Rees Ring Research Articles

Ideals with componentwise linear powers

Abstract Let $S=K[x_1,\ldots ,x_n]$ be the polynomial ring over a field K, and let A be a finitely generated standard graded S-algebra. We show that if the defining ideal of A has a quadratic initial ideal, then all the graded components of A are componentwise linear. Applying this result to the Rees ring $\mathcal {R}(I)$ of a graded ideal I gives a criterion on I to have componentwise linear powers. Moreover, for any given graph G, a construction on G is presented which produces graphs whose cover ideals $I_G$ have componentwise linear powers. This, in particular, implies that for any Cohen–Macaulay Cameron–Walker graph G all powers of $I_G$ have linear resolutions. Moreover, forming a cone on special graphs like unmixed chordal graphs, path graphs, and Cohen–Macaulay bipartite graphs produces cover ideals with componentwise linear powers.

Componentwise linear powers and the $x$-condition

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gröbner basis of the defining ideal $J$ of $A$ we give a condition, called the $x$-condition, which implies that all graded components $A_k$ of $A$ have linear quotients and with additional assumptions are componentwise linear. A typical example of such an algebra is the Rees ring $\mathcal{R}(I)$ of a graded ideal or the symmetric algebra $\textrm{Sym}(M)$ of a module $M$. We apply our criterion to study certain symmetric algebras and the powers of vertex cover ideals of certain classes of graphs.

Bounding the degrees of the defining equations of Rees rings for certain determinantal and Pfaffian ideals

We consider ideals of minors of a matrix, ideals of minors of a symmetric matrix, and ideals of Pfaffians of an alternating matrix. Assuming these ideals are of generic height, we characterize the condition Gs for these ideals in terms of the heights of other ideals of minors or Pfaffians of the same matrix. We additionally obtain bounds on the generation and concentration degrees of the Rees rings of a subclass of such ideals via specialization of the Rees rings in the generic case. We do this by proving, given sufficient height conditions on ideals of minors or Pfaffians of the matrix, the specialization of a resolution of the Rees ring in the generic case is an approximate resolution of the Rees ring in question. We end the paper by giving some explicit generation and concentration degree bounds.

On conjectures of Itoh and of Lipman on the cohomology of normalized blow-ups

Let (R,𝔪,𝕜) be a three-dimensional Cohen–Macaulay analytically unramified local ring and I an 𝔪-primary R-ideal. Write X=Proj(⊕n∈ℕIn¯tn). We prove some consequences of the vanishing of H2(X,𝒪X), whose length equals the constant term e¯3(I) of the normal Hilbert polynomial of I. Firstly, X is Cohen–Macaulay. Secondly, if the extended Rees ring A := ⊕n∈ℤIn¯tn is not Cohen–Macaulay, and either R is equicharacteristic or I¯=m, then e¯2(I)−lengthRI2¯/II¯≥3; this estimate is proved using Boij–Söderberg theory of coherent sheaves on ℙ𝕜2. The two results above are related to a conjecture of S. Itoh (1992). Thirdly, HE2(X,Im𝒪X)=0 for all integers m, where E is the exceptional divisor in X. Finally, if additionally R is regular and X is pseudorational, then the adjoint ideals In ˜, n≥1 satisfy In ˜=IIn−1 ˜ for every n≥3. The last two results are related to conjectures of J. Lipman (1994).

Equations of negative curves of blow-ups of Ehrhart rings of rational convex polygons

Finite generation of the symbolic Rees ring of a space monomial prime ideal of a 3-dimensional weighted polynomial ring is a very interesting problem. Negative curves play important roles in finite generation of these rings. We are interested in the structure of the negative curve. We shall prove that negative curves are rational in many cases.We also see that the Cox ring of the blow-up of a toric variety at the point (1,1,…,1) coincides with the extended symbolic Rees ring of an ideal of a polynomial ring. For example, Roberts' second counterexample to Cowsik's question (and Hilbert's 14th problem) coincides with the Cox ring of some normal projective variety (Remark 2.7).

The socle module of a monomial ideal

For any ideal I in a Noetherian local ring or any graded ideal I in a standard graded K-algebra over a field K, we introduce the socle module Soc(I), whose graded components give us the socle of the powers of I. It is observed that Soc(I) is a finitely generated module over the fiber cone of I. In the case that S is the polynomial ring and all powers of I⊆S have linear resolution, we define the module Soc∗(I), which is a module over the Rees ring of I. For the edge ideal of a graph and for classes of polymatroidal ideals, we study the module structure of their socle modules.

