Rees Algebra Research Articles

A construction of Prüfer rings involving quotients of Rees algebras

A family of quotients of a Rees algebra associated to a ring with respect to a fixed ideal was recently introduced by Barucci, D’Anna and Strazzanti. In this paper, we will classify rings of this family that satisfy certain Prüfer-like properties and, as a particular case, we will extend results obtained for amalgamated duplications and Nagata idealizations.

Associated points and integral closure of modules

Let X:=Spec(R) be an affine Noetherian scheme, and M⊂N be a pair of finitely generated R-modules. Denote their Rees algebras by R(M) and R(N). Let Nn be the nth homogeneous component of R(N) and let Mn be the image of the nth homogeneous component of R(M) in Nn. Denote by Mn‾ be the integral closure of Mn in Nn. We prove that AssX(Nn/Mn‾) and AssX(Nn/Mn) are asymptotically stable, generalizing known results for the case where M is an ideal or where N is a free module. Suppose either that M and N are free at the generic point of each irreducible component of X or N is contained in a free R-module. When X is universally catenary, we prove a generalization of a classical result due to McAdam and obtain a geometric classification of the points appearing in AssX(Nn/Mn‾). Notably, we show that if x∈AssX(Nn/Mn‾) for some n, then x is the generic point of a codimension-one component of the nonfree locus of N/M or x is a generic point of an irreducible set in X where the fiber dimension Proj(R(M))→X jumps. We prove a converse to this result without requiring X to be universally catenary. Our approach is geometric in spirit. Also, we recover, strengthen, and prove a sort of converse of an important result of Kleiman and Thorup about integral dependence of modules.

Castelnuovo–Mumford regularity and Ratliff–Rush closure

We establish strong relationships between the Castelnuovo–Mumford regularity and the Ratliff–Rush closure of an ideal. Our results have several interesting consequences on the computation of the Ratliff–Rush closure, the stability of the Ratliff–Rush filtration, the invariance of the reduction number, and the computation of the Castelnuovo–Mumford regularity of the Rees algebra and the fiber ring. In particular, we prove that the Castelnuovo–Mumford regularity of the Rees algebra and of the fiber ring are equal for large classes of monomial ideals in two variables, thereby verifying a conjecture of Eisenbud and Ulrich for these cases.

Stability results for projective modules over Rees algebras

We provide a class of commutative Noetherian domains R of dimension d such that every finitely generated projective R-module P of rank d splits off a free summand of rank one. On this class, we also show that P is cancellative. At the end we give some applications to the number of generators of a module over the Rees algebras.

Asymptotic prime ideals of S2-filtrations

Let A be a noetherian ring with total ring of fractions Q(A) and S=⊕n∈ZIntn a noetherian graded ring such that A[t−1]⊆S⊆Q(A)[t,t−1] and S satisfies the (S2) property of Serre. Under mild conditions on the ring A, we study the behavior of the sets of associated prime ideals Ass(A/In∩A) for n≥1. In particular, we consider the case when S is the S2-ification of the extended Rees algebra of an ideal I. As applications, we obtain several results regarding the asymptotic behavior of Ass(A/In) for certain ideals of analytic deviation one. We also prove several consequences about the symbolic powers of a prime ideal.

Almost Gorenstein Rees Algebras of pg-Ideals, Good Ideals, and Powers of the Maximal Ideals

Let (A,m) be a Cohen–Macaulay local ring, and let I be an ideal of A. We prove that the Rees algebra R(I) is an almost Gorenstein ring in the following cases: (1) (A,m) is a two-dimensional excellent Gorenstein normal domain over an algebraically closed field K≅A/m, and I is a pg-ideal; (2) (A,m) is a two-dimensional almost Gorenstein local ring having minimal multiplicity, and I=mℓ for all ℓ≥1; (3) (A,m) is a regular local ring of dimension d≥2, and I=md−1. Conversely, if R(mℓ) is an almost Gorenstein graded ring for some ℓ≥2 and d≥3, then ℓ=d−1.

