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Tony J. Puthenpurakal

Publications and source records attributed to Tony J. Puthenpurakal.

At least 55 records · Page 3Linked to original sources
arXiv2024

First Coefficient ideals and $R_1$ property of Rees algebras

Let $(A,\mathfrak{m})$ be an excellent normal local ring of dimension $d \geq 2$ with infinite residue field. Let $I$ be an $\mathfrak{m}$-primary ideal. Then the following assertions are equivalent: (i) The extended Rees algebra $A[It, t^{-1}]$ is $R_1$. (ii) The Rees algebra $A[It]$ is $R_1$. (iii) $Proj(A[It])$ is $R_1$. (iv) $(I^n)^* = (I^n)_1$ for all $n \geq 1$. Here $(I^n)^*$ is the integral closure of $I^n$ and $(I^n)_1$ is the first coefficient ideal of $I^n$.

math.AC↗
arXiv2024

Derived functors and Hilbert polynomials over hypersurface rings

Let $(A,\mathfrak{m})$ be a hypersurface local ring of dimension $d \geq 1$ and let $I$ be an $\mathfrak{m}$-primary ideal. We show that there is a non-negative integer $r_I$ (depending only on $I$) such that if $M$ is any non-free maximal Cohen-Macaulay $A$-module the function $n \rightarrow \ell(Tor^A_1(M, A/I^{n+1}))$ (which is of polynomial type) has degree $r_I$. Analogous results hold for Hilbert polynomials associated to Ext-functors. Surprisingly a key ingredient is the classification of thick subcategories of the stable category of MCM $A$-modules (obtained by Takahashi).

math.AC↗
arXiv2024

On some Rings of differentiable type

Let $K$ be a field of characteristic 0 and $S=K[x_1,\ldots,x_m]/I$ be an affine domain. Consider $R=S_P$ where $P\in Spec(S)$ such that $R$ is regular. In this paper we construct a field $F$ which is contained in $R$ such that (1) The residue field of $R$ is a finite extension of $F$. (2) $D_F(R)$, the ring of $F$-linear differential operators on $R$ is left and right Noetherian with finite global dimension. (3) The Bernstein class of $D_F(R)$ is closed under localization at one element of $R$. We also prove a similar result for $R^h$, the Henselization of $R$. As an application we prove that $\frac{D_F(R)}{D_F(R)P}\cong E(κ(P))$ where $E(κ(P))$ is the injective hull of the residue field of $R$.

math.AC↗
arXiv2024

Dimension filtration of the bounded Derived category of a Noetherian ring

Let $A$ be a Noetherian ring of dimension $d$ and let $\mathcal{D}^b(A)$ be the bounded derived category of $A$. Let $\mathcal{D}_i^b(A)$ denote the thick subcategory of $\mathcal{D}^b(A)$ consisting of complexes $\mathbf{X}_\bullet$ with $\dim H^n(\mathbf{X}_\bullet) \leq i$ for all $n$. Set $\mathcal{D}_{-1}^b(A) = 0$. Consider the Verdier quotients $\mathcal{C}_i(A) = \mathcal{D}_i^b(A)/\mathcal{D}_{i-1}^b(A)$. We show for $i = 0, \ldots, d$, $\mathcal{C}_i(A)$ is a Krull-Remak-Schmidt triangulated category with a bounded $t$-structure. We identify its heart. We also prove that if $A$ is regular then $\mathcal{C}_i(A)$ has AR-triangles. We also prove that $$ \mathcal{C}_i(A) \cong \bigoplus_{\stackrel{P}{\dim A/P = i}} \mathcal{D}_0^b(A_P). $$

math.AC↗
arXiv2024

Categorical Krull-Remak-Schmidt for triangulated categories

Let $R$ be a commutative ring If $\mathcal{C}_1$ and $\mathcal{C}_2$ are $R$-linear triangulated categories then we can give an obvious triangulated structure on $\mathcal{C} = \mathcal{C}_1 \oplus \mathcal{C}_2$ where $Hom_\mathcal{C}(U, V) = 0$ if $U \in \mathcal{C}_i$ and $V \in \mathcal{C}_j$ with $i \neq j$. We say a $R$-linear triangulated category $\mathcal{C}$ is disconnected if $\mathcal{C} = \mathcal{C}_1 \oplus \mathcal{C}_2$ where $\mathcal{C}_i$ are non-zero triangulated subcategories of $\mathcal{C}$. Let $\mathcal{C}_i$ and $\mathcal{D}_j$ be connected triangulated $R$ categories with $i \in Γ$ and $j \in Λ$. Suppose there is an equivalence of triangulated $R$-categories \[ Φ\colon \bigoplus_{i \in Γ}\mathcal{C}_i \xrightarrow{\cong} \bigoplus_{j \in Λ}\mathcal{D}_j \] Then we show that there is a bijective function $π\colon Γ\rightarrow Λ$ such that we have an equivalence $\mathcal{C}_i \cong \mathcal{D}_{π(i)} $ for all $i \in Γ$. We give several examples of connected triangulated categories and also of triangulated subcategories which decompose into utmost finitely many components.

