A note on the subadditivity problem for maximal shifts in free resolutions
We present some partial results regarding subadditivity of maximal shifts in finite graded free resolutions.
math.AC↗
arXiv subjects
Hema Srinivasan
Publications and source records attributed to Hema Srinivasan.
Bounds for the Multiplicity of Gorenstein algebras
We prove upper bounds for the Hilbert-Samuel multiplicity of standard graded Gorenstein algebras. The main tool that we use is Boij-Söderberg theory to obtain a decomposition of the Betti table of a Gorenstein algebra as the sum of rational multiples of symmetrized pure tables. Our bound agrees with the one in the quasi-pure case obtained by Srinivasan [J. Algebra, vol.~208, no.~2, (1998)].
math.AC↗
Periodic Occurance of Complete Intersection Monomial Curves
We study the complete intersection property of monomial curves in the family $Γ_{å+ \jj} = {(t^{a_0 + j}, t^{a_1+j},..., t^{a_n + j}) ~ | ~ j \geq 0, ~ a_0 < a_1 <...< a_n}$. We prove that if $Γ_{å+\jj}$ is a complete intersection for $j \gg0$, then $Γ_{å+\jj+\underline{a_n}}$ is a complete intersection for $j \gg 0$. This proves a conjecture of Herzog and Srinivasan on eventual periodicity of Betti numbers of semigroup rings under translations for complete intersections. We also show that if $Γ_{å+\jj}$ is a complete intersection for $j \gg 0$, then $Γ_å$ is a complete intersection. We also characterize the complete intersection property of this family when $n = 3$.
math.AC↗
Gorenstein Hilbert Coefficients
We prove upper and lower bounds for all the coefficients in the Hilbert Polynomial of a graded Gorenstein algebra $S=R/I$ with a quasi-pure resolution over $R$. The bounds are in terms of the minimal and the maximal shifts in the resolution of $R$ . These bounds are analogous to the bounds for the multiplicity found in \cite{S} and are stronger than the bounds for the Cohen Macaulay algebras found in \cite{HZ}.
math.AC↗
Minimal Graded Free Resolutions for Monomial Curves Defined by Arithmetic Sequences
Let $\mm=(m_0,...,m_n)$ be an arithmetic sequence, i.e., a sequence of integers $m_0<...<m_n$ with no common factor that minimally generate the numerical semigroup $\sum_{i=0}^{n}m_i\N$ and such that $m_i-m_{i-1}=m_{i+1}-m_i$ for all $i\in\{1,...,n-1\}$. The homogeneous coordinate ring $Γ_\mm$ of the affine monomial curve parametrically defined by $X_0=t^{m_0},...,X_n=t^{m_n}$ is a graded $R$-module where $R$ is the polynomial ring $k[X_0,...,X_n]$ with the grading obtained by setting $°{X_i}:=m_i$. In this paper, we construct an explicit minimal graded free resolution for $Γ_\mm$ and show that its Betti numbers depend only on the value of $m_0$ modulo $n$. As a consequence, we prove a conjecture of Herzog and Srinivasan on the eventual periodicity of the Betti numbers of semigroup rings under translation for the monomial curves defined by an arithmetic sequence.
math.AC↗
Minimal free resolutions for certain affine monomial curve
Given an arbitrary field k and an arithmetic sequence of positive integers m_0<...<m_n, we consider the affine monomial curve parameterized by X_0=t^{m_0},...,X_n=t^{m_n}. In this paper, we conjecture that the Betti numbers of its coordinate ring are completely determined by n and the value of m_0 modulo n. We first show that the defining ideal of the monomial curve can be written as a sum of two determinantal ideals. Using this fact, we describe the minimal free resolution of the coordinate ring in the following three cases: when m_0 is 1 modulo n (determinantal), when m_0 is n modulo n (almost determinantal), and when m_0 is 2 modulo n and n=4 (Gorenstein of codimension 4).
math.AC↗
On Unimodality of Hilbert Functions of Gorenstein Artin Algebras of Embedding Dimension Four
We prove that the Hilbert functions of codimension four graded Gorenstein Artin algebras R/I are unimodal provided I has a minimal generator in degree less than five. It is an open question whether all Gorenstein h-vectors in codimension four are unimodal. In this paper, we prove that Hilbert functions of all artinian codimension four Gorenstein algebras starting with (1,4,10, 20, h_4,...), where h_4\leq 34 are unimodal. Combining this with the previously known results, we obtain that all Gorenstein h-vectors (1, h_1, h_2,h_3, h_4, ....) are unimodal if h_4\leq 3.
math.AC↗
Finite Generation of Algebras Associated to Powers of Ideals
We study generalized symbolic powers and form ideals of powers of ideals and compare their growth with the growth of ordinary powers, and we discuss the question of when the graded rings attached to symbolic powers or to form ideals of powers are finitely generated.
math.AC↗
Multiplicity of Codimension Three Almost Complete Intersections
We establish the upper bound in the multiplicity conjecture of Herzog, Huneke and Srinivasan for the codimension three almost complete intersections. We also give some partial results in the case where I is the aci linked to a complete intersection in one step.
math.AC↗
Duality and Tameness
We prove a duality theorem for certain graded algebras and show by various examples different kinds of failure of tameness of local cohomology.
math.AC↗
Factorizations of birational extensions of local rings
We give a proof of local strong factorization of a birational extension of regular local rings (of equicharacteristic zero) along a valuation of rank 1 and maximal rational rank. This gives an alternate proof to the geometric proof of this Theorem by Karu. Our proof uses only linear algebra, and is in the spirit of Christensen's proof of this result in dimension 3.
math.AC↗
Betti Numbers and Degree Bounds for Some Linked Zero-Schemes
In their paper on multiplicity bounds (1998), Herzog and Srinivasan study the relationship between the graded Betti numbers of a homogeneous ideal I in a polynomial ring R and the degree of I. For certain classes of ideals, they prove a bound on the degree in terms of the largest and smallest Betti numbers, generalizing results of Huneke and Miller (1985). The bound is conjectured to hold in general; we study this using linkage. If R/I is Cohen-Macaulay, we may reduce to the case where I defines a zero-dimensional subscheme Y. If Y is residual to a zero-scheme Z of a certain type (low degree or points in special position), then we show that the conjecture is true for I_Y.
math.AC↗
Artinian Gorenstein algebras of embedding dimension four: Components of PGor(H) for H=(1,4,7,..., 1)
We first determine all height four Gorenstein sequences beginning H=(1,4,7,...), and we show that their first differences satisfy $ΔH_{\le j/2}$ is an O-sequence. We then study the family PGor(H) parametrizing all graded Artinian Gorenstein [AG] quotients A=R/I of the polynomial ring R=K[w,x,y,z] having a Hilbert function H as above. We give a structure theorem for such AG quotients with $I_2\cong < wx,wy,wz>$. For most H this subfamily forms an irreducible component of PGor(H), and for a slightly more restrictive set, PGor(H) has several irreducible components. M. Boij and others had already shown that PGor(T) is reducible for certain Gorenstein sequences T in codimensions at least four.
math.AC↗
Asymptotic behaviour of the length of local cohomology
Let k be a field of characteristic 0, R = k[x_1, ..., x_d] be a polynomial ring, and m its maximal homogeneous ideal. Let I be a homogeneous ideal in R. In this paper we investigate asymptotic behaviour of the quotient between the length of local cohomology group H^0_m(R/I^n) and n^d. We show that this quantity always has a limit as n goes to infinity. We also give an example for which the limit is irrational; in particular, this proves that the length of H^0_m(R/I^n) is not asymptotically a polynomial in n.
math.AC↗