Rees Algebra Research Articles

Newton–Okounkov body, Rees algebra, and analytic spread of graded families of monomial ideals

Newton–Okounkov body, Rees algebra, and analytic spread of graded families of monomial ideals

Powers of generalized binomial edge ideals of path graphs

In this paper, we study the powers of the generalized binomial edge ideal [Formula: see text] of a path graph [Formula: see text]. 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 [Formula: see text] utilizing Sagbi basis theory. In particular, we obtain precise formulas for the regularity of these blowup algebras.

Very Well-Covered Graphs via the Rees Algebra

Very Well-Covered Graphs via the Rees Algebra

Equivariant Cohomology and Conditional Oriented Matroids

Abstract We give a cohomological interpretation of the Heaviside filtration on the Varchenko–Gelfand ring of a pair $({\mathcal{A}},{\mathcal{K}})$, where ${\mathcal{A}}$ is a real hyperplane arrangement and ${\mathcal{K}}$ is a convex open subset of the ambient vector space. This builds on work of the first author, who studied the filtration from a purely algebraic perspective, as well as work of Moseley, who gave a cohomological interpretation in the special case where ${\mathcal{K}}$ is the ambient vector space. We also define the Gelfand–Rybnikov ring of a conditional oriented matroid, which simultaneously generalizes the Gelfand–Rybnikov ring of an oriented matroid and the aforementioned Varchenko–Gelfand ring of a pair. We give purely combinatorial presentations of the ring, its associated graded, and its Rees algebra.

On Rees algebras of linearly presented ideals and modules

On Rees algebras of linearly presented ideals and modules

Rees algebras of sparse determinantal ideals

We determine the defining equations of the Rees algebra and of the special fiber ring of the ideal of maximal minors of a 2 × n 2\times n sparse matrix. We prove that their initial algebras are ladder determinantal rings. This allows us to show that the Rees algebra and the special fiber ring are Cohen-Macaulay domains, they are Koszul, they have rational singularities in characteristic zero and are F-rational in positive characteristic.

Normality of Rees algebras of generalized mixed product ideals

Let $K$ be a field and $K[x_1,x_{2}]$ the polynomial ring in two
 variables over $K$ with each $x_i$ of degree $1$. Let $L$ be the
 generalized mixed product ideal induced by a monomial ideal
 $I\subset K[x_1,x_2]$, where the ideals substituting the monomials
 in $I$ are squarefree Veronese ideals. In this paper, we study the
 integral closure of $L$, and the normality of $\mathcal{R}(L)$, the
 Rees algebra of $L$. Furthermore, we give a geometric description of
 the integral closure of $\mathcal{R}(L)$.

Rees algebras of unit interval determinantal facet ideals

Using SAGBI basis techniques, we find Gröbner bases for the presentation ideals of the Rees algebras and special fiber rings of unit interval determinantal facet ideals. In particular, we show that unit interval determinantal facet ideals are of fiber type and that their special fiber rings are Koszul. Moreover, their Rees algebras and special fiber rings are normal Cohen-Macaulay domains and have rational singularities.

Subrings of polynomial rings and the conjectures of Eisenbud and Evans

Let R be a commutative Noetherian ring of dimension d. In 1973, Eisenbud and Evans proposed three conjectures on the polynomial ring R[T]. These conjectures were settled in the affirmative by Sathaye, Mohan Kumar and Plumstead. One of the primary objectives of this article is to investigate the validity of these conjectures over Noetherian subrings of R[T] of dimension d+1, containing R. We formulate a class of such rings, which includes polynomial rings, Rees algebras, Rees-like algebras and Noetherian symbolic Rees algebras, and exhibit that all three conjectures hold for rings belonging to this class.

On Rees algebras of 2‐determinantal ideals

AbstractLet be the ideal of minors of a matrix of linear forms with the expected codimension. In this paper, we prove that the Rees algebra of and its special fiber ring are Cohen–Macaulay and Koszul; in particular, they are quadratic algebras. The main novelty in our approach is the analysis of a stratification of the Hilbert scheme of determinantal ideals. We study degenerations of Rees algebras along this stratification, and combine it with certain squarefree Gröbner degenerations.