О КОНГРУЭНЦ–КОГЕРЕНТНЫХ АЛГЕБРАХ РИСА И АЛГЕБРАХ С ОПЕРАТОРОМ

The paper contains a classification of congruence-coherent Rees algebras and algebras with an operator. The concept of coherence was introduced by D.Geiger. An algebra A is called coherent if each of its subalgebras containing a class of some congruence on A is a union of such classes. In Section 3 conditions for the absence of congruence-coherence property for algebras having proper subalgebras are found. Necessary condition of congruence-coherence for Rees algebras are obtained. Sufficient condition of congruence-coherence for algebras with an operator are obtained. In this section we give a complete classification of congruence-coherent unars. In Section 4 some modification of the congruence-coherent is considered. The concept of weak and locally coherence was introduced by I.Chajda. An algebra A with a nullary operation 0 is called weakly coherent if each of its subalgebras including the kernel of some congruence on A is a union of classes of this congruence. An algebra A with a nullary operation 0 is called locally coherent if each of its subalgebras including a class of some congruence on A also includes a class the kernel of this congruence. Section 4 is devoted to proving sufficient conditions for algebras with an operator being weakly and locally coherent. In Section 5 deals with algebras ⟨ A,d,f ⟩ with one ternary operation d ( x,y,z ) and one unary operation f acting as endomorphism with respect to the operation d ( x,y,z ). Ternary operation d ( x,y,z ) was defined according to the approach offered by V.K. Kartashov. Necessary and sufficient conditions of congruence-coherent for algebras ⟨ A,d,f ⟩ are obtained. Also, necessary and sufficient conditions of weakly and locally coherent for algebras ⟨ A,d,f, 0⟩ with nullary operation 0 for which f (0) = 0 are obtained.

The equations defining blowup algebras of height three Gorenstein ideals

We find the defining equations of Rees rings of linearly presented height three Gorenstein ideals. To prove our main theorem we use local cohomology techniques to bound the maximum generator degree of the torsion submodule of symmetric powers in order to conclude that the defining equations of the Rees algebra and the special fiber ring have the same image in the symmetric algebra. We show that this image is the unmixed part of the ideal generated by the maximal minors of a matrix of linear forms which is annihilated by a vector of indeterminates, and otherwise has maximal possible grade. An important step of the proof is the calculation of the degree of the variety parametrized by the forms generating the grade three Gorenstein ideal.

Discreteness of 𝐹-jumping numbers at isolated non-ℚ-Gorenstein points

We show that the F F -jumping numbers of a pair ( X , a ) (X, \mathfrak {a}) in positive characteristic have no limit points whenever the symbolic Rees algebra of − K X -K_X is finitely generated outside an isolated collection of points. We also give a characteristic zero version of this result, as well as a generalization of the Hartshorne–Speiser–Lyubeznik–Gabber stabilization theorem describing the non- F F -pure locus of a variety.

On bigraded regularities of Rees algebra

For any homogeneous ideal I in $$K[x_1,\ldots ,x_n]$$ of analytic spread $$\ell $$ , we show that for the Rees algebra R(I), $${\text {reg}}_{(0,1)}^{\mathrm{syz}}(R(I))={\text {reg}}_{(0,1)}^{\mathrm{T}}(R(I))$$ . We compute a formula for the (0, 1)-regularity of R(I), which is a bigraded analog of Theorem 1.1 of Aramova and Herzog (Am. J. Math. 122(4) (2000) 689–719) and Theorem 2.2 of Romer (Ill. J. Math. 45(4) (2001) 1361–1376) to R(I). We show that if the defect sequence, $$e_k:= {\text {reg}}(I^k)-k\rho (I)$$ , is weakly increasing for $$k \ge {\text {reg}}^{\mathrm{syz}}_{(0,1)}(R(I))$$ , then $${\text {reg}}(I^j)=j\rho (I)+e$$ for $$j \ge {\text {reg}}^{\mathrm{syz}}_{(0,1)}(R(I))+\ell $$ , where $$\ell ={\text {min}}\{\mu (J)~|~ J\subseteq I \text{ a } \text{ graded } \text{ minimal } \text{ reduction } \text{ of } I\}$$ . This is an improvement of Corollary 5.9(i) of [16].

