Related papers: Negativity Conjecture for the First Hilbert Coeffi…

In this article, we prove a strong version of local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen-Macaulay, and complete local domain of…

Khabibullin's conjecture for integral inequalities has two numeric parameters $n$ and $\alpha$ in its statement, $n$ being a positive integer and $\alpha$ being a positive real number. This conjecture is already proved in the case where…

Let R be a commutative ring, M an R-module, and N a finitely presented R-module such that the intersection of Max(R) and Supp(N) is finite-dimensional and Noetherian. Suppose also that N is homothetic; in other words, suppose that the…

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…

A finitely generated module C over a commutative noetherian ring R is semidualizing if Hom_R(C,C) \cong R and Ext^i_R(C,C) = 0 for all i \geq 1. For certain local Cohen-Macaulay rings (R,m), we verify the equality of Hilbert-Samuel…

Let $k$ be a field and $x,y$ indeterminates over $k$. Let $R=k[x^a,x^{p_1}y^{s_1},\ldots,x^{p_t}y^{s_t},y^b] \subseteq k[x,y]$. We calculate the Hilbert polynomial of $(x^a,y^b)$. The multiplicity of this ideal provides part of a criterion…

We consider a Moebius covariant sector, possibly with infinite dimension, of a local conformal net of von Neumann algebras on the circle. If the sector has finite index, it has automatically positive energy. In the infinite index case, we…

The main aim of this article is to study the relation between $F$-injective singularity and the Frobenius closure of parameter ideals in Noetherian rings of positive characteristic. The paper consists of the following themes, including many…

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d\geq 3$ and $I$ an $\mathfrak{m}$-primary ideal of $R$. Let $r_J(I)$ be the reduction number of $I$ with respect to a minimal reduction $J$ of $I$. Suppose depth $G(I)\geq…

It is proved that for a ring $R$ that is either an affine algebra over a field, or an equicharacteristic complete local ring, some power of the Jacobian ideal of $R$ annihilates $\mathrm{Ext}^{d+1}_{R}(-,-)$, where $d$ is the Krull…

We provide a Hochster type formula for the local cohomology modules of binomial edge ideals. As a consequence we obtain a simple criterion for the Cohen-Macaulayness of these ideals and we describe their Castelnuovo-Mumford regularity and…

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.…

Let $(A,\mathfrak{m})$ be an analytically un-ramified Noetherian local ring of dimension $d \geq 1$, $I$ a regular $\mathfrak{m}$-primary ideal of $A$ and let $\overline{I}$ be integral closure ideal of $I$. If $A$ is of characteristic $p >…

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring, and suppose $R$ is Cohen-Macaulay with canonical module $\omega_R$. We develop new tools for analyzing questions involving annihilators of several homologically defined objects.…

A C*-algebra is n-homogeneous (where n is finite) if every its nonzero irreducible representation acts on an n-dimensional Hilbert space. An elementary proof of Fell's characterization of n-homogeneous C*-algebras (by means of their…

We prove that if the initial ideal of a prime ideal is Borel-fixed and the dimension of the quotient ring is less than or equal to two, then given any non-minimal associated prime ideal of the initial ideal it contains another associated…

Let $R$ be a commutative Noetherian Cohen-Macaulay local ring that has positive dimension and prime characteristic. Li proved that the tensor product of a finitely generated non-free $R$-module $M$ with the Frobenius endomorphism…

For any finitely generated module $M$ with non-zero rank over a commutative one-dimensional Noetherian local domain, we study a numerical invariant $\operatorname{h}(M)$ based on a partial trace ideal of $M$. We study its properties and…

Let $(R, \mathfrak{m}, \Bbbk)$ be a Noetherian three-dimensional Cohen-Macaulay analytically unramified ring and $I$ an $\mathfrak{m}$-primary $R$-ideal. Write $X = \mathrm{Proj}\left(\oplus_{n \in \mathbb{N}} \overline{I^n}t^n\right)$. We…

Let $M$ be a finitely generated module over a Noetherian local ring. This paper reports, for a given parameter ideal $Q$ for $M$, a criterion for the equality…