Jugal Verma - Academia.edu (original) (raw)

Papers by Jugal Verma

arXiv (Cornell University), Jan 18, 2019

Using vanishing of graded components of local cohomology modules of the Rees algebra of the norma... more Using vanishing of graded components of local cohomology modules of the Rees algebra of the normal filtration of an ideal, we give bounds on the normal reduction number. This helps to get necessary and sufficient conditions in Cohen-Macaulay local rings of dimension d ≥ 3, for the vanishing of the normal Hilbert coefficients e k (I) for k ≤ d, in terms of the normal reduction number. Let I be an m-primary ideal of a Noetherian local ring (R, m). The function H I (n) = ℓ(R/I n) is called the normal Hilbert function of I. Using (1.1), Rees showed that if R is an analytically unramified Noetherian local ring of dimension d, then there exists a polynomial P I (x) ∈ Q[x] of degree d such that for all large n, H I (n) = P I (n). The polynomial P I (x) is called the normal Hilbert polynomial of I. We write it as The first author is supported by a UGC fellowship, Govt. of India.

Commutative Algebra, 2021

Let R R be the face ring of a simplicial complex of dimension d − 1 d-1 and R ( n ) {\mathcal R}(... more Let R R be the face ring of a simplicial complex of dimension d − 1 d-1 and R ( n ) {\mathcal R}({\mathfrak {n}}) be the Rees algebra of the maximal homogeneous ideal n {\mathfrak {n}} of R . R. We show that the generalized Hilbert-Kunz function H K ( s ) = ℓ ( R ( n ) / ( n , n t ) [ s ] ) HK(s)=\ell ({\mathcal {R}}({\mathfrak {n}})/({\mathfrak {n}}, {\mathfrak {n}} t)^{[s]}) is given by a polynomial for all large s . s. We calculate it in many examples and also provide a Macaulay2 code for computing H K ( s ) . HK(s).

Communications in Algebra, 2021

We prove that the Hilbert-Kunz function of the ideal (I, It) of the Rees algebra R(I), where I is... more We prove that the Hilbert-Kunz function of the ideal (I, It) of the Rees algebra R(I), where I is an m-primary ideal of a 1-dimensional local ring (R, m), is a quasi-polynomial in e, for large e. For s ∈ N, we calculate the Hilbert-Samuel function of the R-module I [s] and obtain an explicit description of the generalized Hilbert-Kunz function of the ideal (I, It)R(I) when I is a parameter ideal in a Cohen-Macaulay local ring of dimension d ≥ 2, proving that the generalized Hilbert-Kunz function is a piecewise polynomial in this case. The first author is supported by a UGC fellowship, Govt. of India.

arXiv: Commutative Algebra, 2018

In this exposition of the equality and inequality of Minkowski for multiplicity of ideals, we pro... more In this exposition of the equality and inequality of Minkowski for multiplicity of ideals, we provide simple algebraic and geometric proofs. Connections with mixed multiplicities of ideals are explained.

Commutative Algebra, 2021

Cornell University - arXiv, Mar 18, 2022

We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel coefficient... more We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel coefficients of the Frobenius powers of an m-primary ideal exists in a Noetherian local ring (R, m) with prime characteristic p > 0. This, in turn, gives an expression of the Hilbert-Kunz multiplicity of powers of the ideal. We also prove that for a face ring R of a simplicial complex and an ideal J generated by pure powers of the variables, the generalized Hilbert-Kunz function ℓ(R/(J [q]) k) is a polynomial for all q, k and also give an expression of the generalized Hilbert-Kunz multiplicity of powers of J in terms of Hilbert-Samuel multiplicity of J. We conclude by giving a counterexample to a conjecture proposed by I. Smirnov which connects the stability of an ideal with the asymptotic limit of the first Hilbert coefficient of the Frobenius power of the ideal.

A theorem of Macaulay on colons of ideals in polynomial rings is proved for homogeneous Gorenstei... more A theorem of Macaulay on colons of ideals in polynomial rings is proved for homogeneous Gorenstein algebras.

Formulas are obtained in terms of complete reductions for the bigraded components of local cohomo... more Formulas are obtained in terms of complete reductions for the bigraded components of local cohomology modules of bigraded Rees algebras of 0-dimensional ideals in 2-dimensional Cohen-Macaulay local rings. As a consequence, cohomological expressions for the coefficients of the Bhattacharya polynomial of such ideals are obtained.

