimaxness of local cohomology modules. We show that if $M$ is a minimax $R$-module, then the local-global principle is valid for minimaxness of local cohomology modules. This implies that if $n$ is a nonnegative integer such that ${\left(H_I^i\left(M\right)\right)}_{𝔪}$ is a minimax $R_{𝔪}$-module for all $𝔪\in \mathrm{Max}\left(R\right)$ and for all $i<n$, then the set ${\mathrm{Ass}}_R\left(H_I^n\left(M\right)\right)$ is finite. Also, if $H_I^i\left(M\right)$ is minimax for...

We obtain, in a simple way, an estimate for the Noether exponent of an ideal I without embedded components (i.e. we estimate the smallest number μ such that ${\left(radI\right)}^{\mu }\subset I$).

Motivated by the paper by H. Herrlich, E. Tachtsis (2017) we investigate in ZFC the following compactness question: for which uncountable cardinals $\kappa$, an arbitrary nonempty system $S$ of homogeneous $\mathbb{Z}$-linear equations is nontrivially solvable in $\mathbb{Z}$ provided that each of its subsystems of cardinality less than $\kappa$ is nontrivially solvable in $\mathbb{Z}$?