arXiv subjects
Publications and source records attributed to Tony J. Puthenpurakal.
Let $(A,\mathfrak{m})$ be an abstract complete intersection and let $P$ be a prime ideal of $A$. In [1] Avramov proved that $A_P$ is an abstract complete intersection. In this paper we give an elementary proof of this result.
Let $(A, \m, k)$ be a Gorenstein local ring of dimension $ d\geq 1.$ Let $I$ be an ideal of $A$ with $\htt(I) \geq d-1.$ We prove that the numerical function \[ n \mapsto \ell(\ext_A^i(k, A/I^{n+1}))\] is given by a polynomial of degree $d-1 $ in the case when $ i \geq d+1 $ and $\curv(I^n) > 1$ for all $n \geq 1.$ We prove a similar result for the numerical function \[ n \mapsto \ell(\Tor_i^A(k, A/I^{n+1}))\] under the assumption that $A$ is a \CM ~ local ring. \noindent We note that there are many examples of ideals satisfying the condition $\curv(I^n) > 1,$ for all $ n \geq 1.$ We also consider more general functions $n \mapsto \ell(\Tor_i^A(M, A/I_n)$ for a filtration $\{I_n \}$ of ideals in $A.$ We prove similar results in the case when $M$ is a maximal \CM ~ $A$-module and $\{I_n=\overline{I^n} \}$ is the integral closure filtration, $I$ an $\m$-primary ideal in $A.$
Let $A = K[X_1,\ldots, X_d]$ and let $I$, $J$ be monomial ideals in $A$. Let $I_n(J) = (I^n \colon J^\infty)$ be the $n^{th}$ symbolic power of $I$ \wrt \ $J$. It is easy to see that the function $f^I_J(n) = e_0(I_n(J)/I^n)$ is of quasi-polynomial type, say of period $g$ and degree $c$. For $n \gg 0$ say \[ f^I_J(n) = a_c(n)n^c + a_{c-1}(n)n^{c-1} + \text{lower terms}, \] where for $i = 0, \ldots, c$, $a_i \colon \mathbb{N} \rt \mathbb{Z}$ are periodic functions of period $g$ and $a_c \neq 0$. In an earlier paper we (together with Herzog and Verma) proved that $\dim I_n(J)/I^n$ is constant for $n \gg 0$ and $a_c(-)$ is a constant. In this paper we prove that if $I$ is generated by some elements of the same degree and height $I \geq 2$ then $a_{c-1}(-)$ is also a constant.
Set $ A := Q/({\bf z}) $, where $ Q $ is a polynomial ring over a field, and $ {\bf z} = z_1,\ldots,z_c $ is a homogeneous $ Q $-regular sequence. Let $ M $ and $ N $ be finitely generated graded $ A $-modules, and $ I $ be a homogeneous ideal of $ A $. We show that (1) $ \mathrm{reg}\left( \mathrm{Ext}_A^{i}(M, I^nN) \right) \le ρ_N(I) \cdot n - f \cdot \left\lfloor \frac{i}{2} \right\rfloor + b \mbox{ for all } i, n \ge 0 $, (2) $ \mathrm{reg}\left( \mathrm{Ext}_A^{i}(M,N/I^nN) \right) \le ρ_N(I) \cdot n - f \cdot \left\lfloor \frac{i}{2} \right\rfloor + b' \mbox{ for all } i, n \ge 0 $, where $ b $ and $ b' $ are some constants, $ f := \mathrm{min}\{ \mathrm{deg}(z_j) : 1 \le j \le c \} $, and $ ρ_N(I) $ is an invariant defined in terms of reduction ideals of $ I $ with respect to $ N $. There are explicit examples which show that these inequalities are sharp.
