Bounds on Bass numbers of local cohomology modules
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911254202810368 |
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| author | Islam, Sayed Sadiqul Puthenpurakal, Tony J. |
| author_facet | Islam, Sayed Sadiqul Puthenpurakal, Tony J. |
| contents | Let $R=K[x_1,\ldots,x_m]$ where $K$ is an uncountable algebraically closed field of characteristic $0$. For a prime ideal $P$ of $R$, let $μ_j(P,M)$ be the $j$-th Bass number of an $R$-module $M$ with respect to the prime $P$. For $1\leq g\leq m-1$, we construct a set $\mathcal{S}_g(t)$ such that $\mathcal{S}_g(t)\subseteq \mathcal{S}_g(t+1)$ for all $t\geq 1$ and $\bigcup_{t\geq 1} \mathcal{S}_g(t)=\operatorname{Spec}_g(R)=\{P\in \operatorname{Spec}(R)\mid \operatorname{height}P=g\}$. Let $\mathcal{T}$ be a Lyubeznik functor on $\operatorname{Mod}(R)$. We prove that there exists some function $ϕ^g_i: \mathbb{N}^2\rightarrow \mathbb{N}$ which is monotonic in both the variables such that $μ_i(P,\mathcal{T}(R))\leq ϕ^g_i(e(\mathcal{T}(R)),t)$ for all $P\in \mathcal{S}_{g}(t)$. In particular, the result holds for composition of local cohomology functors of the form $ H^{i_1}_{I_1}(H^{i_2}_{I_2}(\dots H^{i_r}_{I_r}(-)\dots)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_05450 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounds on Bass numbers of local cohomology modules Islam, Sayed Sadiqul Puthenpurakal, Tony J. Commutative Algebra Primary 13D45, Secondary 13N10, 13H10 Let $R=K[x_1,\ldots,x_m]$ where $K$ is an uncountable algebraically closed field of characteristic $0$. For a prime ideal $P$ of $R$, let $μ_j(P,M)$ be the $j$-th Bass number of an $R$-module $M$ with respect to the prime $P$. For $1\leq g\leq m-1$, we construct a set $\mathcal{S}_g(t)$ such that $\mathcal{S}_g(t)\subseteq \mathcal{S}_g(t+1)$ for all $t\geq 1$ and $\bigcup_{t\geq 1} \mathcal{S}_g(t)=\operatorname{Spec}_g(R)=\{P\in \operatorname{Spec}(R)\mid \operatorname{height}P=g\}$. Let $\mathcal{T}$ be a Lyubeznik functor on $\operatorname{Mod}(R)$. We prove that there exists some function $ϕ^g_i: \mathbb{N}^2\rightarrow \mathbb{N}$ which is monotonic in both the variables such that $μ_i(P,\mathcal{T}(R))\leq ϕ^g_i(e(\mathcal{T}(R)),t)$ for all $P\in \mathcal{S}_{g}(t)$. In particular, the result holds for composition of local cohomology functors of the form $ H^{i_1}_{I_1}(H^{i_2}_{I_2}(\dots H^{i_r}_{I_r}(-)\dots)$. |
| title | Bounds on Bass numbers of local cohomology modules |
| topic | Commutative Algebra Primary 13D45, Secondary 13N10, 13H10 |
| url | https://arxiv.org/abs/2511.05450 |