Let R be a d-dimensional standard graded ring over an Artinian local ring. Let M be the unique ma... more Let R be a d-dimensional standard graded ring over an Artinian local ring. Let M be the unique maximal homogeneous ideal of R. Let h i (R)n denote the length of the nth graded component of the local cohomology module Hi ( M d−1 q the Eisenbud-Goto invariant EG(R) of R to be the number ∑ d−1 q=0

Let ∆ be an abstract finite simplicial complex with vertices X1, . . . , Xn. Let k be a field thr... more Let ∆ be an abstract finite simplicial complex with vertices X1, . . . , Xn. Let k be a field throughout this chapter. Let R denote the polynomial ring k[X1, X2, . . . , Xn], where, by abuse of notation, we regard the vertices X1, X2, . . . , Xn as indeterminates over k. Let I∆ be the ideal of R generated by the monomials Xi1 . . . Xir , i1 < i2 < . . . < ir such that {Xi1 , . . . , Xir} is not a face of ∆. The face ring of ∆ is the quotient ring k[∆] := R/I∆. Since I∆ is a homogeneous ideal, k[∆] is a graded ring. In this section we will prove Stanley’s formula for the Hilbert series of k[∆]. In some sense, this formula opened up the connection of Commutative Algebra with Combinatorics. We will exhibit the power of Hilbert series methods by giving an elementary proof of Dehn-Sommerville equations towards the end of this section. We begin by establishing the primary decomposition of I∆. (1.1) Definition. Let F be a face of a simplicial complex ∆. Let PF denote the prime ide...

In this exposition of the equality and inequality of Minkowski for multiplicity of ideals, we pro... more In this exposition of the equality and inequality of Minkowski for multiplicity of ideals, we provide simple algebraic and geometric proofs. Connections with mixed multiplicities of ideals are explained.

Two formulas for the multiplicity of the fiber cone F(I)=oplusn=0inftyIn/mInF(I)=\oplus_{n=0}^{\infty} I^n/\m I^nF(I)=oplusn=0inftyIn/mIn of an... more Two formulas for the multiplicity of the fiber cone F(I)=oplusn=0inftyIn/mInF(I)=\oplus_{n=0}^{\infty} I^n/\m I^nF(I)=oplusn=0inftyIn/mIn of an m\mm-primary ideal of a ddd-dimensional Cohen-Macaulay local ring (R,m)(R,\m)(R,m) are derived in terms of the mixed multiplicity ed−1(m∣I),e_{d-1}(\m | I),ed−1(m∣I), the multiplicity e(I)e(I)e(I) and superficial elements. As a consequence, the Cohen-Macaulay property of F(I)F(I)F(I) when III has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of reduction number of III and lengths of certain ideals. We also characterize Cohen-Macaulay and Gorenstein property of fiber cones of m\mm-primary ideals with a ddd-generated minimal reduction JJJ satisfying (i) ell(I2/JI)=1\ell(I^2/JI)=1ell(I2/JI)=1 or (ii) ell(Im/Jm)=1.\ell(I\m/J\m)=1.ell(Im/Jm)=1.

Let R be a 3-dimensional regular local ring. Let p be a dimension one prime of R. We are concerne... more Let R be a 3-dimensional regular local ring. Let p be a dimension one prime of R. We are concerned with two problems: (1) when is p a complete intersection and (2) when is p a set theoretic complete intersection ? We prove a result of Cowsik and Nori to answer (1) and several results answering (2). Cowsik observed that if the symbolic Rees algebra of p is Noetherian then p is a set-theoretic complete intersection. Thus it becomes important to know when the symbolic Rees algebra is Noetherian. We will present Huneke’s elegant criterion for its Noetherian property.

Journal of Commutative Algebra, 2012

Let I be an m-primary ideal of a Noetherian local ring (R, m) of positive dimension. The coeffici... more Let I be an m-primary ideal of a Noetherian local ring (R, m) of positive dimension. The coefficient e1(I) of the Hilbert polynomial of an I-admissible filtration I is called the Chern number of I. A formula for the Chern number has been derived involving Euler characteristic of subcomplexes of a Koszul complex. Specific formulas for the Chern number have been given in local rings of dimension at most two. These have been used to provide new and unified proofs of several results about e1(I).

Journal of Pure and Applied Algebra, 2014

Let (R, m) be an analytically unramified Cohen-Macaulay local ring of dimension 2 with infinite r... more Let (R, m) be an analytically unramified Cohen-Macaulay local ring of dimension 2 with infinite residue field and I be the integral closure of an ideal I in R. Necessary and sufficient conditions are given for I r+1 J s+1 = aI r J s+1 +bI r+1 J s to hold for all r ≥ r0 and s ≥ s0 in terms of vanishing of [H 2 (at 1 ,bt 2) (R ′ (I, J))] (r 0 ,s 0) , where a ∈ I, b ∈ J is a good joint reduction of the filtration {I r J s }. This is used to derive a theorem due to Rees on normal joint reduction number zero. The vanishing of e2(IJ) is shown to be equivalent to Cohen-Macaulayness of R(I, J).

