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On the symmetric and Rees algebra of an ideal generated by a d-sequence

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Craig L. Huneke at University of Virginia
Craig L. Huneke
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... According to [55, p. 30], the terminology was introduced by Robbiano and Valla. Ideals of linear type have been extensively studied in the 1980s [6,7,17,18,19,20,21,27,50,53] but the first important family of ideals of linear type is already given by Micali in the early 1960s [30,31]: if I is a generated by a regular sequence, then it is of linear type. In [20] Huneke introduces the notion of d-sequence, a weaker condition to that of regular sequence, and shows that an ideal generated by a d-sequence is of linear type, a result proved simultaneously by Valla in [53]. ...
... Ideals of linear type have been extensively studied in the 1980s [6,7,17,18,19,20,21,27,50,53] but the first important family of ideals of linear type is already given by Micali in the early 1960s [30,31]: if I is a generated by a regular sequence, then it is of linear type. In [20] Huneke introduces the notion of d-sequence, a weaker condition to that of regular sequence, and shows that an ideal generated by a d-sequence is of linear type, a result proved simultaneously by Valla in [53]. Other classes of ideals with low relation type have been studied. ...
... A.1.6]. Recall that the general version of the Artin-Rees Lemma for modules states that, for any ideal I in A and any two finitely generated A- There is a well-known connection between the Artin-Rees Lemma and the relation type, which roughly speaking establishes that the aforementioned integer s linked to the triple (N, M; I) is bounded above by the relation type of the Rees module R(I; M/N) (see, e.g., [18,20,21,28,41,59] and, in particular, [40,Theorem 2]). We would like to mention that in [18], the authors observe how being of linear type gives the Artin-Rees Lemma 'on the nose' as already observed in [21] for ideals generated by a d-sequence. ...
Sifted degrees of the equations of the Rees module and their connection with the Artin-Rees numbers
Preprint
  • Dec 2024
Let A be a noetherian ring, I an ideal of A and N⊂MN\subset M finitely generated A-modules. The relation type of I with respect to M, denoted by rt (I;M){\bf rt}\,(I;M), is the maximal degree in a minimal generating set of relations of the Rees module R(I;M)=⊕n≥0InM{\bf R}(I;M)=\oplus_{n\geq 0}I^nM. It is a well-known invariant that gives a first measure of the complexity of R(I;M){\bf R}(I;M). To help to measure this complexity, we introduce the sifted type of R(I;M){\bf R}(I;M), denoted by st (I;M){\bf st}\,(I;M), a new invariant which counts the non-zero degrees appearing in a minimal generating set of relations of R(I;M){\bf R}(I;M). Just as the relation type rt (I;M/N){\bf rt}\,(I;M/N) is closely related to the strong Artin-Rees number s (N,M;I){\bf s}\,(N,M;I), it turns out that the sifted type st (I;M/N){\bf st}\,(I;M/N) is closely related to the medium Artin-Rees number m (N,M;I){\bf m}\,(N,M;I), a new invariant which lies in between the weak and the strong Artin-Rees numbers of (N,M;I). We illustrate the meaning, interest and mutual connection of m (N,M;I){\bf m}\,(N,M;I) and st (I;M){\bf st}\,(I;M) with some examples.
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... An ideal in a commutative ring is said to be of linear type if their Rees and symmetric algebras are isomorphic; equivalently, the defining ideal of Rees algebra is generated by linear forms. The notion of d-sequence was introduced and initially studied by Huneke in [12,14]. The author proved that an ideal generated by d-sequence in a commutative ring is of linear type (cf. ...
... The author proved that an ideal generated by d-sequence in a commutative ring is of linear type (cf. [12]). Villarreal characterized graphs for which edge ideals are of the linear type in [32], namely, edge ideals are of linear type if and only if the graph is a tree or has a unique cycle of odd length. ...
... Jayanthan et al. characterized graphs whose binomial edge ideal is an almost complete intersection in [17,Theorems 4.3,and 4.4]. One has the following strict inclusions: almost complete intersection ideal =⇒ d-sequence ideal =⇒ ideal of linear type [12,17] and [13, page no. 341]. ...
Regularity of powers of d-sequence (parity) binomial edge ideals of unicycle graphs
Preprint
Full-text available
  • Sep 2022
We classify all unicycle graphs whose edge-binomials form a d-sequence, particularly linear type binomial edge ideals. We also classify unicycle graphs whose parity edge-binomials form a d-sequence. We study the regularity of powers of (parity) binomial edge ideals of unicycle graphs generated by d-sequence (parity) edge-binomials.
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... Since complete intersection ideal =⇒ d-sequence ideal =⇒ ideal of linear type (cf. [8]), the binomial edge ideal of paths and star graphs are of linear type. It is natural to ask for combinatorial characterization of linear type binomial edge ideals. ...
... Note that there is a strict inclusion among the family of ideals, that is, almost complete intersection ideal =⇒ d-sequence ideal =⇒ ideal of linear type (cf. [8,13] and [9, page no. 341].). ...
... The notion of d-sequence was introduced by Huneke in [8] (see Definition 1.1). An ideal is said to be generated by d-sequence if there is a generating set that forms a d-sequence. ...
d-sequence edge binomials, and regularity of powers of binomial edge ideals of trees
Preprint
Full-text available
  • Sep 2022
We provide the necessary and sufficient conditions for the edge-binomials of the tree forming a d-sequence in terms of the degree sequence notion of a graph. We study the regularity of powers of the binomial edge ideals of trees generated by d-sequence edge binomials.