Rees algebra of maximal order Pfaffians and its diagonal subalgebras

Given a skew-symmetric matrix X, the Pfaffian of X is defined as the square root of the determinant of X. In this article, we give the explicit defining equations of the Rees algebra of a Pfaffian ideal I generated by the maximal order Pfaffians of a generic skew-symmetric matrix. We further prove that all diagonal subalgebras of the corresponding Rees algebra of I are Koszul. We also look at Rees algebras of Pfaffian ideals of linear type associated with certain sparse skew-symmetric matrices. In particular, we consider the tridiagonal matrices and identify the corresponding Pfaffian ideals to be of Gröbner linear type and as the vertex cover ideals of unmixed bipartite graphs. As an application of our results, we conclude that all their ordinary and symbolic powers have linear quotients.

On almost p-standard system of parameters of idealization and applications

Let (R,m) be a Noetherian local ring and M be a finitely generated R-module. In this paper, we construct almost p-standard systems of parameters (a strict subclass of d-sequences) of the idealization R⋉M. As applications, we give a class of Cohen-Macaulay Rees algebras for idealizations, Cohen-Macaulay Rees modules for unmixed modules, and give precise formulas computing all Hilbert coefficients of the idealization with respect to an almost p-standard system of parameters.

On Rees algebras of ideals and modules over hypersurface rings

The acquisition of the defining equations of Rees algebras is a natural way to study these algebras and allows certain invariants and properties to be deduced. In this paper, we consider Rees algebras of codimension 2 perfect ideals of hypersurface rings and produce a minimal generating set for their defining ideals. Then, using generic Bourbaki ideals, we study Rees algebras of modules with projective dimension one over hypersurface rings. We describe the defining ideal of such algebras and determine Cohen-Macaulayness and other invariants.

Rees algebras and generalized depth‐like conditions in prime characteristic

AbstractIn this paper, we address a question concerning nilpotent Frobenius actions on Rees algebras and associated graded rings. We prove a nilpotent analog of a theorem of Huneke for Cohen–Macaulay singularities. This is achieved by introducing a depth‐like invariant which captures as special cases Lyubeznik's F‐depth and the generalized F‐depth from Maddox–Miller and is related to the generalized depth with respect to an ideal. We also describe several properties of this new invariant and identify a class of regular elements for which weak F‐nilpotence deforms.

Expected Resurgence of Ideals Defining Gorenstein Rings

Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence and thus satisfy the stable Harbourne conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided that its symbolic powers are given by saturations with the maximal ideal. Although this property is not suitable for reduction to characteristic p, we show that a similar result holds in equicharacteristic 0 under the additional hypothesis that the symbolic Rees algebra of I is Noetherian.

On coefficient ideals

Let be a Cohen-Macaulay local ring of dimension with infinite residue field and let I be an primary ideal. Let For let be the ith-coefficient ideal of I. Also let denote the Ratliff-Rush closure of A. Let be the associated graded ring of I. We show that if for then for all . In particular if G is generalized Cohen-Macaulay then for all . As a consequence we get that if A is an analytically unramified domain with G generalized Cohen-Macaulay, then the -ification of the Rees algebra A[It] is .

Gorensteinness in Rees algebras of powers of parameter ideals

This paper gives a necessary and sufficient condition for Gorensteinness in Rees algebras of the dth power of parameter ideals in certain Noetherian local rings of dimension d≥2. The main result of this paper produces many Gorenstein Rees algebras over non-Cohen-Macaulay local rings. For example, the Rees algebra R(qd)=⊕i≥0qdi is Gorenstein for every parameter ideal q that is a reduction of the maximal ideal in a d-dimensional Buchsbaum local ring of depth 1 and multiplicity 2.

Symbolic Rees algebras and set-theoretic complete intersections

In this paper we extend a result of Cowsik on set-theoretic complete intersection and a result Huneke, Morales and Goto and Nishida about Noetherian symbolic Rees algebras of ideals. As applications, we show that the symbolic Rees algebras of the following ideals are Noetherian and the ideals are set-theoretic complete intersections: (a) the edge ideal of a complete graph, (b) the Fermat ideal and (c) the Jacobian ideal of a certain hyperplane arrangement.

On quotients of Rees algebras

We study ring theory properties of some quotients of Rees algebras , A f ( a ) , that extend results of [2] and [3] . In particular, we use pullback constructions to describe the prime spectrum of A f ( a ) . Some questions that we discuss in this paper remain open in general.

When are the rings I:I Gorenstein?

Let be an ideal of a d-dimensional Noetherian local ring A with , containing a non-zerodivisor. The problem of when the ring is Gorenstein is studied, in connection with the problem of the Gorensteinness in Rees algebras for certain parameter ideals Q of A, that was closely explored by the preceding paper [8] of the authors. Examples are given. Communicated by Scott Chapman