math.AC↗
arXiv2024

A solution to Itoh's conjecture for integral closure filtration

Let $(A,\mathfrak{m})$ be an analytically unramified Cohen-Macaulay local ring of dimension $d \geq 3$ and let $\mathfrak{a}$ be an $\mathfrak{m}$-primary ideal in $A$. If $I$ is an ideal in $A$ then let $I^*$ be the integral closure of $I$ in $A$. Let $G_{\mathfrak{a}}(A)^* = \bigoplus_{n\geq 0 }(\mathfrak{a}^n)^*/(\mathfrak{a}^{n+1})^*$ be the associated graded ring of the integral closure filtration of $\mathfrak{a}$. Itoh conjectured in 1992 that if third Hilbert coefficient of $G_{\mathfrak{a}}(A)^*$ , i.e., $e_3^{\mathfrak{a}^*}(A) = 0$ and $A$ is Gorenstein then $G_{\mathfrak{a}}(A)^*$ is Cohen-Macaulay. In this paper we prove Itoh's conjecture (more generally for analytically unramified Cohen-Macaulay local rings).

math.AC↗
arXiv2024

Integral closure and local cohomology

Let $A$ be a Noetherian ring and let $I$ be an ideal in $A$. Let $\mathcal{F} = \{ J_n \}_{n \geq 0}$ be a multiplicative filtration of ideals in $A$ such that $\mathcal{R}(\mathcal{F}) = \bigoplus_{n \geq 0} J_n$ is a finitely generated $A$-algebra. Let $\mathcal{R} = A[It]$ and assume $I^n \subseteq J_n$ for all $n \geq 1$. We show the following two assertions are equivalent: (1) For all $i \geq 0$ we have $H^i_{\mathcal{R}_+}(\mathcal{R}(\mathcal{F}))_n = 0$ for all $n \gg 0$. (2) $J_n \subseteq \overline{I^n}$ for all $n \geq 1$. Here $\overline{I^n}$ is the integral closure of $I^n$.

math.AC↗
arXiv2024

$2$-Periodic complexes over regular local rings

Let $(A,\mathfrak{m})$ be a regular local ring of dimension $d \geq 1$. Let $\mathcal{D}^2_{fg}(A)$ denote the derived category of $2$-periodic complexes with finitely generated cohomology modules. Let $\mathcal{K}^2(\proj A) $ denote the homotopy category of $2$-periodic complexes of finitely generated free $A$-modules. We show the natural map $\mathcal{K}^2(\ proj \ A) \longrightarrow \mathcal{D}^2(A)$ is an equivalence of categories. When $A$ is complete we show that $\mathcal{K}^2_f(\ proj \ A)$ ($2$-periodic complexes with finite length cohomology) is Krull-Schmidt with Auslander-Reiten (AR) triangles. We also compute the AR-quiver of $\mathcal{K}^2_f(\ proj \ A)$ when $\ dim \ A = 1$.

math.AC↗
arXiv2024

The Cohen-Macaulay property of invariant rings over ring of integers of a global field

Let $A$ be the ring of integers of global field $K$. Let $G \subseteq GL_2(A)$ be a finite group. Let $G$ act linearly on $R = A[X,Y]$ (fixing $A$). Let $R^G$ be the ring of invariants. In the equi-characteristic case we prove $R^G$ is Cohen-Macaulay. In mixed characteristic case we prove that if for all primes $p$ dividing $|G|$ the Sylow $p$-subgroup of $G$ has exponent $p$ then $R^G$ is Cohen-Macaulay. We prove a similar case if for all primes $p$ dividing $|G|$ the prime $p$ is un-ramified in $K$.