CONTENTS 1. Introduction 2. Milnor numbers and mixed multiplicities 3. Minkowski inequalities and... more CONTENTS 1. Introduction 2. Milnor numbers and mixed multiplicities 3. Minkowski inequalities and equalities 4. Joint reductions 5. Multiplicities of blow-up algebras 6. Cohen-Macaulay Blow-up algebras with minimal multiplicity 7. Applications to number of generators of ideals 8. Questions

Abstract. We settle the negativity conjecture of Vasconcelos for the Chern number of an ideal in ... more Abstract. We settle the negativity conjecture of Vasconcelos for the Chern number of an ideal in certain unmixed quotients of regular local rings by explicit calculation of the Hilbert polynomials of all ideals generated by systems of parameters.

Abstract. Two formulas for the multiplicity of the fiber cone F(I) = ⊕ ∞ n=0I n /mI n of an m-pri... more Abstract. Two formulas for the multiplicity of the fiber cone F(I) = ⊕ ∞ n=0I n /mI n of an m-primary ideal of a d-dimensional Cohen-Macaulay local ring (R,m) are derived in terms of the mixed multiplicity ed−1(m|I), the multiplicity e(I) and superficial elements. As a consequence, the Cohen-Macaulay property of F(I) when I has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of reduction number of I and lengths of certain ideals. We also characterize Cohen-Macaulay and Gorenstein property of fiber cones of m-primary ideals with a d-generated minimal reduction J satisfying (i) ℓ(I 2 /JI) = 1 or (ii) ℓ(Im/Jm) = 1. 1.

Abstract. We present short and elementary proofs of two theorems of Huckaba and Marley, while gen... more Abstract. We present short and elementary proofs of two theorems of Huckaba and Marley, while generalizing them at the same time to the case of a module. The theorems concern a characterization of the depth of the associated graded ring of a Cohen-Macaulay module, with respect to a Hilbert filtration, in terms of the Hilbert coefficient e1. As an application, we derive bounds on the higher Hilbert coefficient ei in terms of e0. 1.

arXiv (Cornell University), Jan 18, 2019

Using vanishing of graded components of local cohomology modules of the Rees algebra of the norma... more Using vanishing of graded components of local cohomology modules of the Rees algebra of the normal filtration of an ideal, we give bounds on the normal reduction number. This helps to get necessary and sufficient conditions in Cohen-Macaulay local rings of dimension d ≥ 3, for the vanishing of the normal Hilbert coefficients e k (I) for k ≤ d, in terms of the normal reduction number. Let I be an m-primary ideal of a Noetherian local ring (R, m). The function H I (n) = ℓ(R/I n) is called the normal Hilbert function of I. Using (1.1), Rees showed that if R is an analytically unramified Noetherian local ring of dimension d, then there exists a polynomial P I (x) ∈ Q[x] of degree d such that for all large n, H I (n) = P I (n). The polynomial P I (x) is called the normal Hilbert polynomial of I. We write it as The first author is supported by a UGC fellowship, Govt. of India.

Commutative Algebra, 2021

Let R R be the face ring of a simplicial complex of dimension d − 1 d-1 and R ( n ) {\mathcal R}(... more Let R R be the face ring of a simplicial complex of dimension d − 1 d-1 and R ( n ) {\mathcal R}({\mathfrak {n}}) be the Rees algebra of the maximal homogeneous ideal n {\mathfrak {n}} of R . R. We show that the generalized Hilbert-Kunz function H K ( s ) = ℓ ( R ( n ) / ( n , n t ) [ s ] ) HK(s)=\ell ({\mathcal {R}}({\mathfrak {n}})/({\mathfrak {n}}, {\mathfrak {n}} t)^{[s]}) is given by a polynomial for all large s . s. We calculate it in many examples and also provide a Macaulay2 code for computing H K ( s ) . HK(s).