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... Relations of this type are called linear relations, and R(I) is called of linear type if the linear relations are the only generating relations of R(I). Complete intersection ideals and d-sequence ideals are some classes of ideals of linear type [6], [7]. We denote by L ⊆ J the ideal generated by the linear relations of R(I). ...
... But the converse is not true in general. Indeed, for I = (x 6 1 , x 1 x 5 2 , x 2 2 x 4 3 , x 3 2 x 3 3 , x 4 ), one can see that C(I) is m-primary, while I is not. ...
On the Rees algebra and the conductor of an ideal
Preprint
Full-text available
  • Jul 2024
For an ideal I in a Noetherian ring R, we introduce and study its conductor as a tool to explore the Rees algebra of I. The conductor of I is an ideal C(I)⊂RC(I)\subset R obtained from the defining ideals of the Rees algebra and the symmetric algebra of I by a colon operation. Using this concept we investigate when adding an element to an ideal preserves the property of being of linear type. In this regard, a generalization of a result by Valla in terms of the conductor ideal is presented. When the conductor of a graded ideal in a polynomial ring is the graded maximal ideal, a criteria is given for when the Rees algebra and the symmetric algebra have the same Krull dimension. Finally, noting the fact that the conductor of a monomial ideal is a monomial ideal, the conductor of some families of monomial ideals, namely bounded Veronese ideals and edge ideals of graphs, are determined.
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... Huneke introduced the notion of d-sequence in [13]. Huneke, and independently Valla, proved that if the ideal is generated by unconditioned d-sequence, then the ideal is of linear type in [12,26]. Costa [5] established that d-sequence condition on I is a sufficient condition for the ideal I to be of linear type. ...
... In particular, if a unicyclic graph G contains C 4,1 as an induced subgraph, then J G is not of linear type. 12 ...
On Conjecture of Binomial Edge Ideals of Linear Type
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Full-text available
  • Jun 2024
An ideal I of a commutative ring R is said to be of {\emph{linear type}} when its Rees algebra and symmetric algebra exhibit isomorphism. In this paper, we investigate the conjecture put forth by Jayanthan, Kumar, and Sarkar (2021) that if G is a tree or a unicyclic graph, then the binomial edge ideal of G is of linear type. Our investigation validates this conjecture for trees. However, our study reveals that not all unicyclic graphs adhere to this conjecture. Furthermore, we provide an explicit description of the defining ideal of the Rees algebra of binomial edge ideals of trees and demonstrate that these algebras are Cohen-Macaulay. Finally, we conclude the equality between symbolic powers and ordinary powers of binomial edge ideals of trees.
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... The case that a is linear type. Recall that an ideal I in a Noetherian ring is linear type if the natural map from the symmetric algebra Sym(I) of I to the Rees algebra R(I) of I is an isomorphism (see, e.g., [Hun80,§2]). Unravelling the definitions involved, one gets Proposition 3.6.1 below. ...
... To state the proposition, we will need the following two facts (both of which can be found in [Hun80,§2]): The kernel of the natural map S[a] → R(a) is J = {h(a 1 , . . . , a n ) : h (f 1 g 1 , . . . ...
Geometry of logarithmic derivations of hyperplane arrangements
Preprint
Full-text available
  • Dec 2021
We study the Hadamard product of the linear forms defining a hyperplane arrangement with those of its dual, which we view as generating an ideal in a certain polynomial ring. We use this ideal, which we call the ideal of pairs, to study logarithmic derivations and critical set varieties of arrangements in a way which is symmetric with respect to matroid duality. Our main result exhibits the variety of the ideal of pairs as a subspace arrangement whose components correspond to cyclic flats of the arrangement. As a corollary, we are able to give geometric explanations of some freeness and projective dimension results due to Ziegler and Kung--Schenck.
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... In general, it is quite a hard task to describe the defining ideals of Rees algebras and symmetric algebras. Huneke proved that if I is generated by d-sequence, then I is of linear type, [12]. We compute the defining ideal of symmetric algebra of LSS ideals of trees and odd unicyclic graphs (Theorems 5. 2, 5.4). ...
... Now, we recall the definition of d-sequence. We refer the reader to [12] for more properties of d-sequences. ...
Lovász–Saks–Schrijver ideals and parity binomial edge ideals of graphs
Article
Full-text available
  • Mar 2021
Let G be a simple graph on n vertices. Let LG and IG denote the Lovász–Saks–Schrijver(LSS) ideal and parity binomial edge ideal of G in the polynomial ring S=K[x1,…,xn,y1,…,yn] respectively. We classify graphs whose LSS ideals and parity binomial edge ideals are complete intersections. We also classify graphs whose LSS ideals and parity binomial edge ideals are almost complete intersections, and we prove that their Rees algebra is Cohen–Macaulay. We compute the second graded Betti number and obtain a minimal presentation of LSS ideals of trees and odd unicyclic graphs. We also obtain an explicit description of the defining ideal of the symmetric algebra of LSS ideals of trees and odd unicyclic graphs.
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... In general, it is quite a hard task to describe the defining ideals of Rees algebras and symmetric algebras. Huneke proved that if I is generated by d-sequence, then I is of linear type, [12]. We compute the defining ideal of symmetric algebra of LSS ideals of trees and odd unicyclic graphs(Theorems 5. 2, 5.4). ...
... Now, we recall definition of d-sequence. We refer the reader to [12] for more properties of d-sequences. ...
