arXiv subjects

Tony J. Puthenpurakal

Publications and source records attributed to Tony J. Puthenpurakal.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotic behavior of invariants of syzygies of maximal Cohen-Macaulay modules

$(A,\mathfrak{m})$ be a complete intersection ring of codimension $c\geq 2$ and dimension $d\geq 1$. Let $M$ be a finitely generated maximal Cohen-Macaulay $A$-module and $M_i=\text{Syz}^A_{i}(M)$ be the $i$-th syzygy of $M$. Let $e^{I}_i(M)$ be the $i$-th Hilbert coefficient of $M$ with respect to an $\mathfrak{m}$-primary ideal $I$. In \cite{Quasipolynomial}, it is proved that the function $i\mapsto e^I_1(M)$ is of quasi-polynomial type with period $2$ and degree $\leq \text{cx}_A(M)-1$, where $\text{cx}_A(M)$ is the complexity of $M.$ We prove that for $I=\mathfrak{m}$, the associated polynomials have the same degree equal to $\text{cx}_A(M)-1$. For $\text{cx}_A(M)=2$ and $j=0,1$, we prove $$\lim_{i\to \infty}\dfrac{e^{\mathfrak{m}}_1(M_{2i+j})}{i}\geq \lim_{i\to \infty}\dfrac{e^{\mathfrak{m}}_0(M_{2i+j})}{i}-\lim_{i\to \infty}\dfrac{μ(M_{2i+j})}{i}.$$ When equality holds, we prove that the set $\{\text{reg }G_{\mathfrak{m}}(M_i)\,|\, i\geq 0\}$ is bounded, where reg $G_{\mathfrak{m}}(M_i)$ is the Castelnuovo-Mumford regularity of $G_{\mathfrak{m}}(M_i).$

math.AC↗

arXiv2026

A lower bound on $\text{depth} \ \mathbf{A/I^n}$

Let $(A,\mathfrak{m})$ be a Noetherian local ring and let $I\subseteq \mathfrak{m}$ be an ideal in $A$. Let $\ell(I)$ be the analytic spread of $I$. Let $G_I(A) = \bigoplus_{n \geq 0}I^n/I^{n+1}$ be the associated graded ring of $I$ and let its depth be $t$. We prove $\text{depth} \ A/I^n \geq t - \ell(I)$ for all $n\geq 1$.

math.AC↗

arXiv2026

Derived functors and Hilbert polynomials over dominant Cohen-Macaulay rings

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$, residue field $k$ and let $I$ be an $\mathfrak{m}$-primary ideal. Assume $k$ is infinite. Let $N$ be a perfect $A$-module of dimension $t \geq 1$. Let $M$ be a MCM $A$-module. The function $n \rightarrow \ell(\text{Tor}^A_j(M, N/I^{n+1}N))$ is of polynomial type and let $r_{I, N}^j(M)$ be its degree. In general we have $r_{I, N}^j(M) \leq r_{I,N}^2(\text{Syz}^d_A(k))$ for $j \geq d + 2$. If $A$ is also dominant in the sense of Takahashi and $M$ is non-free then we show $\limsup_{j \rightarrow \infty} r_{I, N}^j(M) = r_{I,N}^2(\text{Syz}^d_A(k))$. We prove an analogous result when $V$ is a Cohen-Macaulay module of finite injective dimension and the polynomial type function $n \rightarrow \ell(\text{Ext}_A^j(M, V/I^{n+1}V))$

math.AC↗

arXiv2026

Annhilators of local cohomology modules over modular invariant rings and Dickson polynomials

Let $\mathbb{F}_q$ be a finite field with $q = p^s$ elements. Let $V$ be a $d$ dimensional vector space over $\mathbb{F}_q$ and let $G$ be a subgroup of $GL(V)$. Let $R = \mathbb{F}_q[V] = \text{Sym}_{\mathbb{F}_q}(V^*)$ and let $G$ act naturally on $R$. Set $S = R^G$. Let $\mathbf{d}_{d,0}, \mathbf{d}_{d, 1}, \ldots, \mathbf{d}_{d, d-1} \in S$ be the Dickson polynomials with $°\mathbf{d}_{d,i} = q^d - q^i$. Let $I$ be a homogeneous ideal of $S$ and let $H^i_I(S)$ be the $i^{th}$-local cohomology module of $S$ with respect to $I$. Let $J_i = \sqrt{\text{ann} H^i_I(S)}$. Assume $J_i \neq 0$ and $\dim S/J_i = d - g$. Then we show that $\mathbf{d}_{d,0}, \ldots, \mathbf{d}_{d, d - g + 1} \in J_i$. We give several applications of our results. An application is a considerably simpler proof of Landweber-Stong conjecture.

