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Tony J. Puthenpurakal

Publications and source records attributed to Tony J. Puthenpurakal.

At least 127 records · Page 7Linked to original sources
arXiv2013

de Rham cohomology of $H^1_{(f)}(R)$ where $V(f)$ is a smooth hypersurface in $\mathbb{P}^n$

Let $K$ be a field of characteristic zero, $R = K[X_1,\ldots,X_n]$. Let $A_n(K) = K $ be the $n^{th}$ Weyl algebra over $K$. We consider the case when $R$ and $A_n(K)$ is graded by giving $°X_i = ω_i $ and $°\partial_i = -ω_i$ for $i =1,\ldots,n$ (here $ω_i$ are positive integers). Set $ω= \sum_{k=1}^{n}ω_k$. Let $I$ be a graded ideal in $R$. By a result due to Lyubeznik the local cohomology modules $H^i_I(R)$ are holonomic $A_n(K)$-modules for each $i \geq 0$. In this article we compute the de Rham cohomology modules $H^j(\mathbb{\partial}; H^1_{(f)}(R))$ for $j \leq n-2$ when $V(f)$ is a smooth hypersurface in $\mathbb{P}^n$ (equivalently $A = R/(f)$ is an isolated singularity).

math.AC↗
arXiv2013

Bockstein cohomology of associated graded rings

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ and let $I$ be an $\mathfrak{m}$-primary ideal. Let $G$ be the associated graded ring of $A$ \wrt \ $I$ and let $\R = A[It,t^{-1}]$ be the extended Rees ring of $A$ with respect to $I$. Notice $t^{-1}$ is a non-zero divisor on $\R$ and $\R/t^{-1}\R = G$. So we have \textit{Bockstein operators} $β^i \colon H^i_{G_+}(G)(-1) \rt H^{i+1}_{G_+}(G)$ for $i \geq 0$. Since $β^{i+1}(+1)\circ β^i = 0$ we have \textit{Bockstein cohomology} modules $BH^i(G)$ for $i = 0,\ldots,d$. In this paper we show that certain natural conditions on $I$ implies vanishing of some Bockstein cohomology modules.

math.AC↗
arXiv2013

De Rahm Cohomology of Local Cohomology modules II

Let $K$ be an algebraically closed field of characteristic zero and let $R = K[X_1,\ldots,X_n]$. Let $I$ be an ideal in $R$. Let $A_n(K)$ be the $n^{th}$ Weyl algebra over $K$. By a result of Lyubeznik, the local cohomology modules $H^i_I(R)$ are holonomic $A_n(K)$-modules for each $i \geq 0$. In this paper we compute the Euler characteristic of De-Rahm cohomology of $H^{\htt P}_P(R)$ for certain classes of prime ideals $P$ in $R$.

math.AC↗
arXiv2013

A function on the the set of isomorphism classes in the stable category of maximal Cohen-Macaulay modules over a Gorenstein ring: with applications to Liason theory

Let $(A,\m)$ be a Gorenstein local ring of dimension $d \geq 1$. Let $\CMS(A)$ be the stable category of maximal \CM \ $A$-modules and let $\ICMS(A)$ denote the set of isomorphism classes in $\CMS(A)$. We define a function $ξ\colon \ICMS(A) \rt \ZZ$ which behaves well with respect to exact triangles in $\CMS(A)$. We then apply this to (Gorenstein) liason theory. We prove that if $\dim A \geq 2$ and $A$ is not regular then the even liason classes of $\m^n; n\geq 1$ is an infinite set. We also prove that if $A$ is an complete equi-characteristic simple singularity with $A/\m$ uncountable then for each $m \geq 1$ the set $\mathcal{C}_m = \{I \mid I \ \text{is a codim 2 CM-ideal with} \ e_0(A/I) \leq m \}$ is contained in finitely many even liason classes $L_1,\ldots,L_r$ (here $r$ may depend on $m$).

math.AC↗
arXiv2013

De Rahm cohomology of local cohomology modules

Let $K$ be a field of characteristic zero, $R = K[X_1,...,X_n]$ and let $I$ be an ideal in $R$. Let $A_n(K) = K $ be the $n^{th}$ Weyl algebra over $K$. By a result due to Lyubeznik the local cohomology modules $H^i_I(R)$ are holonomic $A_n(K)$-modules for each $i \geq 0$. In this article we compute the Koszul homology modules $H_*(\partial_1,...,\partial_n ; H^*_I(R))$ for certain classes of ideals.