We investigate three cases regarding asymptotic associate primes. First, assume $ (A,\mathfrak{m}) $ is an excellent Cohen-Macaulay (CM) non-regular local ring, and $ M = \operatorname{Syz}^A_1(L) $ for some maximal CM $ A $-module $ L $ which is free on the punctured spectrum. Let $ I $ be a normal ideal. In this case, we examine when $ \mathfrak{m} \notin \operatorname{Ass}(M/I^nM) $ for all $ n \gg 0 $. We give sufficient evidence to show that this occurs rarely. Next, assume that $ (A,\mathfrak{m}) $ is excellent Gorenstein non-regular isolated singularity, and $ M $ is a CM $ A $-module with $\operatorname{projdim}_A(M) = \infty $ and $ \dim(M) = \dim(A) -1 $. Let $ I $ be a normal ideal with analytic spread $ l(I) < \dim(A) $. In this case, we investigate when $\mathfrak{m} \notin \operatorname{Ass} \operatorname{Tor}^A_1(M, A/I^n)$ for all $n \gg 0$. We give sufficient evidence to show that this also occurs rarely. Finally, suppose $ A $ is a local complete intersection ring. For finitely generated $ A $-modules $ M $ and $ N $, we show that if $ \operatorname{Tor}_i^A(M, N) \neq 0 $ for some $ i > \dim(A) $, then there exists a non-empty finite subset $ \mathcal{A} $ of $ \operatorname{Spec}(A) $ such that for every $ \mathfrak{p} \in \mathcal{A} $, at least one of the following holds true: (i) $ \mathfrak{p} \in \operatorname{Ass}\left( \operatorname{Tor}_{2i}^A(M, N) \right) $ for all $ i \gg 0 $; (ii) $ \mathfrak{p} \in \operatorname{Ass}\left( \operatorname{Tor}_{2i+1}^A(M, N) \right) $ for all $ i \gg 0 $. We also analyze the asymptotic behaviour of $\operatorname{Tor}^A_i(M, A/I^n)$ for $i,n \gg 0$ in the case when $I$ is principal or $I$ has a principal reduction generated by a regular element.
We give a short proof of a result due to Katz and West: Let $R$ be a Noetherian ring and $I_1,\ldots,I_t$ ideals of $R$. Let $M$ and $N$ be finitely generated $R$-modules and $N' \subseteq N$ a submodule. For every fixed $i \ge 0$, the sets $\mathrm{Ass}_R\left( \mathrm{Ext}_R^i(M, N/I_1^{n_1}\cdots I_t^{n_t} N') \right)$ and $\mathrm{Ass}_R\left( \mathrm{Tor}_i^R(M, N/I_1^{n_1}\cdots I_t^{n_t} N') \right)$ are independent of $(n_1,\ldots,n_t)$ for all sufficiently large $n_1,\ldots,n_t$.
In this paper we determine the possible Hilbert functions of a Cohen-Macaulay local ring of dimension $d$, multiplicity $e$ and first Hilbert coefficient $e_1$ in the case $e_1 = e + 2$.
Let $ R $ be a $ d $-dimensional Cohen-Macaulay (CM) local ring of minimal multiplicity. Set $ S := R/({\bf f}) $, where $ {\bf f} := f_1,\ldots,f_c $ is an $ R $-regular sequence. Suppose $ M $ and $ N $ are maximal CM $ S $-modules. It is shown that if $ \mathrm{Ext}_S^i(M,N) = 0 $ for some $ (d+c+1) $ consecutive values of $ i \geqslant 2 $, then $ \mathrm{Ext}_S^i(M,N) = 0 $ for all $ i \geqslant 1 $. Moreover, if this holds true, then either $ \mathrm{projdim}_R(M) $ or $ \mathrm{injdim}_R(N) $ is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, we give a number of necessary and sufficient conditions for a CM local ring of minimal multiplicity to be regular or Gorenstein. These conditions are based on vanishing of certain Exts or Tors involving homomorphic images of syzygy modules of the residue field.