math.AC↗

arXiv2026

On the Stable category of maximal Cohen-Macaulay modules over Gorenstein rings-II

Let $(A,\mathfrak{m}), (B,\mathfrak{n}) $ be Gorenstein local rings and let $\underline{CM}(A)$ be its stable category of finitely generated maximal Cohen-Macaulay $A$-modules. Suppose we have an equivalence $Φ\colon\underline{CM}(A) \rightarrow \underline{CM}(A)$ as triangulated categories. We show (1) If $A, B$ are not hypersurfaces then $dim A = dim B$. (2) If $M$ is a maximal \CM \ $A$-module then $$\text{curv}_A(M) = \text{curv}_B (Φ(M)),$$ here $\text{curv}_A(M) = \limsup_n \sqrt[n]{\ell(\text{Tor}^A_n(M,k))}$. (3) $A$ satisfies Serre's condition $R_i$ if and only if $B$ satisfies $R_i$. (4) $A$ is a complete intersection on the punctured spectrum of $A$ if and only if $B$ is a complete intersection on the punctured spectrum of $B$. We also show that $Φ$ imposes constraints of residue field of $B$ in terms of residue field of $A$ and vice-versa. Finally if $I$ is an ideal in $A$ such that the extended Rees algebra $\mathcal{R}(I) = A[It, t^{-1}]$ is Gorenstein then we construct a triangulated functor $Ψ\colon \underline{CM}^\mathbb{Z}(\mathcal{R}(I)) \rightarrow \underline{CM}(A)$ where $\underline{CM}^\mathbb{Z}(\mathcal{R}(I))$ is the stable category of all graded maximal Cohen-Macaulay $\mathcal{R}(I)$-modules. We show that $Ψ$ induces an equivalence $\underline{CM}^\mathbb{Z}(\mathcal{R}(I))/\ker Ψ\rightarrow \underline{CM}(A)$. We give some applications of this map.

math.AC↗

arXiv2026

Bounds on multiplicity of MCM modules having non-extremal growth of betti-numbers

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ and residue field $k$. Let $M$ be a maximal Cohen-Macaulay $A$-module. Let $e(M)$ be the multiplicity of $M$ and let $μ(M)$ denote the number of its minimal generators. (1) Assume $A$ is not a complete intersection. If $\text{curv}(M) < \text{curv}(k)$ then we prove that under mild conditions, $e(M) \geq μ(M)(1 + \text{curv}(k))$. (2) Assume $A$ is a complete intersection. If $\text{cx}(M) < \text{cx}(k)$ then we prove that $e(M) \geq 2μ(M)$. In both cases we give examples which shows our results are sharp.

math.AC↗

arXiv2026

On generalization of two results of Foxby

Let $(A,\mathfrak{m})$ be a Noetherian local ring of dimension $d$ and let $M$ be a finitely generated $A$-module. Assume $M$ has rank $r > 0$. We show that if $M$ is NOT Cohen-Macaulay then $μ_d(\mathfrak{m}, M) > r$. If further $A$ is unmixed and $μ_n(\mathfrak{m}, M) \leq 1$ for some $n \geq d$ then we prove $\text{injdim} \ M < \infty$ and $A$ is Cohen-Macaulay.

