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J. K. Verma

Publications and source records attributed to J. K. Verma.

At least 37 records · Page 2Linked to original sources
arXiv2016

Postulation and reduction vectors of multigraded filtrations of ideals

We study 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 generalisation 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 $\mathcal F.$

math.AC↗
arXiv2014

Local Cohomology of Multi-Rees Algebras with Applications to Joint Reductions and Complete Ideals

In this paper, we obtain a generalization, in dimension $3$, of a theorem of David Rees about joint reductions of the bigraded filtration $\{ \overline{I^rJ^s}\}$ of complete ${\mathfrak m}$-primary ideals and vanishing of the second normal Hilbert coefficient $\overline{e}_2(IJ)$ where $R$ is a two-dimensional Cohen-Macaulay analytically unramified local ring with maximal ideal $\mathfrak m.$ This generalization is obtained as a consequence of a formula for the third local cohomology module of the extended Rees algebras of the $\mathbb Z^3$-graded filtration $\{\overline{I^rJ^sK^t}\}$ with support in the ideal $(x_1t_1,x_2t_2,x_3t_3)$ where $(x_1,x_2,x_3)$ is a good joint reduction of $\{\overline{I^rJ^sK^t}\}.$

math.AC↗
arXiv2013

Local Cohomology of Bigraded Rees Algebras and Normal Hilbert Coefficients

Let $(R,\m)$ be an analytically unramified Cohen-Macaulay local ring of dimension 2 with infinite residue field and $\ov{I}$ be the integral closure of an ideal $I$ in $R$. Necessary and sufficient conditions are given for $\ov{I^{r+1}J^{s+1}}=a\ov{I^rJ^{s+1}}+b\ov{I^{r+1}J^s}$ to hold ${for all}r \geq r_0{and}s \geq s_0$ in terms of vanishing of $[H^2_{(at_1,bt_2)}(\ov{\mathcal{R}^\prime}(I,J))]_{(r_0,s_0)}$, where $a \in I,b \in J$ is a good joint reduction of the filtration $\{\ov{I^rJ^s}\}.$ This is used to derive a theorem due to Rees on normal joint reduction number zero. The vanishing of $\ov{e}_2(IJ)$ is shown to be equivalent to Cohen-Macaulayness of $\ov{\mathcal{R}}(I,J)$.

math.AC↗
arXiv2012

Normal Hilbert Polynomials : A survey

We survey some of the major results about normal Hilbert polynomials of ideals. We discuss a formula due to Lipman for complete ideals in regular local rings of dimension two, theorems of Huneke, Itoh, Huckaba, Marley and Rees in Cohen-Macaulay analytically unramified local rings. We also discuss recent works of Goto-Hong-Mandal and Mandal-Singh-Verma concerning the positivity of the first coefficient of the normal Hilbert polynomial in unmixed analytically unramified local rings. Results of Moral{é}s and Villarreal linking normal Hilbert polynomial of monomial ideals with Ehrhart polynomials of polytopes are also presented.

math.AC↗
arXiv2012

On the Chern number of $I$-admissible filtrations of ideals

Let $I$ be an $\m$-primary ideal of a Noetherian local ring $(R, \m)$ of positive dimension. The coefficient $e_1(\mathcal I)$ of the Hilbert polynomial of an $I$-admissible filtration $\mathcal I$ is called the Chern number of $\mathcal I$. 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 $e_1(\mathcal I)$.

math.AC↗
arXiv2010

On some cconjectures about the Chern numbers of filtrations

Let I be an m-primary ideal of a Noetherian local ring (R,m) of positive dimension. The coefficient $e_1(A)$ of the Hilbert polynomial of an I-admissible filtration A is called the Chern number of A. The Positivity Conjecture of Vasconcelos for the Chern number of the integral closure filtration ${\bar{I^n}}$ is proved for a 2-dimensional complete local domain and more generally for any analytically unramified local ring R whose integral closure in its total ring of fractions is Cohen-Macaulay as an R-module. It is proved that if I is a parameter ideal then the Chern number of the I-adic filtration is non-negative. Several other results on the Chern number of the integral closure filtration are established, especially in the case when R is not necessarily Cohen-Macaulay.

math.AC↗
arXiv2009

On the Chern Number of an Ideal

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.