Lovász-Saks-Schrijver ideals and parity binomial edge ideals of graphs
Preprint
Full-text available
  • Dec 2020
Let G be a simple graph on n vertices. Let LG and IG L_G \text{ and } \mathcal{I}_G \: denote the Lov\'asz-Saks-Schrijver(LSS) ideal and parity binomial edge ideal of G in the polynomial ring S=K[x1,…,xn,y1,…,yn]S = \mathbb{K}[x_1, \ldots, x_n, y_1, \ldots, y_n] respectively. We classify graphs whose LSS ideals and parity binomial edge ideals are complete intersections. We also classify graphs whose LSS ideals and parity binomial edge ideals are almost complete intersection, and we prove that their Rees algebra is Cohen-Macaulay. We compute the second graded Betti number and obtain a minimal presentation of LSS ideals of trees and odd unicyclic graphs. We also obtain an explicit description of the defining ideal of the symmetric algebra of LSS ideals of trees and odd unicyclic graphs.
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... By (e), codim Supp R (I/J) ≥ s. In particular J : I is an s-residual intersection of I, which according to 6 A. Corso et al. ...
The structure of the core of ideals
Preprint
  • Oct 2002
The core of an R-ideal I is the intersection of all reductions of I. This object was introduced by D. Rees and J. Sally and later studied by C. Huneke and I. Swanson, who showed in particular its connection to J. Lipman's notion of adjoint of an ideal. Being an a priori infinite intersection of ideals, the core is difficult to describe explicitly. We prove in a broad setting that: core(I){\rm core}(I) is a finite intersection of minimal reductions; core(I){\rm core}(I) is a finite intersection of general minimal reductions; core(I){\rm core}(I) is the contraction to R of a `universal' ideal; core(I){\rm core}(I) behaves well under flat extensions. The proofs are based on general multiplicity estimates for certain modules.
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... We observe that the implication (a) ⇒ (d) recovers [17,Corollary 3.11]. Moreover, a crucial fact here (which, as far as we know, is new) is given by (b) ⇒ (a), i.e., (R, m) must be regular if m is generated by a d-sequence (see [11], [12] for the definition of this type of sequence and its properties), which in particular solves the problem suggested in [17,Remark 3.12]. Finally, the equivalence between assertions (a) and (e) reveals the curious role played by d 3 (I) in regard to the theory of regular local rings, which can be re-expressed by means of equivalence to the structural assertion (f). ...
Bounds on Dao numbers and applications to regular local rings
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Full-text available
  • May 2024
The so-called Dao numbers are a sort of measure of the asymptotic behaviour of full properties of certain product ideals in a Noetherian local ring R with infinite residue field and positive depth. In this paper, we answer a question of H. Dao on how to bound such numbers. The auxiliary tools range from Castelnuovo-Mumford regularity of appropriate graded structures to reduction numbers of the maximal ideal. In particular, we substantially improve previous results (and answer questions) by the authors. As an application, we provide new characterizations of when R is regular; for instance, we show that this holds if and only if the maximal ideal of R can be generated by a d-sequence (in the sense of Huneke) if and only if the third Dao number of any (minimal) reduction of the maximal ideal vanishes.
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... For every point p ∈ Z the local equations a, b form a regular sequence in the local ring O S,p . In this setting, the presentation of the powers of the ideal can be read off from the Rees algebra, which coincides with the symmetric algebra by [Hun80]. ...
Special triple covers of algebraic surfaces
Article
Full-text available
  • Jan 2022
  • DOC MATH
We study special triple covers f:T\to S of algebraic surfaces, where the Tschirnhausen bundle \mathcal{E}=\left(f_*\mathcal{O}_T/\mathcal{O}_S\right)^\vee is a quotient of a split rank three vector bundle, and we provide several necessary and sufficient criteria for the existence. As an application, we give a complete classification of special triple planes, finding among others two nice families of K3 surfaces.
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... The theory of d-sequence was introduced by Huneke in the 1980's [20,19] in the sense of "weak" regular sequence, which seems to help in the study of depth of asymptotic powers of a homogeneous ideal in a graded ring. Numerous examples of d-sequences are given in [20]. ...
A note on sequences and y-regularity of Rees algebra
Preprint
Full-text available
  • Mar 2023
Given a graded ring A and a homogeneous ideal I, the ideal is said to be of linear type if the Rees algebra of I is isomorphic to the symmetric algebra of I. We examine the relations between different notions of sequences already known in literature which form ideals of linear type. We observe how the vanishing of y-regularity of Rees algebra of an ideal is connected to the ideal being generated by some specific sequences. As an application of this, we discuss Koszulness and Cohen-Macaulayness of the diagonal subalgebra of Rees algebra of ideals generated by some of these sequences. We also mention Macaulay2 algorithms for checking if an ideal is generated by d-sequence and for computing bigraded Betti numbers.
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... An ideal I of a ring A is said to be of linear type if the Rees algebra of I is isomorphic to it's symmetric algebra. We are interested in the study of Rees algebra of ideals generated by d-sequence (a notion introduced by Huneke in [24,25]) as they form a class of ideals of linear type. The motivation to explore Rees algebra corresponding to d-sequence comes from the analogous study in [15,41,12,31] for the ideals generated by regular sequence. ...
Rees algebra of Maximal Order Pfaffians and its diagonal subalgebras
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Full-text available
  • Feb 2023
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 Rees algebra of Pfaffian ideal I generated by maximal order Pfaffians of generic skew-symmetric matrices. We further prove that all diagonal subalgebras of the corresponding Rees algebra of I are Koszul. We also look at the Rees algebra of Pfaffian ideals of linear type associated to certain sparse skew-symmetric matrices. In particular, we consider the tridiagonal matrices and identify the corresponding Pfaffian ideal to be of Gr\"obner linear type and as the vertex cover ideal of an unmixed bipartite graph. As an application of our results, we conclude that all their powers have linear resolutions.