math.AC↗
arXiv2013

De Rahm cohomology of local cohomology modules-The graded case

Let $K$ be a field of characteristic zero, $R = K[X_1,...,X_n]$. Let $A_n(K) = K $ be the $n^{th}$ Weyl algebra over $K$. We consider the case when $R$ and $A_n(K)$ is graded by giving $°X_i = ω_i $ and $°\partial_i = -ω_i$ for $i =1,...,n$ (here $ω_i$ are positive integers). Set $ω= \sum_{k=1}^{n}ω_k$. Let $I$ be a graded ideal in $R$. By a result due to Lyubeznik the local cohomology modules $H^i_I(R)$ are holonomic $A_n(K)$-modules for each $i \geq 0$. In this article we prove that the De Rahm cohomology modules $H^*(\bP ; H^*_I(R))$ is concentrated in degree $- ω$, i.e., $H^*(\bP ; H^*_I(R))_j = 0$ for $j \neq - ω$. As an application when $A = R/(f)$ is an isolated singularity we relate $H^{n-1}(\bP ; H^1_{(f)}(R)$ to $H^{n-1}(\partial(f); A)$, the $(n-1)^{th}$ Koszul cohomology of $A$ \wrt \ $ \partial_1(f),...,\partial_n(f)$.

math.AC↗
arXiv2013

On the intersection of Annihilator of the Valabrega-Valla module

Let $(A,\m)$ be a \CM \ local ring with an infinite residue field and let $I$ be an $\m$-primary ideal. Let $\bx = x_1,\ldots,x_r$ be a $A$-superficial sequence \wrt \ $I$. Set $$\Vc_I(\bx) = \bigoplus_{n\geq 1} \frac{I^{n+1}\cap (\bx)}{\bx I^n}. $$ A consequence of a theorem due to Valabrega and Valla is that $\Vc_I(\bx) = 0$ \ff \ the initial forms $x_1^*,\ldots,x_r^*$ is a $G_I(A)$ regular sequence. Furthermore this holds if and only if $\depth G_I(A) \geq r$. We show that if $\depth G_I(A) < r$ then \[ \af_r(I)= \bigcap_{\substack{\text{$\bx = x_1,\ldots,x_r$ is a} \\ \text{$A$-superficial sequence w.r.t $I$}}} \ann_A \Vc_I(\bx) \quad \ \text{is} \ \m\text{-primary}. \] Suprisingly we also prove that under the same hypotheses, \[ \bigcap_{n\geq 1} \af_r(I^n) \quad \ \text{is also} \ \m\text{-primary}. \]

math.AC↗
arXiv2013

Quasi-finite modules and asymptotic prime divisors

Let $A$ be a Noetherian ring, $J\subseteq A$ an ideal and $C$ a finitely generated $A$-module. In this note we would like to prove the following statement. Let $\{I_n\}_{n\geq 0}$ be a collection of ideals satisfying : (i) $I_n\supseteq J^n$, for all $n$, (ii) $J^s\cdot I_s \subseteq I_{r+s}$, for all $r,s\geq 0$ and (iii) $I_n\subseteq I_m$, whenever $m\leq n$. Then $\Ass_A(I_nC/J^nC)$ is independent of $n$, for $n$ sufficiently large. Note that the set of prime ideals $\cup_{n\geq 1} \Ass_A(I_nC/J^nC)$ is finite, so the issue at hand is the realization that the primes in $\Ass_A(I_nC/J^nC)$ \textit{do not} behave periodically, as one might have expected, say if $\bigoplus _{n\geq 0}I_n$ were a Noetherian $A$-algebra generated in degrees greater than one. We also give a multigraded version of our results.

math.AC↗
arXiv2013

Injective modules over some rings of Differential operators

Let $R$ be a regular domain containing a field $K$ of characteristic zero and let $D$ be the ring of $K$-linear differential operators on $R$. Let $E$ be an injective left $D$-module. We ask the question, when is $E$ injective as a $R$-module? We show that this is indeed the case when $R = K[X_1,...,X_n]$ or $R = K[[X_1,...,X_n]]$ or $R = \mathbb{C}\{z_1,...,z_n\}$. We also give an application of our result to local cohomology.