Communications in Algebra, 2021

We prove that the Hilbert-Kunz function of the ideal (I, It) of the Rees algebra R(I), where I is... more We prove that the Hilbert-Kunz function of the ideal (I, It) of the Rees algebra R(I), where I is an m-primary ideal of a 1-dimensional local ring (R, m), is a quasi-polynomial in e, for large e. For s ∈ N, we calculate the Hilbert-Samuel function of the R-module I [s] and obtain an explicit description of the generalized Hilbert-Kunz function of the ideal (I, It)R(I) when I is a parameter ideal in a Cohen-Macaulay local ring of dimension d ≥ 2, proving that the generalized Hilbert-Kunz function is a piecewise polynomial in this case. The first author is supported by a UGC fellowship, Govt. of India.

arXiv: Commutative Algebra, 2018

In this exposition of the equality and inequality of Minkowski for multiplicity of ideals, we pro... more In this exposition of the equality and inequality of Minkowski for multiplicity of ideals, we provide simple algebraic and geometric proofs. Connections with mixed multiplicities of ideals are explained.

Commutative Algebra, 2021

Cornell University - arXiv, Mar 18, 2022

We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel coefficient... more We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel coefficients of the Frobenius powers of an m-primary ideal exists in a Noetherian local ring (R, m) with prime characteristic p > 0. This, in turn, gives an expression of the Hilbert-Kunz multiplicity of powers of the ideal. We also prove that for a face ring R of a simplicial complex and an ideal J generated by pure powers of the variables, the generalized Hilbert-Kunz function ℓ(R/(J [q]) k) is a polynomial for all q, k and also give an expression of the generalized Hilbert-Kunz multiplicity of powers of J in terms of Hilbert-Samuel multiplicity of J. We conclude by giving a counterexample to a conjecture proposed by I. Smirnov which connects the stability of an ideal with the asymptotic limit of the first Hilbert coefficient of the Frobenius power of the ideal.

A theorem of Macaulay on colons of ideals in polynomial rings is proved for homogeneous Gorenstei... more A theorem of Macaulay on colons of ideals in polynomial rings is proved for homogeneous Gorenstein algebras.

Formulas are obtained in terms of complete reductions for the bigraded components of local cohomo... more Formulas are obtained in terms of complete reductions for the bigraded components of local cohomology modules of bigraded Rees algebras of 0-dimensional ideals in 2-dimensional Cohen-Macaulay local rings. As a consequence, cohomological expressions for the coefficients of the Bhattacharya polynomial of such ideals are obtained.

Let R be a d-dimensional standard graded ring over an Artinian local ring. Let M be the unique ma... more Let R be a d-dimensional standard graded ring over an Artinian local ring. Let M be the unique maximal homogeneous ideal of R. Let h i (R)n denote the length of the nth graded component of the local cohomology module Hi ( M d−1 q the Eisenbud-Goto invariant EG(R) of R to be the number ∑ d−1 q=0

Let ∆ be an abstract finite simplicial complex with vertices X1, . . . , Xn. Let k be a field thr... more Let ∆ be an abstract finite simplicial complex with vertices X1, . . . , Xn. Let k be a field throughout this chapter. Let R denote the polynomial ring k[X1, X2, . . . , Xn], where, by abuse of notation, we regard the vertices X1, X2, . . . , Xn as indeterminates over k. Let I∆ be the ideal of R generated by the monomials Xi1 . . . Xir , i1 < i2 < . . . < ir such that {Xi1 , . . . , Xir} is not a face of ∆. The face ring of ∆ is the quotient ring k[∆] := R/I∆. Since I∆ is a homogeneous ideal, k[∆] is a graded ring. In this section we will prove Stanley’s formula for the Hilbert series of k[∆]. In some sense, this formula opened up the connection of Commutative Algebra with Combinatorics. We will exhibit the power of Hilbert series methods by giving an elementary proof of Dehn-Sommerville equations towards the end of this section. We begin by establishing the primary decomposition of I∆. (1.1) Definition. Let F be a face of a simplicial complex ∆. Let PF denote the prime ide...

In this exposition of the equality and inequality of Minkowski for multiplicity of ideals, we pro... more In this exposition of the equality and inequality of Minkowski for multiplicity of ideals, we provide simple algebraic and geometric proofs. Connections with mixed multiplicities of ideals are explained.

Two formulas for the multiplicity of the fiber cone F(I)=oplusn=0inftyIn/mInF(I)=\oplus_{n=0}^{\infty} I^n/\m I^nF(I)=oplusn=0inftyIn/mIn of an... more Two formulas for the multiplicity of the fiber cone F(I)=oplusn=0inftyIn/mInF(I)=\oplus_{n=0}^{\infty} I^n/\m I^nF(I)=oplusn=0inftyIn/mIn of an m\mm-primary ideal of a ddd-dimensional Cohen-Macaulay local ring (R,m)(R,\m)(R,m) are derived in terms of the mixed multiplicity ed−1(m∣I),e_{d-1}(\m | I),ed−1(m∣I), the multiplicity e(I)e(I)e(I) and superficial elements. As a consequence, the Cohen-Macaulay property of F(I)F(I)F(I) when III has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of reduction number of III and lengths of certain ideals. We also characterize Cohen-Macaulay and Gorenstein property of fiber cones of m\mm-primary ideals with a ddd-generated minimal reduction JJJ satisfying (i) ell(I2/JI)=1\ell(I^2/JI)=1ell(I2/JI)=1 or (ii) ell(Im/Jm)=1.\ell(I\m/J\m)=1.ell(Im/Jm)=1.