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... This is a contradiction, so it must be that I " J, whence ℓ " µpIq. Therefore, by [4, 4.1 and 3.6(b)] it follows that I is generated by a d-sequence, hence it is of linear type (see [17,Theorem 3.1]). ...
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... Therefore by [HSV, Corollary 2.2 and Theorem 2.3] JS i is of linear type. (See also [H1,Theorem 3.1] or [V,Theorem 3.15].) ...
Residual Intersections of 2×n2\times n Determinantal Ideals
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Full-text available
  • Jul 2022
Schemes defined by residual intersections have been extensively studied in the case when they are Cohen-Macaulay, but this is a very restrictive condition. In this paper we make the first study of a class of natural examples far from satisfying this condition, the rank 1 loci of generic 2×n2\times n matrices. Here we compute their depths and many other properties. These computations require a number of novel tools.
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... The functoriality of the formation of P X (Y ) follows from its universal property. If g : Y ′ → Y is p-completely flat, then each g n : Y ′ n → Y n is flat (see Proposition 7. 13), and if in addition ...
Crystalline prisms: Reflections on the present and past
Preprint
  • Apr 2022
Let Y/S be a morphism of crystalline prisms, i.e., a p-torsion free p-adic formal schemes endowed with a Frobenius lift, and let Y‾/S‾\overline Y/\overline S denote its reduction modulo p. We show that the category of crystals on the prismatic site of Y‾/S\overline Y/S is equivalent to the category of OY\mathcal{O}_Y-modules with integrable and quasi-nilpotent p-connection and that the cohomology of such a crystal is computed by the associated p-de Rham complex. If X is a closed subscheme of Y‾\overline Y, smooth over S‾\overline S, then the prismatic envelope ΔX(Y)\Delta_X(Y) of X in Y admits such a p-connection, and the category of prismatic crystals on X/S is equivalent to the category of OΔX(Y)\mathcal{ O}_ {\Delta_X(Y)}-modules with compatible p-connection, and cohomology is again computed by p-de Rham complexes. Our main tools are a detailed study of prismatic envelopes and a formal smoothness property for Y/S when working with prisms in the p-completely flat topology. We also explain how earlier work by several authors relating Higgs fields, p-connections, and connections can be placed in the prismatic context.
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... (i) This is a folklore fact in the literature. The statement is, e.g., written in [Hu80,page 270] without proof. ...
Reductions towards a characteristic free proof of the Canonical Element Theorem
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Full-text available
  • Mar 2022
We reduce Hochster's Canonical Element Conjecture (theorem since 2016) to a localization problem in a characteristic free way. We prove the validity of a new variant of the Canonical Element Theorem (CET) and explain how a characteristic free deduction of the new variant from the original CET would provide us with a characteristic free proof of the CET. We show that also the Balanced Big Cohen-Macaulay Module Theorem can be settled by a characteristic free proof if the big Cohen-Macaulayness of Hochster's modification module can be deduced from the existence of a maximal Cohen-Macaulay complex in a characteristic free way.
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... For every point p ∈ Z the local equations a, b form a regular sequence in the local ring O S,p . In this setting, the presentation of the powers of the ideal can be read off from the Rees algebra, which coincides with the symmetric algebra by [Hun80]. ...
SOME SPECIAL TRIPLE COVERS OF ALGEBRAIC SURFACES
Article
Full-text available
  • Dec 2022
  • DOC MATH
We study \emph{special triple covers} f ⁣:T→Sf\colon T \to S, where the Tschirnhausen bundle \ke = \left(f_*\ko_T/\ko_S\right)^\vee is a quotient of a split rank three vector bundle, and we provide several necessary and sufficient criteria for the existence. As an application, we give a complete classification of special triple planes, finding among others two nice families of K3 surfaces.
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... For every point ∈ the local equations , form a regular sequence in the local ring O , . In this setting, the presentation of the powers of the ideal can be read off from the Rees algebra, which coincides with the symmetric algebra by [Hun80]. ...
Special triple covers of algebraic surfaces
Preprint
Full-text available
  • Feb 2022
We study special triple covers f ⁣:T→Sf\colon T \to S of algebraic surfaces, where the Tschirnhausen bundle E=(f∗OT/OS)∨\mathcal E = \left(f_*\mathcal O_T/\mathcal O_S\right)^\vee is a quotient of a split rank three vector bundle, and we provide several necessary and sufficient criteria for the existence. As an application, we give a complete classification of special triple planes, finding among others two nice families of K3 surfaces.
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... It is enough to prove Ker ϕ ⊇ Ker ϕ. According to [20] and [31], Ker ϕ is generated by linear forms, since the sequence a 1 , a 2 , . . . , a d is a d-sequence on A by 2.14. ...
Gorensteinness in Rees algebras of powers of parameter ideals
Preprint
  • Dec 2021
This paper gives a necessary and sufficient condition for Gorensteinness in Rees algebras of the d-th power of parameter ideals in certain Noetherian local rings of dimension d≥2d\ge 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\mathcal{R}(\mathfrak{q}^d)=\oplus_{i\ge 0}\mathfrak{q}^{di} is Gorenstein for every parameter ideal q\mathfrak{q} that is a reduction of the maximal ideal in a d-dimensional Buchsbaum local ring of depth 1 and multiplicity 2.