math.AC↗
arXiv2008

Hilbert Coefficients and Depth of the Associated Graded Ring of an Ideal

In this expository paper we survey results that relate Hilbert coefficients of an m-primary ideal I in a Cohen-Macaulay local ring (R, m) with depth of the associated graded ring G(I). Several results in this area follow from two theorems of S. Huckaba and T. Marley. These were proved using homological techniques. We provide simple proofs using superficial sequences.

math.AC↗
arXiv2007

Hilbert polynomials and powers of ideals

The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal $I$ in the polynomial ring $S=K[x_1,...,x_n]$ and a finitely generated graded $S$-module, the Hilbert coefficients $e_i(M/I^kM)$ are polynomial functions. Given two families of graded ideals $(I_k)_{k\geq 0}$ and $(J_k)_{k\geq 0}$ with $J_k\subset I_k$ for all $k$ with the property that $J_kJ_\ell\subset J_{k+\ell}$ and $I_kI_\ell\subset I_{k+\ell}$ for all $k$ and $\ell$, and such that the algebras $A=\Dirsum_{k\geq 0}J_k$ and $B=\Dirsum_{k\geq 0}I_k$ are finitely generated, we show the function $k \mapsto_0(I_k/J_k)$ is of quasi-polynomial type, say given by the polynomials $P_0,..., P_{g-1}$. If $J_k = J^k$ for all $k$ then we show that all the $P_i$ have the same degree and the same leading coefficient. As one of the applications it is shown that $\lim_{k\to \infty}\length(Γ_\mm(S/I^k))/k^n \in \mathbb{Q}.$ We also study analogous statements in the local case.

math.AC↗
arXiv2006

On fiber cones of ${\mathfrak m}$-primary ideals

Two formulas for the multiplicity of the fiber cone $F(I)=\oplus_{n=0}^{\infty} I^n/\m I^n$ of an $\m$-primary ideal of a $d$-dimensional Cohen-Macaulay local ring $(R,\m)$ are derived in terms of the mixed multiplicity $e_{d-1}(\m | I),$ 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) $\ell(I^2/JI)=1$ or (ii) $\ell(I\m/J\m)=1.$

math.AC↗
arXiv2003

Hilbert coefficients and depth of fiber cones

Criteria are given in terms of certain Hilbert coefficients for the fiber cone F(I) of an m-primary ideal I in a Cohen-Macaulay local ring (R,m) so that it is Cohen-Macaulay or has depth at least dim(R)-1. A version of Huneke's fundamental lemma is proved for fiber cones. S. Goto's results concerning Cohen-Macaulay fiber cones of ideals with minimal multiplicity are obtained as consequences.

math.AC↗
arXiv2003

Hilbert coefficients and depths of form rings

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 e_1. As an application, we derive bounds on the higher Hilbert coefficient e_i in terms of e_0.

math.AC↗
arXiv2002

Fiber cones of ideals with almost minimal multiplicity

Fiber cones of 0-dimensional ideals with almost minimal multiplicity in Cohen-Macaulay local rings are studied. Ratliff-Rush closure of filtration of ideals with respect to another ideal is introduced. This is used to find a bound on the reduction number with respect to an ideal. Rossi's bound on reduction number in terms of Hilbert coefficients is obtained as a consequence. Sufficient conditions are provided for the fiber cone of 0-dimensional ideals to have almost maximal depth. Hilbert series of such fiber cones are also computed.

math.AC↗
arXiv2002

Cohen-Macaulay fiber cones

Cohen Macaulay property of fiber cones of ideals is characterized in terms of its Hilbert series. Hilbert series of fiber cones of ideals with minimal mixed multiplicity is calculated. It is proved that the fiber cone of an m-primary ideal I with minimal mixed multiplicity is Cohen-Macaulay if and only if its reduction number is atmost one. Hilbert series of fiber cones of ideals generated by quadratic sequences in standard graded rings is computed by deforming it to a face ring of a simplicial complex. Applications are given to defining ideals of monomial projective space curves lying on the quadric xw-yz=0, straightening-closed ideals in graded algebras with straightening law and Huckaba-Huneke ideals of analytic spread 1 and 2.

math.AC↗
arXiv2002

Local cohomology modules of bigraded Rees algebras

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.

math.AC↗