math.AC↗
arXiv2011

The Hilbert function of a maximal Cohen-Macaulay module Part II

Let $(A,\m)$ be a strict complete intersection of positive dimension and let $M$ be a maximal \CM \ $A$-module with bounded betti-numbers. We prove that the Hilbert function of $M$ is non-decreasing. We also prove an analogous statement for complete intersections of codimension two.

math.AC↗
arXiv2011

Analytic Deviation One Ideals and Test Modules

Let A be a Cohen-Macaulay local ring of dimension d and I an ideal in A. Let M be a finitely generated maximal Cohen-Macaulay A-module. Let I be a locally complete intersection ideal of analytic deviation one and reduction number at most one. We prove that the polynomial given by $length(Tor^{A}_{1}(M,A/I^{n+1}))$ either has degree d-1 or $F_I(M) $ is a free$F(I)-$$module.

math.AC↗
arXiv2008

Properties of Koszul homology modules

We investigate various module-theoretic properties of Koszul homology under mild conditions. These include their depth, $S_2$-property and their Bass numbers

math.AC↗
arXiv2008

The dual Hilbert-Samuel function of a Maximal Cohen-Macaulay module

Let $R$ be a Cohen-Macaulay local ring with a canonical module $ω_R$. Let $I$ be an $\m$-primary ideal of $R$ and $M$, a maximal Cohen-Macaulay $R$-module. We call the function $n\longmapsto \ell (\Hom_R(M,{ω_R}/{I^{n+1} ω_R}))$ the dual Hilbert-Samuel function of $M$ with respect to $I$. By a result of Theodorescu this function is a polynomial function. We study its first two normalized coefficients.

math.AC↗
arXiv2008

On the finite generation of a family of Ext modules

Let $Q$ be a Noetherian ring with finite Krull dimension and let $\mathbf{f}= f_1,... f_c$ be a regular sequence in $Q$. Set $A = Q/(\mathbf{f})$. Let $I$ be an ideal in $A$, and let $M$ be a finitely generated $A$-module with $\projdim_Q M$ finite. Set $\R = \bigoplus_{n\geq 0}I^n$, the Rees-Algebra of $I$. Let $N = \bigoplus_{j \geq 0}N_j$ be a finitely generated graded $\R$-module. We show that \[\bigoplus_{j\geq 0}\bigoplus_{i\geq 0} \Ext^{i}_{A}(M,N_j) \] is a finitely generated bi-graded module over $\Sc = \R[t_1,...,t_c]$. We give two applications of this result to local complete intersection rings.

math.AC↗
arXiv2008

Ratliff-Rush Filtration, regularity and depth of Higher Associated graded modules: Part II

Let $(A,\m)$ be a Noetherian local ring, let $M$ be a finitely generated \CM $A$-module of dimension $r \geq 2$ and let $I$ be an ideal of definition for $M$. Set $L^I(M) = \bigoplus_{n\geq 0}M/I^{n+1}M$. In part one of this paper we showed that $L^I(M)$ is a module over $\R$, the Rees algebra of $I$ and we gave many applications of $L^I(M)$ to study the associated graded module, $G_I(M)$. In this paper we give many further applications of our technique; most notable is a reformulation of a classical result due to Narita in terms of the Ratliff-Rush filtration. This reformulation can be extended to all dimensions $\geq 2$.

math.AC↗
arXiv2008

An elementary proof of Grothendieck's Non-vanishing Theorem

We give an elementary proof of Grothendieck's non-vanishing Theorem: For a finitely generated non-zero module $M$ over a Noetherian local ring $A$ with maximal ideal $\m$, the local cohomology module $H^{\dim M}_{\m}(M)$ is non-zero.

math.AC↗
arXiv2007

On the Upper bound of the Multiplicity Conjecture

Let $A = K[X_1,...,X_n]$ and let $I$ be a graded ideal in $A$. We show that the upper bound of Multiplicity conjecture of Herzog, Huneke and Srinivasan holds asymptotically (i.e., for $I^k$ and all $k \gg 0$) if $I$ belongs to any of the following large classes of ideals: \begin{enumerate}[\rm (1)] \item radical ideals. \item monomial ideals with generators in different degrees. \item zero-dimensional ideals with generators in different degrees. \end{enumerate} Surprisingly, our proof uses local techniques like analyticity, reductions, equimultiplicity and local results like Rees's theorem on multiplicities.

math.AC↗