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... In general, it is quite a hard task to describe the defining ideals of Rees algebras. Huneke proved that the defining ideal of the Rees algebra of an ideal generated by a d-sequence has a linear generating set, [11] (see [21] for a simple proof). We show that homogeneous almost complete intersection ideals in polynomial rings over an infinite field are generated by a d-sequence, Proposition 4.10. ...
Almost complete intersection binomial edge ideals and their Rees algebras
Article
Full-text available
  • Nov 2020
  • J PURE APPL ALGEBRA
Let G be a simple graph on n vertices and JGJ_G denote the binomial edge ideal of G in the polynomial ring S = \K[x_1, \ldots, x_n, y_1, \ldots, y_n]. In this article, we compute the second graded Betti numbers of JGJ_G, and we obtain a minimal presentation of it when G is a tree or a unicyclic graph. We classify all graphs whose binomial edge ideals are almost complete intersection, prove that they are generated by a d-sequence and that the Rees algebra of their binomial edge ideal is Cohen-Macaulay. We also obtain an explicit description of the defining ideal of the Rees algebra of those binomial edge ideals.
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... Huneke introduced the notion of d-sequence, in [17], and proved that the symmetric algebra and Rees algebra of an ideal generated by a d-sequence in a Noetherian ring are isomorphic, [15] (see [38] for a simple proof). He used the theory of d-sequence to study the depth of powers of ideals in a Noetherian ring R, [17]. ...
Regularity of powers of quadratic sequences with applications to binomial ideals
Article
Full-text available
  • Aug 2020
In this article, we obtain an upper bound for the Castelnuovo-Mumford regularity of powers of an ideal generated by a homogeneous quadratic sequence in a polynomial ring in terms of the regularity of its related ideals and degrees of its generators. As a consequence, we compute upper bounds for the regularity of powers of several binomial ideals. We generalize a result of Matsuda and Murai to show that the regularity of JGs is bounded below by 2s+ℓ(G)−1 for all s≥1, where JG denotes the binomial edge ideal of a graph G and ℓ(G) is the length of a longest induced path in G. We compute the regularity of powers of binomial edge ideals of cycle graphs, star graphs, and balloon graphs explicitly. Also, we give sharp bounds for the regularity of powers of almost complete intersection binomial edge ideals and parity binomial edge ideals.
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... For a more general situation when the Rees algebra is isomorphic with Sym(I) see[37]. ...
Double transgressions and Bott-Chern duality
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Full-text available
  • Mar 2020
We present a general framework for obtaining currential double transgression formulas on complex manifolds which can be seen as manifestations of Bott-Chern Duality. These results complement on one hand the simple transgression formulas obtained by Harvey -Lawson and on the other hand the double transgression formulas of Bismut-Gillet-Soul\'e. An explicit general double transgression for sections of the projectivization of a vector bundle endowed with a weighted homogeneous C∗\mathbb{C}^* action is obtained. Several applications are given. We mention a Gysin isomorphism for Bott-Chern cohomology, an abstract Poincar\'e-Lelong formula for sections of holomorphic and Hermitian vector bundles implying Andersson's generalization of the standard Poincar\'e-Lelong, a Bott-Chern duality formula for the Chern-Fulton classes of singular varieties or a refinement of the first author's simple transgression formula for the Chern character of a Quillen superconnection associated to a self-adjoint, odd endomorphism. The existence of a Bismut-Gillet-Soul\'e double transgression without the hypothesis of degeneration along a submanifold stands out and is based on an extension to linear correspondences of the operation of morphism addition. Finally, as a by-product we also obtain a statement about the {pointwise localization} of the Samuel multiplicity of an analytic subvariety of a regular variety (manifold) along an irreducible component.
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... Huneke introduced the notion of d-sequences, in [12], and proved that the Symmetric Algebra and Rees Algebra of ideals generated by a d-sequence in a Noetherian ring are isomorphic. He later used the theory of d-sequences to study the depth of powers of ideals in a Noetherian ring R, [14]. ...
Regularity of Powers of Quadratic Sequences with applications to binomial edge ideals
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Full-text available
  • Oct 2019
In this article we obtain an upper bound for the regularity of powers of ideals generated by a homogeneous quadratic sequence in a polynomial ring in terms of regularity of its related ideals and degrees of its generators. As a consequence we compute upper bounds for reg(JGs)reg(J_G^s) for some classes of graphs G. We generalize a result of Matsuda and Murai to show that the Castelnuovo-Mumford regularity of JGsJ^s_G is bounded below by 2s+ℓ(G)−12s+\ell(G)-1, where ℓ(G)\ell(G) is the longest induced path in any graph G. Using these two bounds we compute explicitly the regularity of powers of binomial edge ideals of cycle graphs, star graphs and balloon graphs. Also we give a sharp upper bound for the regularity of powers of almost complete intersection binomial edge ideals.
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... In particular, if x-regularity of R(I) is zero, then each power of I has a linear resolution. For a homogeneous ideal I, the defining ideal of R(I) is studied by many authors, see [10,11,16], and D. Taylor in [19] studied it for a monomial ideal. Further, Villarreal in [21] gives an explicit description of the defining ideal of the Rees algebra of any square free monomial ideal generated in degree 2. ...
