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Tony J. Puthenpurakal

Publications and source records attributed to Tony J. Puthenpurakal.

At least 109 records · Page 6Linked to original sources
arXiv2016

Asymptotic prime divisors over complete intersection rings

Let $A$ be a local complete intersection ring. Let $M,N$ be two finitely generated $A$-modules and $I$ an ideal of $A$. We prove that \[ \bigcup_{i\geqslant 0}\bigcup_{n \geqslant 0}\mathrm{Ass}_A\left(\mathrm{Ext}_A^i(M,N/I^n N)\right) \] is a finite set. Moreover, we prove that there exist $i_0,n_0\geqslant 0$ such that for all $i\geqslant i_0$ and $n \geqslant n_0$, we have \[ \mathrm{Ass}_A\left(\mathrm{Ext}_A^{2i}(M,N/I^nN)\right) = \mathrm{Ass}_A\left(\mathrm{Ext}_A^{2 i_0}(M,N/I^{n_0}N)\right), \] \[ \mathrm{Ass}_A\left(\mathrm{Ext}_A^{2i+1}(M,N/I^nN)\right) = \mathrm{Ass}_A\left(\mathrm{Ext}_A^{2 i_0 + 1}(M,N/I^{n_0}N)\right). \] We also prove the analogous results for complete intersection rings which arise in algebraic geometry. Further, we prove that the complexity $\mathrm{cx}_A(M,N/I^nN)$ is constant for all sufficiently large $n$.

math.AC↗
arXiv2015

On a generalization of a result of Peskine and Szpiro

Let $(R,\mathfrak{m})$ be a regular local ring containing a field $K$. Let $I$ be a Cohen-Macaulay ideal of height $g$. If $\text{char } K = p > 0$ then by a result of Peskine and Szpiro the local cohomology modules $H^i_I(R)$ vanish for $i > g$. This result is not true if $\text{char } K = 0$. However we prove that the Bass numbers of the local cohomology module $H^g_I(R)$ completely determine whether $H^i_I(R)$ vanish for $i > g$.

math.AC↗
arXiv2015

On the ring of differential operators of certain regular domains

Let $(A,\mathfrak{m})$ be a complete equicharacteristic Noetherian domain of dimension $d + 1 \geq 2$. Assume $k = A/\mathfrak{m}$ has characteristic zero and that $A$ is not a regular local ring. Let $Sing(A)$ the singular locus of $A$ be defined by an ideal $J$ in $A$. Note $J \neq 0$. Let $ f \in J$ with $f \neq 0$. Set $R = A_f$. Then $R$ is a regular domain of dimension $d$. We show $R$ contains naturally a field $\ell \cong k((X))$. Let $\mathfrak{g}$ be the set of $\ell$-linear derivations of $R$ and let $D(R)$ be the subring of $Hom_\ell(R,R)$ generated by $\mathfrak{g}$ and the multiplication operators defined by elements in the ring $R$. We show that $D(R)$, the ring of $\ell$-linear differential operators on $R$, is a left, right Noetherian ring of global dimension $d$. This enables us to prove Lyubeznik's conjecture on $R$ modulo a conjecture on roots of Bernstein-Sato polynomials over power series rings.

math.AC↗
arXiv2015

Invariants of Linkage of modules

Let $(A,\mathfrak{m})$ be a Gorenstein local ring and let $M, N$ be two Cohen-Macaulay \ $A$-modules with $M$ linked to $N$ via a Gorenstein ideal $\mathfrak{q}$. Let $L$ be another finitely generated $A$-module. We show that $Ext^i_A(L,M) = 0 $ for all $i \gg 0$ if and only if $Tor^A_i(L,N) = 0$ for all $i \gg 0$. If $D$ is Cohen-Macaulay then we show that $Ext^i_A(M, D) = 0 $ for all $i \gg 0$ if and only if $Ext^i_A(D^\dagger, N) = 0$ for all $i \gg 0$, where $D^\dagger = Ext^r_A(D,A)$ and $r = codim \ D$. As a consequence we get that $Ext^i_A(M, M) = 0 $ for all $i \gg 0$ if and only if $Ext^i_A(N, N) = 0$ for all $i \gg 0$. We also show that $End_A(M)/rad \ End_A(M) \cong (End_A(N)/rad \ End_A(N))^{op}$. We also give a negative answer to a question of Martsinkovsky and Strooker.