math.AC↗
arXiv2024

A convenient category to study asymptotic primes and related questions

Let $A$ be a Noetherian ring and let $\mathcal{R} = \bigoplus_{n \geq 0}\mathcal{R}_n$ be a standard graded ring with $\mathcal{R}_0 = A$. We define a category $\mathfrak{A}(\mathcal{R})$ of graded $\mathcal{R}$-modules (not necessarily finitely generated) with the following properties: if $X = \bigoplus_{n \in \mathbb{Z}} X_n \in \mathfrak{A}(\mathcal{R}) $ then (1) $X_i$ is finitely generated $A$-module for all $i \in \mathbb{Z}$ and $X_i = 0$ for $i \ll 0$. (2) There exists $n_0$ such that $\text{Ass}_A X_n = \text{Ass}_A X_{n_0}$ for all $n \geq n_0$. (3) If $X_n$ has finite length as an $A$-module for all $n$ then there exists $P_X(z) \in \mathbb{Q}[z]$ such that $P_X(n) = \ell_A(X_n)$ for all $n \gg 0$. (4) If $F$ is a coherent functor on the category of finitely generated $A$-modules then $F(X) = \bigoplus_{n \in \mathbb{Z}} F(X_n) \in \mathfrak{A}(\mathcal{R})$. (5) For an ideal $J$ in $A$, there exists $c_J^X$ such that $\text{grade}(J, X_n) = \text{grade}(J, X_{c_J^X})$ for all $n \geq c_J^X$. We give a unified proof of several results in theory of associate primes and related areas.

math.AC↗
arXiv2023

Derived functors and Hilbert polynomials over regular local rings

Let $(A,\mathfrak{m})$ be a regular local ring of dimension $d \geq 1$, $I$ an $\mathfrak{m}$-primary ideal. Let $N$ be a non-zero finitely generated $A$-module. Consider the functions \[ t^I(N, n) = \sum_{i = 0}^{ d}\ell(\text{Tor}^A_i(N, A/I^n)) \ \text{and}\ e^I(N, n) = \sum_{i = 0}^{ d}\ell(\text{Ext}_A^i(N, A/I^n)) \] of polynomial type and let their degrees be $t^I(N) $ and $e^I(N)$. We prove that $t^I(N) = e^I(N) = \max\{ \dim N, d -1 \}$.

math.AC↗
arXiv2023

Bockstein cohomology of Maximal Cohen-Macaulay modules over Gorenstein isolated singularities

Let $(A,\mathfrak{m})$ be an excellent equi-charateristic Gorenstein isolated singularity of dimension $d \geq 2$. Assume the residue field of $A$ is perfect. Let $I$ be any $\mathfrak{m}$-primary ideal. Let $G_I(A) = \bigoplus_{n \geq 0}I^n/I^{n+1}$ be the associated graded ring of $A$ with respect to $I$ and let $\mathcal{R}_I(A) = \bigoplus_{n \in \mathbb{Z}}I^n$ be the extended Rees algebra of $A$ with respect to $I$. Let $M$ be a finitely generated $A$-module. Let $G_I(M) = \bigoplus_{n \geq 0}I^nM/I^{n+1}M$ be the associated graded ring of $M$ with respect to $I$ (considered as a $G_I(A)$-module). Let $BH^i(G_I(M))$ be the $i^{th}$-Bockstein cohomology of $G_I(M)$ with respect to $\mathcal{R}_I(A)_+$-torsion functor. We show there exists $a \geq 1$ depending only on $A$ such that if $I$ is any $\mathfrak{m}$-primary ideal with $I \subseteq \mathfrak{m}^a$ and $G_I(A) $ generalized Cohen-Macaulay then the Bockstein cohomology $BH^i(G_I(M))$ has finite length for $i = 0, \ldots, d-1$ for any maximal Cohen-Macaulay $A$-module $M$.

math.AC↗
arXiv2023

Quasi-pure resolutions and some lower bounds of Hilbert coefficients of Cohen-Macaulay modules

Let $(A,\mathfrak{m})$ be a Gorenstein local ring and let $M$ be a finitely generated Cohen Macaulay $A$ module. Let $G(A)=\bigoplus_{n\geq 0}\mathfrak{m}^n/\mathfrak{m}^{n+1}$ be the associated graded ring of $A$ and $G(M)=\bigoplus_{n\geq 0}\mathfrak{m}^nM/\mathfrak{m}^{n+1}M$ be the associated graded module of $M$. If $A$ is regular and if $G(M)$ has a quasi-pure resolution then we show that $G(M)$ is Cohen-Macaulay. If $G(A)$ is Cohen-Macaulay and if $M$ has finite projective dimension then we give lower bounds on $e_0(M)$ and $e_1(M)$. Finally let $A = Q/(f_1, \ldots, f_c)$ be a strict complete intersection with $\text{ord}(f_i) = s$ for all $i$. Let $M$ be an Cohen-Macaulay module with $\text{cx}_A(M) = r < c$. We give lower bounds on $e_0(M)$ and $e_1(M)$.