Postulation and reduction vectors of multigraded filtrations of ideals

We study the relationship between postulation and reduction vectors of admissible multigraded filtrations $\mathcal{F}= \{\mathcal{F} (\underline{n})\}_{\underline{n} \in \mathbb{Z} ^s}$ of ideals in Cohen-Macaulay local rings of dimension at most two. This is enabled by a suitable generalization of the Kirby-Mehran complex. An analysis of its homology leads to an analogue of Huneke's fundamental lemma which plays a crucial role in our investigations. We also clarify the relationship between the Cohen-Macaulay property of the multigraded Rees algebra of $\mathcal{F} $ and reduction vectors with respect to complete reductions of $\mathcal{F} $.

Off-Shell Supersymmetry and Filtered Clifford Supermodules

An off-shell representation of supersymmetry is a representation of the super Poincare algebra on a dynamically unconstrained space of fields. We describe such representations formally, in terms of the fields and their spacetime derivatives, and we interpret the physical concept of engineering dimension as an integral grading. We prove that formal graded off-shell representations of one-dimensional N-extended supersymmetry, i.e., the super Poincare algebra $\mathfrak {p}^{1|N}$ , correspond to filtered Clifford supermodules over Cl(N). We also prove that formal graded off-shell representations of two-dimensional (p,q)-supersymmetry, i.e., the super Poincare algebra $\mathfrak {p}^{1,1|p,q}$ , correspond to bifiltered Clifford supermodules over Cl(p + q). Our primary tools are Rees superalgebras and Rees supermodules, the formal deformations of filtered superalgebras and supermodules, which give a one-to-one correspondence between filtered spaces and graded spaces with even degree-shifting injections. This generalizes the machinery used by Gerstenhaber to prove that every filtered algebra is a deformation of its associated graded algebra. Our treatment extends the notion of Rees algebras and modules to filtrations which are compatible with a supersymmetric structure. We also describe the analogous constructions for bifiltrations and bigradings.

One-dimensional Gorenstein local rings with decreasing Hilbert function

In this paper we solve a problem posed by M.E. Rossi: Is the Hilbert function of a Gorenstein local ring of dimension one not decreasing? More precisely, for any integer h>1, h∉{14+22k,35+46k|k∈N}, we construct infinitely many one-dimensional Gorenstein local rings, included integral domains, reduced and non-reduced rings, whose Hilbert function decreases at level h; moreover, we prove that there are no bounds to the decrease of the Hilbert function. The key tools are numerical semigroup theory, especially some necessary conditions to obtain decreasing Hilbert functions found by the first and the third author, and a construction developed by V. Barucci, M. D'Anna and the second author, that gives a family of quotients of the Rees algebra. Many examples are included.

Invariants of Cohen–Macaulay rings associated to their canonical ideals

The purpose of this paper is to introduce new invariants of Cohen–Macaulay local rings. Our focus is the class of Cohen–Macaulay local rings that admit a canonical ideal. Attached to each such ring R with a canonical ideal C, there are integers–the type of R, the reduction number of C–that provide valuable metrics to express the deviation of R from being a Gorenstein ring. We enlarge this list with other integers–the roots of R and several canonical degrees. The latter are multiplicity based functions of the Rees algebra of C.