math.AC↗
arXiv2015

Asymptotic primes and the Chow group

In this paper we present an unexpected connection between the theory of asymptotic prime divisors and Chow groups. As an application we show that the Chow group $A_1(R)$ is a torsion group when $R$ is any graded ring such that we have an inclusion of graded rings $T \subseteq R \subseteq S$ where $S = k[X_1, X_2, Y]/(X_1^m + X_2^m + Y^n)$ where $k$ is algebraically field, $(m,n) = 1$, $(mn)^{-1} \in k$, $m, n \geq 2$. We consider $S$ graded with $deg \ X_1 = deg \ X_2 = n$ and $deg \ Y = m$. Also $T = k[X_1, X_2]$ where $deg \ X_1 = deg \ X_2 = n$. We also consider higher dimensional analogues of this result.

math.AC↗
arXiv2015

Relative Hilbert co-efficients

Let $(A,\m)$ be a \CM \ local ring of dimension $d$ and let $I \subseteq J$ be two $\m$-primary ideals with $I$ a reduction of $J$. For $i = 0,\ldots,d$ let $e_i^J(A)$ ($e_i^I(A)$) be the $i^{th}$ Hilbert coefficient of $J$ ($I$) respectively. We call the number $c_i(I,J) = e_i^J(A) - e_i^I(A)$ the $i^{th}$ relative Hilbert coefficient of $J$ \wrt \ $I$. If $G_I(A)$ is \CM \ then $c_i(I,J)$ satisfy various constraints. We also show that vanishing of some $c_i(I,J)$ has strong implications on $\depth G_{J^n}(A)$ for $n \gg 0$.

math.AC↗
arXiv2015

Characterizations of regular local rings via syzygy modules of the residue field

Let $R$ be a commutative Noetherian local ring with residue field $k$. We show that if a finite direct sum of syzygy modules of $k$ surjects onto `a semidualizing module' or `a non-zero maximal Cohen-Macaulay module of finite injective dimension', then $R$ is regular. We also prove that $R$ is regular if and only if some syzygy module of $k$ has a non-zero direct summand of finite injective dimension.

math.AC↗
arXiv2015

On the Monotonicity of Hilbert functions

In this paper we show that a large class of one-dimensional Cohen-Macaulay local rings $(A,\mathfrak{m})$ has the property that if $M$ is a maximal Cohen-Macaulay $A$-module then the Hilbert function of $M$ ( with respect to $\mathfrak{m}$) is non-decreasing. Examples include (1) Complete intersections $A = Q/(f,g)$ where $(Q,\mathfrak{n})$ is regular local of dimension three and $f \in \mathfrak{n}^2 \setminus \mathfrak{n}^3$. (2) One dimensional Cohen-Macaulay quotients of a two dimensional Cohen-Macaulay local ring with pseudo-rational singularity.

math.AC↗
arXiv2015

Growth of Hilbert coefficients of Syzygy modules

Let $(A,\mathfrak{m})$ be a complete intersection ring of dimension $d$ and let $I$ be an $\mathfrak{m}$-primary ideal. Let $M$ be a maximal \CM \ $A$-module. For $i = 0,1,\cdots,d$, let $e_i^I(M)$ denote the $i^{th}$ Hilbert -coefficient of $M$ with respect to $I$. We prove that for $i = 0, 1, 2$, the function $j \mapsto e_i^I(Syz_j^A(M))$ is of quasi-polynomial type with period $2$. Let $G_I(M)$ be the associated graded module of $M$ with respect to $I$. If $G_I(A)$ is Cohen-Macaulay and $\dim A \leq 2$ we also prove that the functions $j \mapsto depth \ G_I(Syz^A_{2j+i}(M))$ are eventually constant for $i = 0, 1$. Let $ξ_I(M) = \lim_{l \rightarrow \infty} depth \ G_{I^l}(M)$. Finally we prove that if $\dim A = 2$ and $G_I(A)$ is Cohen-Macaulay then the functions $j \mapsto ξ_I(Syz^A_{2j + i}(M))$ are eventually constant for $i = 0, 1$.