math.AC↗

arXiv2026

Derived functors and Hilbert polynomials over Gorenstein rings

Let $(A,\mathfrak{m},k)$ be a Gorenstein ring of dimension $d\ge 1$, $N$ a perfect module of dimension $t\ge 1$ and $I$ an ideal of definition of $N$. For a non-free maximal Cohen-Macaulay (=MCM) $A$-module $M$ and an integer $i\ge 1$, it is well known that the functions $n \mapsto \ell(Tor_i^A(M,N/I^{n+1}N))$ and $n \mapsto \ell(Ext^i_A(M,N/I^{n+1}N))$ are of polynomial types of degrees $r_i^{I,N}(M)$ and $s_{I,N}^i(M)$, respectively. We prove that $r_i^{I,N}(M)\le t-1$ and $s^i_{I,N}(M)\le t-1$ and when $I$ is the maximal ideal $\mathfrak{m}$, both the inequalities become equalities. We also show that $r_i^{I,N}(M)\le r_1^{I,N}(Ω^dk)$, $s^i_{I,N}(M)\le s^1_{I,N}(Ω^dk)$ and $r_i^{I,N}(Ω^dk)=r_1^{I,N}(Ω^dk)=s^1_{I,N}(Ω^dk)=s^i_{I,N}(Ω^dk)$. \end

math.AC↗

arXiv2026

A Generalization of a Theorem of Nakajima-Landweber-Stong

The main result of this paper is a generalization of a theorem of Nakajima-Landweber-Stong to the modular invariant rings of transvection groups over Dedekind domains. More precisely, let $A$ be a Dedekind domain and $K$ be its field of fractions. Assume that $A$ contains a finite field $\mathbb{F}_q$ with $q=p^r$ elements for a prime $p$. Let $n\geq 2$ and consider a finite subgroup $G$ of $\mathrm{GL}(A^n)$ such that every non-identity element of $G$ inside $\mathrm{GL}(K^n)$ is a transvection. Consider the ring $A[X_1,X_2,\dots, X_n]$ and let $G$ act linearly on the ring (fixing $A$). Then $(A[X_1,X_2,\dots, X_n])^{G}$ is regular.

math.AC↗

arXiv2026

On unmixed and equi-dimensional associated graded rings

Let $(A,\mathfrak{m})$ be an analytically un-ramified Noetherian local ring of dimension $d \geq 1$, $I$ a regular $\mathfrak{m}$-primary ideal of $A$ and let $\overline{I}$ be integral closure ideal of $I$. If $A$ is of characteristic $p > 0$ then let $I^*$ denote the tight closure of $I$. Let $G_I(A)=\bigoplus_{n\geq 0}I^n/I^{n+1}$ be the associated graded ring of $A$ with respect to $I$. Assume $G_I(A)$ is unmixed and equi-dimensional. We show that either the function $P_{\overline{I}} :\,n\mapsto λ(\overline{I^n}/I^n)$ is a polynomial type of degree $d-1$ or $\overline{I^n}=I^n$ for all $n\geq 1.$ We prove an analogus result for the tight closure filtration if $A$ is of characteristic $p > 0$. When $A$ is generalized Cohen-Macaulay and $I$ is generated by standard system of parameters we give bounds for the first Hilbert coefficients of the integral closure filtration of $I$ and the tight closure filtration of $I$.

math.AC↗

arXiv2026

On Bass numbers of graded components of local cohomology modules supported on $\mathfrak{C}$-monomial ideals in mixed characteristic

Let $A$ be a Dedekind domain of characteristic zero such that for each height one prime ideal $\mathfrak{p}$ in $A$, the local ring $A_{\mathfrak{p}}$ has mixed characteristic with finite residue field. Suppose that $R=A[X_1,\ldots,X_n]$ is a standard $\mathbb{N}^n$-graded polynomial ring over $A$, i.e., $\operatorname{deg} A=\underline{0}\in \mathbb{N}^n$ and $\operatorname{deg}(X_j)=e_j\in \mathbb{N}^n$. Let $I$ be a $\mathfrak{C}$-monomial ideal of $R$ and let $M:= H^i_I(R)=\bigoplus_{\underline{u}\in \mathbb{Z}^n}M_{\underline{u}}$. Recently, the second author and S. Roy [2025, J. Algebra 681, 1-21] proved that for a fixed $\underline{u}\in\mathbb{Z}^n$, the Bass numbers $μ_i(\mathfrak{p},M_{\underline{u}})$ are finite for each prime ideal $\mathfrak{p}$ in $A$ and for every $i\geq 0$. Let for a subset of $U$ of $\mathcal{S}=\{1, \ldots, n\}$, define a block to be the set $\displaystyle\mathcal{B}(U)=\{\underline{u} \in \mathbb{Z}^n \mid u_i \geq 0 \mbox{ if } i \in U \mbox{ and } u_i \leq -1 \mbox{ if } i \notin U \}$. Note that $\bigcup_{U\subseteq \mathcal{S}}\mathcal{B}(U)=\mathbb{Z}^n$. In this article, the main result we establish is that for a fixed prime ideal $\mathfrak{p}$ in $A$ and $i\geq 0$, the set of Bass numbers $\{μ_i(\mathfrak{p},M_{\underline{u}})\mid \underline{u}\in \mathbb{Z}^n\}$ is constant on $\mathcal{B}(U)$ for each subset $U$ of $\{1, \ldots, n\}$. Our idea is to prove this by carrying out a comprehensive study of the structure theorem for the graded components of $M$ when $A$ is a complete DVR of mixed characteristic with finite residue field.