Matching Numbers and the Regularity of the Rees Algebra of an Edge Ideal

The regularity {text {reg}}R(I(G)) of the Rees ring R(I(G)) of the edge ideal I(G) of a finite simple graph G is studied. It is shown that, if R(I(G)) is normal, one has {text {mat}}(G) le {text {reg}}R(I(G)) le {text {mat}}(G) + 1, where {text {mat}}(G) is the matching number of G. In general, the induced matching number is a lower bound for the regularity, which can be shown by applying the squarefree divisor complex.

BOUNDING THE DEGREES OF THE DEFINING EQUATIONSOF REES RINGS FOR CERTAIN DETERMINANTAL AND PFAFFIAN IDEALS

We consider ideals of minors of a matrix, ideals of minors of a symmetric matrix, and ideals of Pfaffians of an alternating matrix. Assuming these ideals are of generic height, we characterize the condition $G_{s}$ for these ideals in terms of the heights of smaller ideals of minors or Pfaffians of the same matrix. We additionally obtain bounds on the generation and concentration degrees of the defining equations of Rees rings for a subclass of such ideals via specialization of the Rees rings in the generic case. We do this by proving that, given sufficient height conditions on ideals of minors or Pfaffians of the matrix, the specialization of a resolution of a graded component of the Rees ring in the generic case is an approximate resolution of the same component of the Rees ring in question. We end the paper by giving some examples of explicit generation and concentration degree bounds.

Prereductions of ideals in local rings

This paper contributes several results to the analytic theory of ideals. The main new concept is that of a prereduction (together with the closely related type-prereduction): if R is a ring and I (proper subset) are proper ideals in R, then A is called a prereduction of I in case A is not a reduction of I, but each ideal between A and I is a reduction of I. It is shown that each non-nilpotent proper ideal in a Noetherian ring has at least one prereduction, and a number of the basic properties of the ideals in the set of all prereductions of I are proved. Also, if I is a non-nilpotent proper ideal in a local (Noetherian) ring (R, M), then:There is a natural one-to-one correspondence between the set of the equivalence classes of the type-prereductions of I and the set of the maximal relevant ideals in the Rees ring of I (that is, the homogeneous prime ideals Q in the ring (t an indeterminate) such that and the Krull dimension of is equal to one). is the set of maximal elements in for all positive integers n, and, for each for all large integers m.A complete description is given of the prereductions and the elements in for each ideal I that is generated by analytically independent elements.If R/M is algebraically closed and I is a non-nilpotent and non-principal ideal in R, then there is a natural one-to-one correspondence between the sets and and for all positive integer n.

Blowup algebras of rational normal scrolls

We determine the equations of the blowup of P n \mathbb {P}^{n} along a d d -fold rational normal scroll S \mathcal {S} , and we prove that the Rees ring and special fiber ring of S ⊆ P n \mathcal {S}\subseteq \mathbb {P}^{n} are Koszul algebras.

On the Degree of the Kodiyalam Polynomials of a Graded Ideal in the Polynomial Ring

In this paper, we compute the degree of the Kodiyalam polynomials of an ideal in the case where its Rees ring is Cohen—Macaulay and its fiber ring is a domain. We apply this result to some classes of polymatroidal ideals.

A family of reflexive vector bundles of reduction number one

A difficult issue in modern commutative algebra asks for examples of modules (more interestingly, reflexive vector bundles) having prescribed reduction number $r\geq 1$. The problem is even subtler if in addition we are interested in good properties for the Rees algebra. In this note we consider the case $r=1$. Precisely, we show that the module of logarithmic vector fields of the Fermat divisor of any degree in projective $3$-space is a reflexive vector bundle of reduction number $1$ and Gorenstein Rees ring.

Infinitely Generated Symbolic Rees Rings of Space Monomial Curves Having Negative Curves

In this paper, we study finite generation of symbolic Rees rings of the defining ideal p of the space monomial curve (ta,tb,tc) for pairwise coprime integers a, b, c. Suppose that the base field is of characteristic 0, and the ideal p is minimally generated by three polynomials. In Theorem 1.1, under the assumption that the homogeneous element ξ of the minimal degree in p is a negative curve, we determine the minimal degree of an element η such that the pair {ξ,η} satisfies Huneke’s criterion in the case where the symbolic Rees ring is Noetherian. By this result we can decide whether the symbolic Rees ring Rs(p) is Notherian using computers. We give a necessary and sufficient condition for finite generation of the symbolic Rees ring of p in Proposition 4.10 under some assumptions. We give an example of an infinitely generated symbolic Rees ring of p in which the homogeneous element of the minimal degree in p(2) is a negative curve in Example 5.7. We give a simple proof to (generalized) Huneke’s criterion.

