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

Papers by Jugal Verma

arXiv (Cornell University), Jan 18, 2019

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

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

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.

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

Journal of Pure and Applied Algebra, 2014

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

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

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

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.

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

Journal of Pure and Applied Algebra, 2014

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.