Rees Algebra of a Squarefree Monomial Ideal
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Full-text available
  • Jun 2019
Let S=k[X1,…,Xn]S={\sf k}[X_1,\dots, X_n] be a polynomial ring, where k{\sf k} is a field. This article deals with the defining ideal of the Rees algebra of squarefree monomial ideal generated in degree n−2n-2. As a consequence, we prove that Betti numbers of powers of the cover ideal of the complement graph of a tree do not depend on the choice of tree. Further, we study the regularity and Betti numbers of powers of cover ideals associated to certain graphs.
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... In general, it is quite a hard task to describe the defining ideals of Rees algebras. Huneke proved that the defining ideal of the Rees algebra of an ideal generated by a d-sequence has a linear generating set, [8]. Such ideals are said to be of linear type. ...
Almost complete intersection binomial edge ideals and their Rees algebras
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Full-text available
  • Apr 2019
Let G be a simple graph on n vertices and JGJ_G denote the corresponding binomial edge ideal in the polynomial ring S=K[x1,…,xn,y1,…,yn].S = K[x_1, \ldots, x_n, y_1, \ldots, y_n]. In this article, we compute the second Betti number and obtain a minimal presentation of trees and unicyclic graphs. We also classify all graphs whose binomial edge ideals are almost complete intersection and we prove that the Rees algebra of their binomial edge ideal is Cohen-Macaulay. We also obtain an explicit description of the defining ideal of the Rees algebra of those binomial edge ideals.
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... To study its presentation, we will use simply Φ| RrX 1 ,...,Xms . Since r I is generated by a d-sequence, it follows from [10] ...
On the conjecture of Vasconcelos for Artinian almost complete intersection monomial ideals
Preprint
  • Feb 2019
In this short note, we confirm a conjecture of Vasconcelos which states that the Rees algebra of any Artinian almost complete intersection monomial ideal is almost Cohen--Macaulay.
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... This is a contradiction, so it must be that I " J, whence ℓ " µpIq. Therefore, by [4, 4.1 and 3.6(b)] it follows that I is generated by a d-sequence, hence it is of linear type (see [17,Theorem 3.1]). ...
On the Cohen-Macaulayness and defining ideal of Rees algebras of modules
Preprint
  • Nov 2018
In the first part of this paper, we consider a finite, torsion-free, orientable module E over a Gorenstein local ring. We provide a sufficient condition for the Rees algebra R(E)\mathcal{R}(E) of E to be Cohen-Macaulay. In the second part, we consider a finite module E of projective dimension one over k[X1,…,Xn]k[X_1, \ldots, X_n]. We describe the defining ideal of R(E)\mathcal{R}(E), under the assumption that the presentation matrix φ\varphi of E is almost linear, i.e. the entries of all but one column of φ\varphi are linear.
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... Proof. By [Hun80], ideals generated by d-sequences are of linear type, and I is generated by a d-sequence. ...
A stable version of Harbourne's Conjecture and the containment problem for space monomial curves
Preprint
Full-text available
  • Sep 2018
The symbolic powers I(n)I^{(n)} of a radical ideal I in a polynomial ring consist of the functions that vanish up to order n in the variety defined by I. These do not necessarily coincide with the ordinary algebraic powers InI^n, but it is natural to compare the two notions. The containment problem consists of determining the values of n and m for which I(n)⊆ImI^{(n)} \subseteq I^m holds. When I is an ideal of height 2 in a regular ring, I(3)⊆I2I^{(3)} \subseteq I^2 may fail, but we show that this containment does hold for the defining ideal of the space monomial curve (ta,tb,tc)(t^a, t^b, t^c). More generally, given a radical ideal I of big height h, while the containment I(hn−h+1)⊆InI^{(hn-h+1)} \subseteq I^n conjectured by Harbourne does not necessarily hold for all n, we give sufficient conditions to guarantee such containments for n≫0n \gg 0.
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Bounds on Dao numbers and applications to regular local rings
Article
  • Dec 2024
  • ANN MAT PUR APPL
The so-called Dao numbers are a sort of measure of the asymptotic behaviour of full properties of certain product ideals in a Noetherian local ring R with infinite residue field and positive depth. In this paper, we answer a question of H. Dao on how to bound such numbers. The auxiliary tools range from Castelnuovo–Mumford regularity of appropriate graded structures to reduction numbers of the maximal ideal. In particular, we substantially improve previous results (and answer questions) by the authors. Finally, as an application of the theory of Dao numbers, we provide new characterizations of when R is regular; for instance, we show that this holds if and only if the maximal ideal of R can be generated by a d-sequence (in the sense of Huneke) if and only if the third Dao number of any (minimal) reduction of the maximal ideal vanishes.
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Normal Sally modules of rank one
Preprint
  • Jun 2015
In this paper, we explore the structure of the normal Sally modules of rank one with respect to an m-primary ideal in a Nagata reduced local ring which is not necessary Cohen-Macaulay. As an application of this result, when the base ring is Cohen-Macaulay analytically unramified, the extremal bound on the first normal Hilbert coefficient leads to the Cohen-Macaulayness of the associated graded rings with respect to a normal filtration.
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Rees Algebras of Diagonal Ideals
Preprint
  • Jun 2011
There is a natural epimorphism from the symmetric algebra to the Rees algebra of an ideal. When this epimorphism is an isomorphism, we say that the ideal is of linear type. Given two determinantal rings over a field, we consider the diagonal ideal, the kernel of the multiplication map. We prove that the diagonal ideal is of linear type and recover the defining ideal of the Rees algebra in some special cases. The special fiber ring of the diagonal ideal is the homogeneous coordinate ring of the join variety.