math.AC↗

arXiv2026

Asymptotic behaviour of bigraded components of local cohomology modules

Let $C$ be a commutative Noetherian ring containing a field $K$ of characteristic zero. Let $R=C[X_1, \ldots, X_n, Y_1, \ldots, Y_m]$ be a polynomial ring over $C$ with $\mathrm{bideg}~ c=(0,0)$ for all $c \in C$, $\mathrm{bideg}~ X_i=(1,0)$ and $\mathrm{bideg}~ Y_j=(0,1)$ for $i=1, \ldots, n$ and $j=1, \ldots, m$. Let $I$ be a bihomogeneous ideal in $R$. In this article, we study asymptotic behaviour of bigraded pieces of the local cohomology module $H^i_I(R)$. Moreover, under the extra assumption that $C$ is regular, we investigate the asymptotic stability of invariants associated to its bigraded components. Consequently, we obtain certain properties of components of the bigraded local cohomology module $H^i_I(R)$, where $C=K$ is a field and $I$ is a binomial edge ideal.

math.AC↗

arXiv2026

The Regular property of Invariant Rings over Regular Domains

The main result of this paper is a generalization of the theorem of Chevalley-Shephard-Todd to the rings of invariants of pseudo-reflection groups over regular domains. More precisely, let $A$ be a regular domain and let $K$ be its field of fractions. Let $G\subseteq GL_n(A)$ be a finite group. Let $G$ act linearly on $A[X_1,X_2,\dots, X_n]$ (fixing $A$). Assume that $|G|$ is invertible in $A$. We prove that $G\subseteq GL_n(K)$ is generated by pseudo-reflections if and only if $(A[X_1,X_2,\dots, X_n])^G$ is regular.

math.AC↗

arXiv2025

Quasi-Hilbert rings and Ratliff-Rush filtrations

Let $A$ be a non Gorenstein Cohen Macaulay ring of dimension $d\geq 1$, $I$ an ideal of $A$, and suppose $ω_A$ is a canonical $A$-module. Set $$r(I,ω_A) = \bigcup_{n \geq 0} (I^{n+1} ω_A : I^{n} ω_A) \subseteq A .$$ We show that the ideal $r(I,-)$ is $ω_A$ invariant. Motivated by this property, we introduce a new class of rings, which we call quasi Hilbert rings. We provide several examples of quasi Hilbert rings and discuss a number of their applications. Let $A$ be a local ring with maximal ideal $\mathfrak{m}$. We prove that $A$ is quasi Hilbert iff $\widehat{A}$ is quasi Hilbert, where $\widehat{A}$ is the completion of $A$ w.r.t. $\mathfrak{m}.$ If $d\geq 2$ and $x\in \mathfrak{m}\setminus \mathfrak{m}^2$ is an $A\bigoplus ω_A$ superficial element, we prove that if $A$ is quasi Hilbert, then so is $A/(x)$. Writing $\widetilde{I}$ for the Ratliff Rush closure of an ideal $I$, we also provide sufficient conditions ensuring the vanishing of $r(I^n,ω_A)/\widetilde{I^n}$ for all $n\geq 1.$

