arXiv subjects

Tony J. Puthenpurakal

Publications and source records attributed to Tony J. Puthenpurakal.

135 records · Page 8Linked to original sources

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↗

arXiv2007

Ratliff-Rush Filtrations associated with ideals and modules over a Noetherian ring

Let $R$ be a commutative Noetherian ring, $M$ a finitely generated $R$-module and $I$ a proper ideal of $R$. In this paper we introduce and analyze some properties of $r(I, M)=\bigcup_{k\geqslant 1} (I^{k+1}M: I^kM)$, {\it the Ratliff-Rush ideal associated with $I$ and $M$}. When $M= R$ (or more generally when $M$ is projective) then $r(I, M)= \widetilde{I}$, the usual Ratliff-Rush ideal associated with $I$. If $I$ is a regular ideal and $\ann M=0$ we show that $\{r(I^n,M) \}_{n\geqslant 0}$ is a stable $I$-filtration. If $M_{\p}$ is free for all ${\p}\in \spec R\setminus \mspec R,$ then under mild condition on $R$ we show that for a regular ideal $I$, $\ell(r(I,M)/{\widetilde I})$ is finite. Further $r(I,M)=\widetilde I $ if $A^*(I)\cap \mspec R =\emptyset $ (here $A^*(I)$ is the stable value of the sequence $\Ass (R/{I^n})$). Our generalization also helps to better understand the usual Ratliff-Rush filtration. When $I$ is a regular $\m$-primary ideal our techniques yield an easily computable bound for $k$ such that $\widetilde{I^n} = (I^{n+k} \colon I^k)$ for all $n \geqslant 1$. For any ideal $I$ we show that $\widetilde{I^nM}=I^nM+H^0_I(M)\quad\mbox{for all} n\gg 0.$ This yields that $\widetilde {\mathcal R}(I,M)=\bigoplus_{n\geqslant 0} \widetilde {I^nM}$ is Noetherian if and only if $\depth M>0$. Surprisingly if $\dim M=1$ then $\widetilde G_I(M)=\bigoplus_{n\geqslant 0} \widetilde{I^nM}/{\widetilde{I^{n+1}M}}$ is always a Noetherian and a Cohen-Macaulay $G_I(R)$-module. Application to Hilbert coefficients is also discussed.

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↗

arXiv2005

Hilbert-Samuel functions of modules over Cohen-Macaulay rings

For a finitely generated, non-free module $M$ over a CM local ring $(R,\fm,k)$, it is proved that for $n\gg 0$ the length of $\tor 1RM{R/\fm^{n+1}}$ is given by a polynomial of degree $\dim R-1$. The vanishing of $\tor iRM{N/\fm^{n+1}N}$ is studied, with a view towards answering the question: if there exists a finitely generated $R$-module $N$ with $\dim N\ge 1$ such that the projective dimension or the injective dimension of $N/\fm^{n+1}N$ is finite, then is $R$-regular? Upper bounds are provided for $n$ beyond which the question has an affirmative answer.

math.AC↗

arXiv2005

The Hilbert Function of a Maximal Cohen-Macaulay Module

We study Hilbert functions of maximal Cohen-Macaulay(=CM) modules over CM local rings. We show that if $A$ is a hypersurface ring with dimension $d > 0$ then the Hilbert function of $M$ \wrt $\m$ is non-decreasing. If $A = Q/(f)$ for some regular local ring $Q$, we determine a lower bound for $e_0(M)$ and $e_1(M)$. We analyze the case when equality holds and prove that in this case $G(M)$ is CM. Furthermore in this case we also determine the Hilbert function of $M$. When $A$ is Gorenstein then $M$ is the first syzygy of $S^A(M) = (\Syz^{A}_{1}(M^*))^*$. A relation between the second Hilbert coefficient of $M$, $A$ and $S^A(M)$ is found when $G(M)$ is \CM and $\depth G(A) \geq d-1$. We give bounds for the first Hilbert coefficients of the canonical module of a CM local ring and analyse when equality holds. We also give good bounds on Hilbert coefficients of $M$ when $M$ is maximal CM and $G(M)$ is CM.

math.AC↗

arXiv2004

Ratliff-Rush Filtration, Regularity and depth of Higher Associated graded modules: Part I

In this paper we introduce a new technique to study associated graded modules. Let $(A,\m)$ be a Noetherian local ring with $\depth A \geq 2$. Our techniques gives a necessary and sufficient condition for $\depth G_{\m^n}(A) \geq 2$ for all $n \gg 0$. Other applications are also included; most notable is an upper bound regarding the Ratliff-Rush filtration.

math.AC↗

arXiv2004

Invariance of a length associated to a reduction

Let $(A, \mathfrak{m})$ be a $d$-dimensional Cohen-Macaulay local ring with infinite residue field and let $J$ be a minimal reduction of $\mathfrak{m}$. We show that $λ(\mathfrak{m}^3/J\mathfrak{m}^2)$ is independent of $J$.

math.AC↗

arXiv2004

A short note on the non-negativity of partial Euler characteristics

Let $(A,\mathfrak{m})$ be a Noetherian local ring, $M$ a finite $A$-module and $x_1,...,x_n\in \m$ such that $λ(M/\x M)$ is finite. Serre proved that all partial Euler characteristics of $M$ with respect to $\x$ is non-negative. This fact is easy to show when $A$ contains a field. We give an elementary proof of Serre's result when $A$ does not contain a field.

math.AC↗

arXiv2004

An analogue of a theorem due to Levin and Vasconcelos

Let $(R,\m)$ be a Noetherian local ring. Consider the notion of homological dimension of a module, denoted H-dim, for H= Reg, CI, CI$_*$, G, G$^*$ or CM. We prove that, if for a finite $R$-module $M$ of positive depth, $\Hd_R({\m}^iM)$ is finite for some $i \geq \reg(M)$, then the ring $R$ has property H.

math.AC↗