Bounds on Bass numbers of local cohomology modules

TL;DR

This paper establishes bounds on the Bass numbers of local cohomology modules over polynomial rings, relating them to geometric properties of prime ideals and providing a systematic way to estimate their size.

Contribution

It constructs a hierarchy of prime sets and proves the existence of monotonic functions bounding Bass numbers of Lyubeznik functors, extending to compositions of local cohomology.

Findings

Existence of sets _{g}(t) covering prime ideals of fixed height.

Bounded Bass numbers by monotonic functions of geometric invariants.

Applicable to complex compositions of local cohomology functors.

Abstract

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 $\mu_j(P,M)$ 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 $\phi^g_i: \mathbb{N}^2\rightarrow \mathbb{N}$ which is monotonic in both the variables such that $\mu_i(P,\mathcal{T}(R))\leq \phi^g_i(e(\mathcal{T}(R)),t)$ 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)$.