math.AC↗
arXiv2023

On certain DG-algebra resolutions

In this paper we give several classes of Non-Gorenstein local rings $A$ which satisfy the property that $\text{Ext}^i_A(M, A) = 0$ for $i \gg 0$ then $\text{projdim}_A M$ is finite. We also show that if $\text{injdim}_A M = \infty$ then over such rings the bass-numbers of $M$ (with respect to $\mathfrak{m}$) are unbounded. When $A$ is a hypersurface ring we give an alternate proof of a result due to Takahashi regarding thick subcategories of the stable category of maximal Cohen-Macaulay $A$-modules. This result of Takahashi implies some results due to Avramov, Buchweitez, Huneke and Wiegand. The technique used to prove our results is that the minimal resolution of the relevant rings have an appropriate DG-algebra structure (philosophically this technique is due to Nasseh, Ono, and Yoshino).

math.AC↗
arXiv2023

Gorenstein rings via homological dimensions, and symmetry in vanishing of Ext and Tate cohomology

The aim of this article is to consider the spectral sequences induced by tensor-hom adjunction, and provide a number of new results. Let $R$ be a commutative Noetherian local ring of dimension $d$. In the 1st part, it is proved that $R$ is Gorenstein if and only if it admits a nonzero CM (Cohen-Macaulay) module $M$ of finite Gorenstein dimension $g$ such that ${\rm type}(M) \le μ( {\rm Ext}_R^g(M,R) )$ (e.g., ${\rm type}(M)=1$). This considerably strengthens a result of Takahashi. Moreover, we show that if there is a nonzero $R$-module $M$ of depth $\ge d - 1$ such that the injective dimensions of $M$, ${\rm Hom}_R(M,M)$ and ${\rm Ext}_R^1(M,M)$ are finite, then $M$ has finite projective dimension and $R$ is Gorenstein. In the 2nd part, we assume that $R$ is CM with a canonical module $ω$. For CM $R$-modules $M$ and $N$, we show that the vanishing of one of the following implies the same for others: ${\rm Ext}_R^{\gg 0}(M,N^{+})$, ${\rm Ext}_R^{\gg 0}(N,M^{+})$ and ${\rm Tor}_{\gg 0}^R(M,N)$, where $M^{+}$ denotes ${\rm Ext}_R^{d-\dim(M)}(M,ω)$. This strengthens a result of Huneke and Jorgensen. Furthermore, we prove a similar result for Tate cohomologies under the additional condition that $R$ is Gorenstein.

math.AC↗
arXiv2023

Lengths of modules over short Artin local rings

Let $(A,\mathfrak{m})$ be a short Artin local ring (i.e., $\mathfrak{m}^3 = 0$ and $\mathfrak{m}^2 \neq 0$). Assume $A$ is not a hypersurface ring. We show there exists $c_A \geq 2$ such that if $M$ is any finitely generated module with bounded betti-numbers then $c_A $ divides $\ell(M)$, the length of $M$. If $A$ is not a complete intersection then there exists $b_A \geq 2$ such that if $M$ is any module with $curv(M) < \ curv(k)$ then $b_A$ divides $\ell(Ω^i_A(M))$ for all $i \geq 1$ (here $Ω^i_A(M)$ denotes the $i^{th}$-syzygy of $M$).

math.AC↗
arXiv2023

Koszul homology of $F$-finite module and applications

Let $k$ be an infinite field of characteristic $p > 0$ and let $R = k[Y_1,\ldots, Y_d]$ (or $R = k[[Y_1,\ldots, Y_d]]$). Let $F \colon \text{Mod}(R) \rightarrow \text{Mod}(R)$ be the Frobenius functor and let $\mathcal{M}$ be a $F_R$-finite module (in the sense of Lyubeznik \cite{Lyu-2}). We show that if $r \geq 1$ then the Koszul homology modules $H_i(Y_1,\ldots, Y_r; \mathcal{M})$ are $F_{\overline{R}}$-finite modules where $\overline{R} = R/(Y_1,\ldots, Y_r)$ for $i = 0, \ldots, r$. As an application if $A$ is a regular ring containing a field of characteristic $p > 0$ and $S = A[X_1,\ldots, X_m]$ is standard graded and $I$ is an arbitrary graded ideal in $S$ then we give a comprehensive study of graded components of local cohomology modules $H^i_I(S)$. This extends in positive characteristic results we proved in \cite{P}. We study $H^i_I(S) $ when $A$ is local and prove that if $S/I$ is equidimensional and $Proj(S/I)$ is Cohen-Macaulay then $H^i_I(S)_n = 0$ for all $n \geq 0$ and for all $i > \ height \ I$. If $B$ is a equicharacteristic local Noetherian ring with infinite residue field and with a surjective map $π\colon T \rightarrow B$ where $(T,\mathfrak{n})$ is regular local then we show that the Koszul cohomology modules $H^j(\mathfrak{n}, H^{\dim T - i}_{\ker π}(T))$ depend only on $A, i, j$ and not on $T$ and $π$.

math.AC↗