Let $(A,\mathfrak{m})$ be an excellent normal domain of dimension two. We define an $\mathfrak{m}$-primary ideal $I$ to be a $p_g$-ideal if the Rees algebra $A[It]$ is a Cohen-Macaulay normal domain. When $A$ contains an algebraically closed field $k \cong A/\mathfrak{m}$ then Okuma, Watanabe and Yoshida proved that $A$ has $p_g$-ideals and furthermore product of two $p_g$-ideals is a $p_g$ ideal. In this article we show that if $A$ is an excellent normal domain of dimension two containing a field $k \cong A/\mathfrak{m}$ of characteristic zero then also $A$ has $p_g$-ideals. Furthermore product of two $p_g$-ideals is $p_g$.
Let $A$ be a regular ring containing a field $K$ of characteristic zero and let $R = A[X_1,\ldots, X_m]$. Consider $R$ as standard graded with $°A = 0$ and $°X_i = 1$ for all $i$. Let $G$ be a finite subgroup of $GL_m(A)$. Let $G$ act linearly on $R$ fixing $A$. Let $S = R^G$. In this paper we present a comprehensive study of graded components of local cohomology modules $H^i_I(S)$ where $I$ is an \emph{arbitrary} homogeneous ideal in $S$. We prove stronger results when $G \subseteq GL_m(K)$. Some of our results are new even in the case when $A$ is a field.
Let $(A,\mathfrak{m})$ be a Henselian Cohen-Macaulay local ring and let CM(A) be the category of maximal Cohen-Macaulay $A$-modules. We construct $T \colon CM(A)\times CM(A) \rightarrow mod(A)$, a subfunctor of $Ext^1_A(-, -)$ and use it to study properties of associated graded modules over $G(A) = \bigoplus_{n\geq 0} \mathfrak{m}^n/\mathfrak{m}^{n+1}$, the associated graded ring of $A$. As an application we give several examples of complete Cohen-Macaulay local rings $A$ with $G(A)$ Cohen-Macaulay and having distinct indecomposable maximal Cohen-Macaulay modules $M_n$ with $G(M_n)$ Cohen-Macaulay and the set $\{e(M_n)\}$ bounded (here $e(M)$ denotes multiplicity of $M$).
Let $(A,\mathfrak{m})$ be a local complete intersection ring and let $I$ be an ideal in $A$. Let $M, N$ be finitely generated $A$-modules. Then for $l = 0,1$, the values $depth \ Ext^{2i+l}_A(M, N/I^nN)$ become independent of $i, n$ for $i,n \gg 0$. We also show that if $\mathfrak{p}$ is a prime ideal in $A$ then the $j^{th}$ Bass numbers $μ_j\big(\mathfrak{p},\ Ext^{2i+l}_A(M,N/{I^nN})\big)$ has polynomial growth in $(n,i)$ with rational coefficients for all sufficiently large $(n,i)$.
Let $A$ be a regular domain containing a field $K$ of characteristic zero, $G$ be a finite subgroup of the group of automorphisms of $A$ and $B=A^G$ be the ring of invariants of $G$. Let $S= A[X_1,\ldots, X_m]$ and $R= B[X_1, \ldots, X_m]$ be standard graded with $\ deg \ A=0$, $\ deg \ B=0$ and $\ deg \ X_i=1$ for all $i$. Extend the action of $G$ on $A$ to $S$ by fixing $X_i$. Note $S^G=R$. Let $I$ be an arbitrary homogeneous ideal in $R$. The main goal of this paper is to establish a comparative study of graded components of local cohomology modules $H_I^i(R)$ that would be analogs to those proven in a previous paper of the first author for $H_J^i(S)$ where $J$ is an arbitrary homogeneous ideal in $S$.
Let $(A,\mathfrak{m})$ be an analytically unramified formally equidimensional Noetherian local ring with $\ depth \ A \geq 2$. Let $I$ be an $\mathfrak{m}$-primary ideal and set $I^*$ to be the integral closure of $I$. Set $G^*(I) = \bigoplus_{n\geq 0} (I^n)^*/(I^{n+1})^*$ be the associated graded ring of the integral closure filtration of $I$. We prove that $\ depth \ G^*(I^n) \geq 2$ for all $n \gg 0$. As an application we prove that if $A$ is also an excellent normal domain containing an algebraically closed field isomorphic to $A/\m$ then there exists $s_0$ such that for all $s \geq s_0$ and $J$ is an integrally closed ideal \emph{strictly} containing $(\mathfrak{m}^s)^*$ then we have a strict inequality $μ(J) < μ((\mathfrak{m}^s)^*)$ (here $μ(J)$ is the number of minimal generators of $J$).
