arXiv subjects

Siddhartha Pramanik

Publications and source records attributed to Siddhartha Pramanik.

3 recordsLinked to original sources

arXiv2026

Asymptotic prime divisors and Vasconcelos invariant

Let $R$ be a Noetherian ring, $I$ an ideal of $R$, and $M$ a finitely generated $R$-module. In this article, we prove that $$\mathrm{Ass}_R(M/I^{n} M) = \mathrm{Ass}_R(0:_{M} I) \cup \mathrm{Ass}_R(I^{n-1} M/I^{n} M) \text{ for all } n \gg 0.$$ We then investigate the asymptotic behaviour of the (local) Vasconcelos invariant of $M/I^{n} M$ as a function of $n$, when $R$ is $\mathbb{N}$-graded, $I$ is homogeneous, and $M$ is $\mathbb{Z}$-graded. When $I$ is generated by elements of positive degree, we show that, for sufficiently large n, the (local) Vasconcelos invariant of $M/I^{n} M$ either coincides with that of the colon submodule $(0 :_{M} I)$, or is a polynomial in $n$ of degree one whose leading coefficient is one of the degrees of the generators of $I$. This dichotomy depends exclusively on two cases determined by $(0:_{M} I)$. Thus, we recover and considerably strengthen the main results of Fiorindo-Ghosh [Nagoya Math. J. 258 (2025), 296-310.], where asymptotic linearity was shown under the additional assumption that $(0:_{M} I)=0$.

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 $λ_R(F(G_{\underline{n}}))$ is finite for all $\underline{n} \gg 0$, then there exists a polynomial $P$ in $r$ variables over $\mathbb{Q}$ such that $λ_R(F(G_{\underline{n}})) = P(\underline{n})$ 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 $β_i^R(F(G_{\underline{n}}))$ and Bass number $μ^i_R(F(G_{\underline{n}}))$ 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↗

arXiv2025

Asymptotic v-numbers of graded (co)homology modules involving powers of an ideal

Let $R$ be a Noetherian $\mathbb{N}$-graded ring. Let $L$, $M$ and $N$ be finitely generated graded $R$-modules with $N \subseteq M$. For a homogeneous ideal $I$, and for each fixed $k \in \mathbb{N}$, we show the asymptotic linearity of v-numbers of the graded modules $ {\rm Ext}_R^{k}(L,{I^{n}M}/{I^{n}N})$ and ${\rm Tor}_k^{R}(L,{I^{n}M}/{I^{n}N})$ as functions of $n$. Moreover, under some conditions on ${\rm Ext}_R^k(L,M)$ and ${\rm Tor}_k^R(L,M)$ respectively, we prove similar behaviour for v-numbers of ${\rm Ext}_R^{k}(L,{M}/{I^{n}N})$ and $ {\rm Tor}_k^{R}(L,{M}/{I^{n}N})$. The last result is obtained by proving the asymptotic linearity of v-number of $(U+I^{n}V)/I^{n}W$, where $U$, $V$ and $W$ are graded submodules of a finitely generated graded $R$-module such that $W \subseteq V$ and $(0:_{U}I) = 0$.

math.AC↗