math.AC↗
arXiv2014

Associated primes of Local cohomology modules over Regular rings

Let $R$ be an excellent regular ring of dimension $d$ containing a field $K$ of characteristic zero. Let $I$ be an ideal in $R$. We show that $Ass \ H^{d-1}_I(R)$ is a finite set. As an application we show that if $I$ is an ideal of height $g$ with $height \ Q = g$ for all minimal primes of $I$ then for all but finitely many primes $P \supseteq I$ with $height \ P \geq g +2$, the topological space $Spec^\circ(R_P/IR_P)$ is connected.

math.AC↗
arXiv2014

On associated graded modules having a pure resolution

Let $A = K[[X_1,\cdots,X_n]]$ and let $\mathfrak{m} = (X_1,\cdots,X_n)$. Let $M$ be a Cohen-Macaulay $A$-module of codimension $p$. In this paper we give a necessary and sufficient condition for the associated graded module $G_{\mathfrak{m}}(M)$ to have a pure resolution over the polynomial ring $G_{\mathfrak{m}}(A) \cong K[X_1,\cdots,X_n]$.

math.AC↗
arXiv2014

On functors that detect $S_n$

Let $A$ be a Noetherian ring. For each $k$ where $0 \leq k \leq \dim A$ we construct left exact functors $D_k$ on $Mod(A)$. Let $D^i_k$ be the $i^{th}$-right derived functor of $D_k$. Let $M$ be a finitely generated $A$-module. Under mild conditions on $A$ and $M$ we prove that vanishing of some finitely many $D^i_k(M)$ is equivalent to $M$ satisfying $S_n$.

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↗
arXiv2014

Examples of non-commutative crepant resolutions of Cohen Macaulay normal domains

Let $A$ be a Cohen-Macaulay normal domain. A non commutative crepant resolution (NCCR) of $A$ is an $A$-algebra $Γ$ of the form $Γ= End_A(M)$, where $M$ is a reflexive $A$-module, $Γ$ is maximal Cohen-Macaulay as an $A$-module and $gldim(Γ)_P = \dim A_P $ for all primes $P$ of $A$. We give bountiful examples of equi-characteristic Cohen-Macaulay normal local domains and mixed characteristic Cohen-Macaulay normal local domains having NCCR. We also give plentiful examples of affine Cohen-Macaulay normal domains having NCCR.

math.AC↗
arXiv2014

Local cohomology modules of invariant rings

Let $K$ be a field and let $R$ be a regular domain containing $K$. Let $G$ be a finite subgroup of the group of automorphisms of $R$. We assume that $|G|$ is invertible in $K$. Let $R^G$ be the ring of invariants of $G$. Let $I$ be an ideal in $R^G$. Fix $i \geq 0$. If $R^G$ is Gorenstein then, \begin{enumerate} \item $injdim_{R^G} H^i_I(R^G) \leq \dim \ Supp \ H^i_I(R^G).$ \item $H^j_{\mathfrak{m}}(H^i_I(R^G))$ is injective, where $\mathfrak{m}$ is any maximal ideal of $R^G$. \item $μ_j(P, H^i_I(R^G)) = μ_j(P^\prime, H^i_{IR}(R))$ where $P^\prime$ is any prime in $R$ lying above $P$. \end{enumerate} We also prove that if $P$ is a prime ideal in $R^G$ with $R^G_P$ \textit{not Gorenstein} then either the bass numbers $μ_j(P, H^i_I(R^G)) $ is zero for all $j$ or there exists $c$ such that $μ_j(P, H^i_I(R^G)) = 0 $ for $j < c$ and $μ_j(P, H^i_I(R^G)) > 0$ for all $j \geq c$.

math.AC↗
arXiv2013

On the Lowey length of modules of finite projective dimension

Let $(A,\mathfrak{m})$ be a local Gorenstein local ring and let $M$ be an $A$ module of finite length and finite projective dimension. We prove that the Lowey length of $M$ is greater than or equal to order of $A$. This generalizes a result of Avramov, Buchweitz, Iyengar and Miller

math.AC↗
arXiv2013

On injective resolutions of local cohomology modules

Let $K$ be a field of characteristic zero and let $R = K[X_1,\ldots,X_n]$. Let $I$ be an ideal in $R$ and let $M = H^i_I(R)$ be the $i^{th}$-local cohomology module of $R$ with respect to $I$. Let $c = \ injdim \ M$. We prove that if $P$ is a prime ideal in $R$ with Bass number $μ_c(P,M) > 0$ then $P$ is a maximal ideal in $R$.

math.AC↗