arXiv subjects

Samarendra Sahoo

Publications and source records attributed to Samarendra Sahoo.

7 recordsLinked to original sources

arXiv2026

Bounds on Hilbert coefficients of Cohen-Macaulay modules having finite projective dimension

Let $(A,\mathfrak{m})$ be a Gorenstein local ring with $G(A)$ Cohen-Macaulay, and let $M$ be a Cohen-Macaulay $A$-module of finite projective dimension. In \cite{Quasipure}, the authors proved that $e_1(M)\geq \binom{c+1}{2}$, where $c=\operatorname{reg}G(A)$ and $e_i(M)$ is the $i$th Hilbert coefficient of $M$. We first show that this bound remains valid when $A$ is Cohen-Macaulay. We then study upper bounds for $e_2(M)$ when $e_1(M)=\binom{c+1}{2}+i$ for $i=1,2$, and investigate the consequences of equality. In particular, we obtain depth properties and explicit descriptions of the $h$-polynomial of $G(M)$. Finally, we extend these results to strict complete intersection rings without assuming that $M$ has finite projective dimension.

math.AC↗

arXiv2026

Integrally closed ideals with $e_{2}(I)=e_{1}(I)-e_{0}(I)+\lambda(A/I)$

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d.$ We introduce and study the notion of the generalized type of $A$ with respect to an $\mathfrak{m}$-primary ideal $I$ denoted by $\operatorname{type}_I(A).$ Let $e_i(I)$ denote $i$th Hilbert coefficients of $A$ w.r.t. $I$. Assuming $I$ is integrally closed and $e_{2}(I)=e_{1}(I)-e_{0}(I)+\lambda(A/I) \neq 0$, we establish a sharp lower bound for $\operatorname{type}_I(A)$ in terms of the multiplicity and certain lengths associated to $I.$ We further show that when this lower bound is attained, the associated graded ring $G(I)$, is Cohen Macaulay. In the case of Buchsbaum local rings of dimension $d$ and depth at least $d-1$, we obtain an optimal lower bound for $e_{2}(\mathfrak{m})$ using the technique of $S_{2}$-fication. Additionally, for an integrally closed $\mathfrak{m}$-primary ideal $I,$ we also study the second extremal case $e_{2}(I)=e_{1}(I)-e_{0}(I)+\lambda(A/I)+1$ and its consequences on $G(I).$ We also investigate bounds on $e_3(I)$ and for $d=3,$ we study the consequences when these bounds are attained for.

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 $\omega_A$ is a canonical $A$-module. Set $$r(I,\omega_A) = \bigcup_{n \geq 0} (I^{n+1} \omega_A : I^{n} \omega_A) \subseteq A .$$ We show that the ideal $r(I,-)$ is $\omega_A$ 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 $A\bigoplus \omega_A$ 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 $r(I^n,\omega_A)/\widetilde{I^n}$ for all $n\geq 1.$

math.AC↗

arXiv2025

Coherent functors, powers of ideals, and asymptotic stability

Let $R$ be a Noetherian ring, $I_1,\ldots,I_r$ be ideals of $R$, and $N\subseteq M$ be finitely generated $R$-modules. Let $S = \bigoplus_{\underline{n} \in \mathbb{N}^r} S_{\underline{n}}$ be a Noetherian standard $\mathbb{N}^r$-graded ring with $S_{\underline{0}} = R$, and $\mathcal{M} $ be a finitely generated $\mathbb{Z}^r$-graded $S$-module. For $ \underline{n} = (n_1,\dots,n_r) \in \mathbb{N}^r$, set $G_{\underline{n}} := \mathcal{M}_{\underline{n}}$ or $G_{\underline{n}} := M/{\bf I}^{\underline{n}} N$, where ${\bf I}^{\underline{n}} = I_1^{n_1} \cdots I_r^{n_r}$. Suppose $F$ is a coherent functor on the category of finitely generated $R$-modules. We prove that the set $\rm{Ass}_R \big(F(G_{\underline{n}}) \big)$ of associate primes and $\rm{grade}\big(J, F(G_{\underline{n}})\big)$ stabilize for all $\underline{n} \gg 0$, where $J$ is a non-zero ideal of $R$. Furthermore, if the length $\lambda_R(F(G_{\underline{n}}))$ is finite for all $\underline{n} \gg 0$, then there exists a polynomial $P$ in $r$ variables over $\mathbb{Q}$ such that $\lambda_R(F(G_{\underline{n}})) = P(\underline{n})$ for all $\underline{n}\gg 0$. When $R$ is a local ring, and $G_{\underline{n}} = M/{\bf I}^{\underline{n}} N$, we give a sharp upper bound of the total degree of $P$. As applications, when $R$ is a local ring, we show that for each fixed $i \geq 0$, the $i$th Betti number $\beta_i^R(F(G_{\underline{n}}))$ and Bass number $\mu^i_R(F(G_{\underline{n}}))$ are given by polynomials in $\underline{n}$ for all $\underline{n} \gg 0$. Thus, in particular, the projective dimension $\rm{pd}_R(F(G_{\underline{n}}))$ (resp., injective dimension $\rm{id}_R(F(G_{\underline{n}}))$) is constant for all $\underline{n}\gg 0$.

math.AC↗

arXiv2024

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

Let $(A,\mathfrak{m})$ 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. Set $M_i=\text{Syz}^A_{i}(M)$. Let $e^{\mathfrak{m}}_i(M)$ be the $i$-th Hilbert coefficient of $M$ with respect to $\mathfrak{m}$. We prove for all $i\gg0$, the function $i\mapsto e^{\mathfrak{m}}_j(M_i)$ is a quasi-polynomial type with period $2$ and degree $\text{cx}(M)-1$ for $j=0,1$, where $\text{cx}(M)$ is the complexity of $M.$ For $\text{cx}(M)=2,$ we prove $$\lim_{n\to \infty}\dfrac{e^{\mathfrak{m}}_1(M_{2n+j})}{n}\geq \lim_{n\to \infty}\dfrac{e^{\mathfrak{m}}_0(M_{2n+j})}{n}-\lim_{n\to \infty}\dfrac{\mu(M_{2n+j})}{n}$$ for $j=0,1$. When equality holds, we prove that the Castelnuovo-Mumford regularity of the associated graded ring of $M_i$ with respect to the maximal ideal $\mathfrak{m}$ is bounded for all $i\geq 0$.

math.AC↗

arXiv2024

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 $P_{\overline{I}} :\,n\mapsto \lambda(\overline{I^n}/I^n)$ 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↗

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↗