Let R be a 3-dimensional regular local ring. Let p be a dimension one prime of R. We are concerne... more Let R be a 3-dimensional regular local ring. Let p be a dimension one prime of R. We are concerned with two problems: (1) when is p a complete intersection and (2) when is p a set theoretic complete intersection ? We prove a result of Cowsik and Nori to answer (1) and several results answering (2). Cowsik observed that if the symbolic Rees algebra of p is Noetherian then p is a set-theoretic complete intersection. Thus it becomes important to know when the symbolic Rees algebra is Noetherian. We will present Huneke’s elegant criterion for its Noetherian property.

Journal of Commutative Algebra, 2012

Let I be an m-primary ideal of a Noetherian local ring (R, m) of positive dimension. The coeffici... more Let I be an m-primary ideal of a Noetherian local ring (R, m) of positive dimension. The coefficient e1(I) of the Hilbert polynomial of an I-admissible filtration I is called the Chern number of I. A formula for the Chern number has been derived involving Euler characteristic of subcomplexes of a Koszul complex. Specific formulas for the Chern number have been given in local rings of dimension at most two. These have been used to provide new and unified proofs of several results about e1(I).

Journal of Pure and Applied Algebra, 2014

Let (R, m) be an analytically unramified Cohen-Macaulay local ring of dimension 2 with infinite r... more Let (R, m) be an analytically unramified Cohen-Macaulay local ring of dimension 2 with infinite residue field and I be the integral closure of an ideal I in R. Necessary and sufficient conditions are given for I r+1 J s+1 = aI r J s+1 +bI r+1 J s to hold for all r ≥ r0 and s ≥ s0 in terms of vanishing of [H 2 (at 1 ,bt 2) (R ′ (I, J))] (r 0 ,s 0) , where a ∈ I, b ∈ J is a good joint reduction of the filtration {I r J s }. This is used to derive a theorem due to Rees on normal joint reduction number zero. The vanishing of e2(IJ) is shown to be equivalent to Cohen-Macaulayness of R(I, J).

CONTENTS 1. Introduction 2. Milnor numbers and mixed multiplicities 3. Minkowski inequalities and... more CONTENTS 1. Introduction 2. Milnor numbers and mixed multiplicities 3. Minkowski inequalities and equalities 4. Joint reductions 5. Multiplicities of blow-up algebras 6. Cohen-Macaulay Blow-up algebras with minimal multiplicity 7. Applications to number of generators of ideals 8. Questions

Abstract. We settle the negativity conjecture of Vasconcelos for the Chern number of an ideal in ... more Abstract. We settle the negativity conjecture of Vasconcelos for the Chern number of an ideal in certain unmixed quotients of regular local rings by explicit calculation of the Hilbert polynomials of all ideals generated by systems of parameters.

Abstract. Two formulas for the multiplicity of the fiber cone F(I) = ⊕ ∞ n=0I n /mI n of an m-pri... more Abstract. Two formulas for the multiplicity of the fiber cone F(I) = ⊕ ∞ n=0I n /mI n of an m-primary ideal of a d-dimensional Cohen-Macaulay local ring (R,m) are derived in terms of the mixed multiplicity ed−1(m|I), the multiplicity e(I) and superficial elements. As a consequence, the Cohen-Macaulay property of F(I) when I has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of reduction number of I and lengths of certain ideals. We also characterize Cohen-Macaulay and Gorenstein property of fiber cones of m-primary ideals with a d-generated minimal reduction J satisfying (i) ℓ(I 2 /JI) = 1 or (ii) ℓ(Im/Jm) = 1. 1.

Abstract. We present short and elementary proofs of two theorems of Huckaba and Marley, while gen... more Abstract. We present short and elementary proofs of two theorems of Huckaba and Marley, while generalizing them at the same time to the case of a module. The theorems concern a characterization of the depth of the associated graded ring of a Cohen-Macaulay module, with respect to a Hilbert filtration, in terms of the Hilbert coefficient e1. As an application, we derive bounds on the higher Hilbert coefficient ei in terms of e0. 1.