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Torsion-free Aluffi Algebras
Preprint
  • May 2017
A pair of ideals J⊆I⊆RJ\subseteq I\subseteq R has been called Aluffi torsion-free if the Aluffi algebra of I/J is isomorphic with the corresponding Rees algebra. We give necessary and sufficient conditions for the Aluffi torsion-free property in terms of the first syzygy module of the form ideal J∗J^* in the associated graded ring of I. For two pairs of ideals J1,J2⊆IJ_1,J_2\subseteq I such that J1−J2∈I2J_1-J_2\in I^2, we prove that if one pair is Aluffi torsion-free the other one is so if and only if the first syzygy modules of J1J_1 and J2J_2 have the same form ideals. We introduce the notion of strongly Aluffi torsion-free ideals and present some results on these ideals.
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Geometry of logarithmic derivations of hyperplane arrangements
Chapter
  • Jan 2023
We study the Hadamard product of the linear forms defining a hyperplane arrangement with those of its dual, which we view as generating an ideal in a certain polynomial ring. We use this ideal, which we call the ideal of pairs, to study logarithmic derivations and critical set varieties of arrangements in a way which is symmetric with respect to matroid duality. Our main result exhibits the variety of the ideal of pairs as a subspace arrangement whose components correspond to cyclic flats of the arrangement. As a corollary, we are able to give geometric explanations of some freeness and projective dimension results due to Ziegler and Kung–Schenck.
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Graph rings and ideals: Wolmer Vasconcelos contributions
Preprint
Full-text available
  • May 2023
This is a survey article featuring some of Wolmer Vasconcelos' contributions to commutative algebra, and explaining how Vasconcelos' work and insights have contributed to the development of commutative algebra and its interaction with other areas to the present. We show that the regularity of subrings of normal k-uniform monomial ideals is a monotone function, and we give a normality criterion for edge ideals of graphs using Ehrhart rings.
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d-Sequence edge binomials, and regularity of powers of binomial edge ideals of trees
Article
  • May 2023
In this paper, we provide the necessary and sufficient conditions for the edge binomials of the tree forming a [Formula: see text]-sequence in terms of the degree sequence notion of a graph. We study the regularity of powers of the binomial edge ideals of trees generated by [Formula: see text]-sequence edge binomials.
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Residual intersections and modules with Cohen-Macaulay Rees algebra
Article
  • Dec 2021
  • J ALGEBRA
In this paper, we consider a finite, torsion-free module E over a Gorenstein local ring. We provide sufficient conditions for E to be of linear type and for the Rees algebra R(E) of E to be Cohen-Macaulay. Our results are obtained by constructing a generic Bourbaki ideal I of E and exploiting properties of the residual intersections of I.
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Defining Equations of Blowup Algebras
Chapter
  • Jan 2021
  • Lect Notes Math
This is the content of four lectures on Rees rings and their defining equations delivered at the CIME workshop Recent developments in Commutative Algebra held at Levico Terme (Trento), Italy on July 1–5, 2019. The Rees ring and the special fiber ring of an ideal arise in the process of blowing up a variety along a subvariety. Rees rings and special fiber rings also describe, respectively, the graphs and the images of rational maps between projective spaces. A difficult open problem in commutative algebra, algebraic geometry, elimination theory, and geometric modeling is to determine explicitly the equations defining graphs and images of rational maps. In these lectures we will discuss this topic in several situations.
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Buchsbaumness of the associated graded rings of filtration
Preprint
Full-text available
  • Feb 2021
Let (A,m)(A,\mathfrak m) be a Noetherian local ring of dimension d>0d>0 with infinite residue field and I an m\mathfrak{m}-primary ideal. Let I\mathcal I be an I-good filtration. We study an equality of Hilbert coefficients, first given by Elias and Valla, versus passage of Buchsbaum property from the local ring to the blow-up algebras. Suppose e1(I)−e1(Q)=2e0(I)−2ℓ(A/I1)−ℓ(I1/(I2+Q))e_1(\mathcal I)-e_1(Q)=2e_0(\mathcal I)-2\ell(A/I_1)-\ell(I_1/(I_2+Q)) where Q⊆IQ\subseteq I, a minimal reduction of I\mathcal I, is a standard parameter ideal. Under some mild conditions, we prove that if A is Buchsbaum (generalized Cohen-Macaulay respectively), then the associated graded ring G(I)G(\mathcal I) is Buchsbaum (generalized Cohen-Macaulay respectively). Our results settle a question of Corso in general for an I-good filtration. Further, let f0(I)=e1(I)−e0(I)−e1(Q)+ℓ(A/I)+μ(I)−d+1f_0(I)= e_1(I)-e_0(I)-e_1(Q)+\ell(A/I)+\mu(I)-d+1 and e1(I)−e1(Q)=2e0(I)−2ℓ(A/I)−ℓ(I/(I2+Q))e_1(I)-e_1(Q)=2e_0(I)-2\ell(A/I)-\ell(I/(I^2+Q)). We prove, under mild conditions, that (1) if A is generalized Cohen-Macaulay, then the special fiber ring Fm(I)F_{\mathfrak{m}}(I) is generalized Cohen-Macaulay; In addition, if depth of A is positive, then depth of Fm(I)F_{\mathfrak {m}}(I) is same as depth of A and (2) if A is Buchsbaum and depth A≥d−1\geq d-1, then Fm(I)F_{\mathfrak{m}}(I) is Buchsbaum and the I-invariant of Fm(I)F_{\mathfrak{m}}(I) is same as that of A.
