Bounds on Bass numbers of local cohomology modules

Abstract

Let $R=K[x_1,\dots ,x_m]$ where $K$ is an uncountable algebraically closed field of characteristic $0$. For a prime ideal $P$ of $R$, let ${\mu }_j(P,M)$ be the $j$-th Bass number of an $R$-module $M$ with respect to the prime $P$. For $1\le g\le m-1$, we construct a set ${\mathcal{S}}_g(t)$ such that ${\mathcal{S}}_g(t)\subseteq {\mathcal{S}}_g(t+1)$ for all $t\ge 1$ and ${\bigcup }_{t\ge 1}{\mathcal{S}}_g(t)={\mathrm{Spec}}_g(R)=\{P\in \mathrm{Spec}(R)\mid \mathrm{height}P=g\}$. Let $\mathcal{T}$ be a Lyubeznik functor on $\mathrm{Mod}(R)$. We prove that there exists some function ${\varphi }_i^g:{\mathbb{N}}^2\to \mathbb{N}$ which is monotonic in both the variables such that ${\mu }_i(P,\mathcal{T}(R))\le {\varphi }_i^g(e(\mathcal{T}(R)),t)$ for all $P\in {\mathcal{S}}_g(t)$. In particular, the result holds for composition of local cohomology functors of the form $H_{I_1}^{i_1}(H_{I_2}^{i_2}(\dots H_{I_r}^{i_r}(-)\dots )$.

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