Narrow Results

In [D. Rees, Hilbert functions and pseudo-rational local rings of dimension two, J. London Math. Soc. (2) 24 (1981) 467–479], Rees gave a characterization for the normal joint reduction number zero of two $𝔪$-primary ideals in an analytically unramified Cohen–Macaulay local ring of dimension two. Rees’ result is a generalization of Zariski’s product theorem for complete ideals in a regular local ring of dimension two. The aim of this paper is to extend Rees’ theorem for the ordinary powers of $𝔪$-primary ideals $I$ and $J$ in a Cohen–Macaulay local ring of dimension two. Following Rees’ approach, we define the modified Koszul homology modules $M_{r,s}^1(a^k,b^k)$ for a joint reduction $(a,b)$ of $I$ and $J$. Under the additional assumption that the associated graded rings of $I$ and $J$ have positive depth, we obtain a characterization of the joint reduction number zero of $I$ and $J$ in terms of the vanishing of the module $M_{0,0}^1(a,b)$, as well as in terms of the Hilbert coefficients and the bigraded Hilbert coefficients. More generally, we introduce the joint reduction lattice and study the vanishing of $M_{r,s}^1(a,b)$ for any $r,s\ge 0$. This gives a characterization for a vector $(r,s)$ to be in the joint reduction lattice of $I$ and $J$. We also give a cohomological interpretation of these theorems by investigating the local cohomology modules of the bigraded extended Rees algebra. This gives another characterization for a vector $(r,s)$ to be in the joint reduction lattice and also extends a recent result of Masuti and Verma in [Local cohomology of bigraded Rees algebras and normal Hilbert coefficients, J. Pure Appl. Algebra218(5) (2014) 904–918] for ordinary powers of ideals.

In this paper, we study the powers of the generalized binomial edge ideal ${{𝒥}}_{K_m,P_n}$ of a path graph $P_n$. We explicitly compute their regularities and determine the limit of their depths. We also show that these ordinary powers coincide with their symbolic powers. Additionally, we study the Rees algebra and the special fiber ring of ${{𝒥}}_{K_m,P_n}$ utilizing Sagbi basis theory. In particular, we obtain precise formulas for the regularity of these blowup algebras.

Let f₀, f₁, f₂, f₃ be linearly independent nonzero homogeneous polynomials in the standard ℤ-graded ring R ≔ 𝕂[s, t, u] of the same degree d, and gcd(f₀, f₁, f₂, f₃) = 1. This defines a rational map ℙ² → ℙ³. The Rees algebra Rees(I) = R ⊕ I ⊕ I² ⊕ ⋯ of the ideal I = 〈f₀, f₁, f₂, f₃〉 is the graded R-algebra which can be described as the image of the R-algebra homomorphism h: R[x, y, z, w ] → Rees(I). This paper discusses one result concerning the structure of the kernel of the map h when I is a saturated local complete intersection ideal with V(I) ≠ ∅ and μ-basis of degrees (1,1,d - 2).

The aim of this paper is to prove some results about the one-fibered ideals (i.e. ideals with only one Rees valuation). We will introduce and study the notion of nth roots of this type of ideals and we will present several properties related to its integral closure by using the one-fiberedness criterion of Hübl and Swanson.

This work deals with the notion of Newton complementary duality as raised originally in the work of the second author and B. Costa. A conceptual revision of the main steps of the notion is accomplished which then leads to a vast simplification and improvement of several statements concerning rational maps and their images. A ring-homomorphism like map is introduced that allows for a close comparison between the respective graphs of a rational map and its Newton dual counterpart.

We provide a characterization of the Arf property in both the numerical duplication of a numerical semigroup and in a member of a family of quotients of the Rees algebra studied in [V. Barucci, M. D’Anna and F. Strazzanti, A family of quotients of the Rees algebra, Commun. Algebra 43(1) (2015) 130–142].

In this paper, we focus on the initial degree and the vanishing of the Valabrega–Valla module of a pair of monomial ideals $J\subseteq I$ in a polynomial ring over a field $𝕂$. We prove that the initial degree of this module is bounded above by the maximum degree of a minimal generators of J. For edge ideal of graphs, a complete characterization of the vanishing of the Valabrega–Valla module is given. For higher degree ideals, we find classes, where the Valabrega–Valla module vanishes. For the case that J is the facet ideal of a clutter ${𝒞}$ and I is the defining ideal of singular subscheme of J, the non-vanishing of this module is investigated in terms of the combinatorics of ${𝒞}$. Finally, we describe the defining ideal of the Rees algebra of I/J provided that the Valabrega–Valla module is zero.

In this paper, we study a family of rational monomial parametrizations. We investigate a few structural properties related to the corresponding monomial ideal J generated by the parametrization. We first find the implicit equation of the closure of the image of the parametrization. Then we provide a minimal graded free resolution of the monomial ideal J, and describe the minimal graded free resolution of the symmetric algebra of J. Finally, we provide a method to compute the defining equations of the Rees algebra of J using three moving planes that follow the parametrization.

In this paper, we aim to generalize the ideal duplication defined for numerical semigroups to commutative rings with unity. We introduce the semitrivial ideal extension, a construction that, starting with an ideal of a commutative ring $R$ with unity and a submodule of a module $M$ over $R$, under specific assumptions, produces an ideal of the semitrivial extension $R{⋉}_{\varphi }M$. Using this tool we characterize the homogeneous prime ideals of a semitrivial extension and we completely describe the family of the maximal ideals. Furthermore, as it was done for the numerical duplication, using the semitrivial ideal extension, we characterize the modules $M$ such that $R{⋉}_{\varphi }M$ is nearly Gorenstein.

Let f₀, f₁, f₂, f₃ be linearly independent homogeneous quadratic forms in the standard ℤ-graded ring R:= 𝕂[s,t,u], and gcd(f₀,f₁,f₂,f₃)=1. This defines a rational map φ: ℙ²→ ℙ³. The Rees algebra Rees(I)=R ⊕ I ⊕ I²⊕ ⋯ of the ideal I= 〈f₀,f₁,f₂,f₃〉 is the graded R-algebra which can be described as the image of an R-algebra homomorphism h: R[x,y,z,w] → Rees(I). This paper discusses the free resolutions of I, and the structure of ker(h).

This paper presents a generalization to the higher-dimensional situation of the main results of the first author about the normality of one-fibered monomial ideals [2, Théorèmes 2.4 and 3.8]. Precisely, we show that if I is a monomial ideal of R = k[x₁,x₂,…,x_d], then I is normal one-fibered if and only if for all positive integers n and all x, y in R such that xy ∈ I²ⁿ, either x or y belongs to Iⁿ.

In a Cohen-Macaulay local ring $(A,\mathfrak{m})$, we study the Hilbert function of an integrally closed $\mathfrak{m}$-primary ideal $I$ whose reduction number is three. With a mild assumption we give an inequalityi ${\ell }_A(A/I)\ge {\mathrm{e}}_0(I)-{\mathrm{e}}_1(I)+({\mathrm{e}}_2(I)+{\ell }_A(I^2/QI))/2$, where ${\mathrm{e}}_i(I)$ denotes the $i$th Hilbert coefficient and $Q$ denotes a minimal reduction of $I$. The inequality is located between inequalities of Itoh and Elias-Valla. Furthermore, this inequality becomes an equality if and only if the depth of the associated graded ring of $I$ is larger than or equal to $\dim A-1$. We also study the Cohen-Macaulayness of the associated graded rings of determinantal rings.