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Regularity, Rees algebra, and Betti numbers of certain cover ideals
Article
  • Jul 2020
Let S=k[X1,…,Xn]S={\textsf {k}}[X_1,\dots , X_n] be a polynomial ring, where k{\textsf {k}} is a field. This article deals with the defining ideal of the Rees algebra of a squarefree monomial ideal generated in degree n−2n-2. As a consequence, we prove that Betti numbers of powers of the cover ideal of the complement graph of a tree do not depend on the choice of the tree. Further, we study the regularity and Betti numbers of powers of cover ideals associated to certain graphs.
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A stable version of Harbourne's Conjecture and the containment problem for space monomial curves
Article
  • May 2020
  • J PURE APPL ALGEBRA
The symbolic powers I(n) of a radical ideal I in a polynomial ring consist of the functions that vanish up to order n in the variety defined by I. These do not necessarily coincide with the ordinary algebraic powers In, but it is natural to compare the two notions. The containment problem consists of determining the values of n and m for which I(n)⊆Im holds. When I is an ideal of height 2 in a regular ring, I(3)⊆I2 may fail, but we show that this containment does hold for the defining ideal of the space monomial curve (ta,tb,tc). More generally, given a radical ideal I of big height h, while the containment I(hn−h+1)⊆In conjectured by Harbourne does not necessarily hold for all n, we give sufficient conditions to guarantee I(hn−h+1)⊆In for n≫0.
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Symbolic blowup algebras and invariants associated to certain monomial curves in P3
Article
Full-text available
  • Apr 2020
In this paper we explicitly describe the symbolic powers of the ideal defining the curve C(q,m) in P3 parametrized by (xd+2m,xd+mym,xdy2m,yd+2m), where q, m are positive integers, d=2q+1 and gcd(d,m)=1. We show that the symbolic blowup algebra is Noetherian and Gorenstein. An explicit formula for the resurgence and the Waldschmidt constant of the prime ideal p:=pC(q,m) defining the curve C(q,m) is computed. We also give a formula for the Castelnuovo-Mumford regularity of the symbolic powers p(n) for all n≥1. Communicated by Uli Walther
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ON THE CONJECTURE OF VASCONCELOS FOR ARTINIAN ALMOST COMPLETE INTERSECTION MONOMIAL IDEALS
Article
  • Dec 2019
In this short note, we confirm a conjecture of Vasconcelos which states that the Rees algebra of any Artinian almost complete intersection monomial ideal is almost Cohen–Macaulay.
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Symbolic blowup algebras and invariants associated to certain monomial curves in ${\mathbb P}^3
Preprint
  • Apr 2019
In this paper we explicitly describe the symbolic powers of curves C(q,m){\mathcal C}(q,m) in P3{\mathbb P}^3 parametrized by (xd+2m,xd+mym,xdy2m,yd+2m)( x^{d+2m}, x^{d+m} y^m, x^{d} y^{2m}, y^{d+2m}), where q,m are positive integers, d=2q+1 and gcd⁡(d,m)=1\gcd(d,m)=1. The defining ideal of these curves is a set-theoretic complete intersection. We show that the symbolic blowup algebra is Noetherian and Gorenstein. An explicit formula for the resurgence and the Waldschmidt constant of the prime ideal p:=pC(q,m){\mathfrak p}:={\mathfrak p}_{ { \mathcal C}(q,m) } defining the curve C(q,m){\mathcal C}(q,m) is computed. We also give a formula for the Castelnuovo-Mumford regularity of the symbolic powers p(n){\mathfrak p}^{(n)} for all n≥1n \geq 1.
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Conditions for Ker(R [ X ] →R [ c/b ]) to Have a Linear Base
Article
  • Aug 1973
A number of necessary and/or sufficient conditions are given for the kernel of the homomorphism in the title to be generated by linear polynomials. Two applications are given.
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The theory of -sequences and powers of ideals
Article
  • Dec 1982
Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/23781/1/0000019.pdf
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Noncommutative Complete Intersections
Article
  • Feb 2013
Several generalizations of a commutative ring that is a graded complete intersection are proposed for a noncommutative graded k-algebra; these notions are justified by examples from noncommutative invariant theory.
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Young Diagrams and Determinantal Varieties.|*^ W ЯП I*)
Article
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Are complete intersections complete intersections?
Article
  • Sep 2011
A commutative local ring is generally defined to be a complete intersection if its completion is isomorphic to the quotient of a regular local ring by an ideal generated by a regular sequence. It has not previously been determined whether or not such a ring is necessarily itself the quotient of a regular ring by an ideal generated by a regular sequence. In this article, it is shown that if a complete intersection is a one dimensional integral domain, then it is such a quotient. However, an example is produced of a three dimensional complete intersection domain which is not a homomorphic image of a regular local ring, and so the property does not hold in general.
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Sulle algebre simmetricle e di Rees di un ideale
  • P Salmon
  • P Salmon
Sur les algébras symétrique et de Rees d'un idéal
  • A Micali
A. MICALI, Sur les algebras symetrique et de Rees d'un ideal, Ann. Itut. Fourier.
Sulle graduate relative ad un ideale
  • P Salmon
P. SALMON, Sulle graduate relative ad un ideale, Symposia Mathematicn 8, 269-293.
Modules rtflexifs et anneaux facoriels, in " SCminare Dubreil 1963-1964
  • Jan 1964
  • P Samuel
P.'SAMUEL, Modules rtflexifs et anneaux facoriels, in " SCminare Dubreil 1963-1964, " Expose du 27 Janvier (1964).