АЛГЕБРЫ РИСА И КОНГРУЭНЦ-АЛГЕБРЫ РИСА В ОДНОМ КЛАССЕ АЛГЕБР С ОПЕРАТОРОМ И ОСНОВНОЙ ОПЕРАЦИЕЙ ПОЧТИ ЕДИНОГЛАСИЯ

The concept of Rees congruence was originally introduced for semigroups. R. Tichy generalized this concept to universal algebras. Let A be an universal algebra. Denote by △ the identity relation on A . Any congruence of the form B 2 ∪△ on A for some subalgebra B of A is called a Rees congruence. Subalgebra B of A is called a Rees subalgebra whenever B 2 ∪△ is a congruence on A . An algebra A is called a Rees algebra if its every subalgebra is a Rees one. In this paper we introduce concepts of Rees simple algebra and Rees congruence algebra. A non-one-element universal algebra A is called Rees simple algebra if any Rees congruence on A is trivial. An universal algebra A is called Rees congruence algebra if any congruence on A is Rees congruence. Universal algebra is called an algebra with operators if it has an additional set of unary operations acting as endomorphisms with respect to basic operations. For algebras with one operator and an arbitrary basic signature some conditions to be Rees algebra are obtained. Necessary condition under which algebra of the same class is Rees congruence algebra is given. For algebras with one operator and a connected unary reduct that has a loop element and does not contain the nodal elements, except, perhaps, the loop element necessary condition for their Rees simplicity are obtained. A n-ary operation ( n > 3 ) is called near-unanimity operation if it satisfies the identities (x , . . . , x , y ) = ( x , . . . , x , y , x ) = . . . = ( y , x , . . . , x ) = x . If n = 3 then operation is called a majority operation. Rees algebras and Rees congruence algebras of class algebras with one operator and basic near-unanimity operation g ( n ) which defined as follows g (3) ( x 1 , x 2 , x 3 ) = m ( x 1 , x 2 , x 3 ) , g ( n ) ( x 1 , x 2 , . . . , x n ) = m ( g ( n − 1) ( x 1 , x 2 , . . . , xn − 1 ) , xn − 1 , xn ) ( n > 3) are fully described. Under m ( x 1 , x 2 , x 3 ) we mean here a majority operation which permutable with unary operation and which was defined by the author on arbitrary unar according to the approach offered by V.K. Kartashov.

On the center of quiver Hecke algebras

We compute the equivariant cohomology ring of the moduli space of framed instantons over the affine plane. It is a Rees algebra associated with the center of cyclotomic degenerate affine Hecke algebras of type A. We also give some related results on the center of quiver Hecke algebras and cohomology of quiver varieties.

An intrinsic definition of the Rees algebra of a module

This paper concerns a generalization of the Rees algebra of ideals due to Eisenbud, Huneke and Ulrich that works for any finitely generated module over a noetherian ring. Their definition is in terms of maps to free modules. We give an intrinsic definition using divided powers.

Bounds on degrees of birational maps with arithmetically Cohen–Macaulay graphs

A rational map whose source and image are projectively embedded varieties is said to have an arithmetically Cohen–Macaulay graph if the Rees algebra of one (hence any) of its base ideals is a Cohen–Macaulay ring. The main objective of this paper is to obtain an upper bound for the degree of a representative of the map in case it is birational onto the image with an arithmetically Cohen–Macaulay graph. In the plane case a complete classification is given of the Cremona maps with arithmetically Cohen–Macaulay graph, while in arbitrary dimension n it is shown that a Cremona map with arithmetically Cohen–Macaulay graph has degree at most n2.

Sequentially Cohen–Macaulay Rees algebras

This paper studies the question of when the Rees algebras associated to arbitrary filtration of ideals are sequentially Cohen–Macaulay. Although this problem has been already investigated by [CGT], their situation is quite a bit of restricted, so we are eager to try the generalization of their results.

On Rees algebras of linearly presented ideals in three variables

Let I be a height two perfect ideal with a linear presentation matrix in a polynomial ring R=k[x,y,z]. Assume furthermore that after modulo an ideal generated by two variables, the presentation matrix has rank one. We describe the defining ideal of the Rees algebra R(I) explicitly and we show that R(I) is Cohen–Macaulay.