In this paper, we introduce the notion of the strong Rees property (SRP) for $\mathfrak{m}$-primary ideals of a Noetherian local ring and prove that any power of the maximal ideal $\mathfrak{m}$ has its property if the associated graded ring $G$ of $\mathfrak{m}$ satisfies $\text{depth} \ G \ge 2$. As its application, we characterize two-dimensional excellent normal local domains so that $\mathfrak{m}$ is a $p_g$-ideal. Finally we ask what $\mathfrak{m}$-primary ideals have SRP and state a conjecture which characterizes the case when $\mathfrak{m}^n$ are the only ideals which have SRP.
Let $k$ be a field and $G \subseteq Gl_n(k)$ be a finite group with $|G|^{-1} \in k$. Let $G$ act linearly on $A = k[X_1, \ldots, X_n]$ and let $A^G$ be the ring of invariant's. Suppose there does not exist any non-trivial one-dimensional representation of $G$ over $k$. Then we show that if $Q$ is a $G$-invariant homogeneous ideal of $A$ such that $A/Q$ is a Gorenstein ring then $A^G/Q^G$ is also a Gorenstein ring.
Let $K$ be a field of characteristic zero and let $R=K[X_1, \ldots,X_n ]$, with standard grading. Let $\mathfrak{m}= (X_1, \ldots, X_n)$ and let $E$ be the $^*$injective hull of $R/\mathfrak{m}.$ Let $A_n(K)$ be the $n^{th}$ Weyl algebra over $K$. Let $I, J$ be homogeneous ideals in $R$. Fix $i,j \geq 0$ and set $M = H^i_I(R)$ and $N = H^j_J(R)$ considered as left $A_n(K)$-modules. We show the following two results for which no analogous result is known in charactersitc $p > 0$. \begin{enumerate} $H^l_\mathfrak{m}(\Tor^R_ν(M, N)) \cong E(n)^{a_{l,ν}}$ for some $a_{l,ν} \geq 0$. For all $ν\geq 0$; the finite dimensional vector space $\Tor^{A_n(K)}_ν( M^\sharp, N)$ is concentrated in degree $-n$ (here $M^\sharp$ is the standard right $A_n(K)$-module associated to $M$). \end{enumerate} We also conjecture that for all $i \geq 0$ the finite dimensional vector space $\Ext^i_{A_n(K)}(M, N)$ is concentrated in degree zero. We give a few examples which support this conjecture.
Let $(A,\mathfrak{m})$ be a commutative complete equicharacteristic Gorenstein isolated singularity of dimension $d $ with $k = A/\mathfrak{m}$ algebraically closed. Let $Γ(A)$ be the AR (Auslander-Reiten) quiver of $A$. Let $\mathcal{P}$ be a property of maximal Cohen-Macaulay $A$-modules. We show that some naturally defined properties $\mathcal{P}$ define a union of connected components of $Γ(A)$. So in this case if there is a maximal Cohen-Macaulay module satisfying $\mathcal{P}$ and if $A$ is not of finite representation type then there exists a family $\{ M_n \}_{n \geq 1}$ of maximal Cohen-Macaulay indecomposable modules satisfying $\mathcal{P}$ with multiplicity $e(M_n) > n$. Let $\underline{Γ(A)}$ be the stable quiver. We show that there are many symmetries in $\underline{Γ(A)}$. As an application we show that if $(A,\mathfrak{m})$ is a two dimensional Gorenstein isolated singularity with multiplicity $e(A) \geq 3$ then for all $n \geq 1$ there exists an indecomposable self-dual maximal Cohen-Macaulay $A$-module of rank $n$.