Primary Decomposition of Powers of the Prime Ideal of a Numerical Semigroup Ring

Let R=k[t^{n_{1}},ldots ,t^{n_{s}}]=k[x_{1},ldots ,x_{s}]/P be a numerical semigroup ring and let P(n) = PnRP ∩ R be the symbolic power of P and Rs(P) = ⊕i≥ 0P(n)tn the symbolic Rees ring of P. It is hard to determine symbolic powers of P; there are even non-Noetherian symbolic Rees rings for 3-generated semigroups. We determine the primary decomposition of powers of P for some classes of 3-generated numerical semigroups.

Weakly factorial property of a generalized Rees ring $D[X,d/X

Let $D$ be an integral domain, $X$ an indeterminate over $D$, $d \in D$, and $R = D[X, {d}/{X}]$ a subring of $D[X,{1}/{X}]$. In this paper, we show that $R$ is a weakly factorial domain if and only if $D$ is a weakly factorial GCD-domain and $d=0$, $d$ is a unit of $D$ or $d$ is a prime element of $D$. We also show that, if $D$ is a weakly factorial GCD-domain, $p$ is a prime element of $D$, and $n \geq 2$ is an integer, then $D[X, {p^n}/{X}]$ is an almost weakly factorial domain with $Cl(D[X, {p^n}/{X}]) = \mathbb {Z}_n$.

Uniform annihilators of local cohomology of extended Rees rings

The purpose of the paper is to study the annihilators of local cohomology of the extended Rees rings. We will prove that if (A,m) is a noetherian local ring with infinite residue field, and a is a uniform local cohomological annihilator of A, then there is an integer n such that (at)n is a uniform local cohomological annihilator of the extended Rees ring A[t,t−1I] for any m-primary ideal I of A.

On the Betti polynomials of certain graded ideals

ABSTRACTLet be a polynomial ring over a field K and I be a nonzero graded ideal of S. Then, for t≫0, the Betti number is a polynomial in t, which is denoted by . It is proved that is vanished or of degree ℓ(I)−1 provided I is a monomial ideal generated in a single degree or grade(𝔪R(I)) = codim(𝔪R(I)) where and R(I) is the Rees ring of I. One lower bound for the leading coefficient of is given. When I is a Borel principal monomial ideal, is calculated explicitly.

On Aluffi's problem and blowup algebras of certain modules

This paper deals with blowup algebras of certain classical modules related to a quasi-homogeneous hypersurface in characteristic zero. Motivated by Aluffi's problem, which asks for an appropriate adaptation of his quasi-symmetric algebra into the context of modules (or sheaves), we propose and study our general candidate – which is shown to actually retrieve Aluffi's definition in the situation of ideals – and further we compute it explicitly in the case of the module of tangent vector fields on the (possibly singular) given divisor. Along the way, we work out the Rees ring of the corresponding logarithmic derivation module and we apply a structural result of Corso–Polini–Ulrich in order to determine its core (intersection of all reductions) under certain conditions.

David Rees, FRS 1918–2013

David Rees, FRS 1918–2013

Monoidal extensions of a locally quasi-unmixed UFD

Let R be a locally quasi-unmixed domain, a,b1,…,bn an asymptotic sequence in R, I=(a,b1,…,bn)R and S=R[b1/a,…,bn/a]=R[I/a], the monoidal transform of R with respect to I. It is shown that S is a locally quasi-unmixed domain, a,b1/a,…,bn/a is an asymptotic sequence in S and there is a one-to-one correspondence between the asymptotic primes Aˆ⁎(I) of I and the asymptotic primes Aˆ⁎(aS) of aS=IS. Moreover, if a,b1,…,bn is an R-sequence, this extends to a one-to-one correspondence between AssR(R/I) and AssS(S/aS). In the case that R is a unique factorization domain, the height one prime ideals of S are examined to determine how far S is from being a UFD. A complete description is given of which height one prime ideals P of S are principal or have a principal primary ideal in the case that P∩R has height 1. If the prime divisors of a satisfy a mild condition, a similar description is given in the case that P∩R has height >1. These are applied to give similar results for the Rees ring R[1/t,It] where t is an indeterminate.