math.AC↗

arXiv2025

Top dickson class and annihilators of cohomology over invariant rings

Let $\mathbb{F}_q$ denote the finite field with $q = p^r$ elements. Let $V$ be a finite dimensional vector space of dimension $d$ over $\mathbb{F}_q$ and let $G \subseteq GL(V)$ be a group. Let $R = \mathbb{F}_q[V] = \text{Sym}(V^*)$ and let $S = R^G$. Let $\mathbf{d}_{d,0} $ be the top Dickson class, i.e., $\mathbf{d}_{d,0} = \prod_{0\neq v \in V^*}v$. Surprisingly (a power of) $\mathbf{d}_{d,0}$ annihilates many cohomological modules. (a) Let $H^i(G, R)$ be the $i^{th}$-group cohomology of $R$ considered as a $S$-module. Set $J_i = \text{ann}_S \ H^i(G, R)$. We show that $\mathbf{d}_{d,0} \in \sqrt{J_i}$ for all $i \geq 1$. (b) We also show that $\mathbf{d}_{d,0} \in \sqrt{ \text{ann}_S \ H^j_{S_+}(S)}$ for all $ 0 \leq j \leq d - 1$ (here $H^j_{S_+}(S)$ is the $j^{th}$ local cohomology of $S$ with respect to $S_+$). As an application we get that there exists a fixed power of $\mathbf{d}_{d,0} $ which works as a cohomological annihilator.

math.AC↗

arXiv2025

Fixed points and grade of Hilbert polynomial of invariant rings

Let $k$ be a field and let $V$ be a $k$-vector space of dimension $d$. Let $G \subseteq GL(V)$ be a finite group. Let $r = \dim_k (V^*)^G$. Assume $r \geq 1$. Let $R = k[V]^G$ be the ring of invariants of $G$. Let $H_R(n) = a_{d-1}(n)n^{d-1} + \cdots a_1(n)n + a_0(n)$ be the Hilbert polynomial of $R$ where $a_i(-)$ are periodic functions. We show $a_{d-1}(-), \ldots, a_{d-r}(-)$ are constants. In the terminology of Erhart, $\text{grade} H_R \leq d - r-1$. We also give an example which shows that our result is sharp.

math.AC↗

arXiv2025

Obstructions to curvature of modules over Cohen-Macaulay rings

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring with residue field $k$. If $M$ is a finitely generated $A$-module then set $\text{curv}(M) = \limsup_n\sqrt[n]{β_n^A(M)}$. We show that under mild hypotheses the existence of a single module $M$ with $1 \leq \text{curv}(M) < \text{curv}(k)$ imposes obstructions to both $\text{curv}(k)$ and $\text{curv}(M)$. Similarly we show that the condition $\text{Tor}^A_n(M, N) = 0$ for $n \gg 0$ imposes constraints on both $\text{curv}(M)$ and $\text{curv}(N)$.

math.AC↗

arXiv2025

André-Quillen homology of Rees algebras and extended Rees algebras

Let $(A,\mathfrak{m})$ be an excellent local complete intersection ring and let $I = (a_1, \ldots, a_r)$ be an ideal of positive height. Let $\mathcal{R}(I) = A[It]$ be the Rees algebra of $I$. Consider the map $ψ\colon S = A[X_1, \ldots, X_r] \rightarrow \mathcal{R}(I)$ which maps $X_i \mapsto a_it$ for all $i$. Let $J = \ker ψ$ and let $H_*(J)$ be the Koszul homology of $J$. We prove that the following assertions are equivalent: (i) $\text{Proj} \ \mathcal{R}(I)$ is a complete intersection. (ii) (a) $D_3(\mathcal{R}(I)|A, \mathcal{R}(I))_n = 0$ for $n \gg 0$ and, (ii) (b) For $P \in \text{Proj} \ \mathcal{R}(I)$ we have $H_1(J)_P$ is a free $\mathcal{R}(I)_P$-module. Here $D_3(\mathcal{R}(I)|A, \mathcal{R}(I))$ is the third André-Quillen homology of $\mathcal{R}(I)$ with respect to $A \rightarrow \mathcal{R}(I)$. We prove an analogous result for the extended Rees algebra $\widehat{\mathcal{R}} = A[It, t^{-1}]$. When $A$ is a Cohen-Macaulay domain (not necessarily a complete intersection) we compute that rank of $H_1(J)$ and hence compute its free locus.

math.AC↗