Bounds on Bass numbers of local cohomology modules
Sayed Sadiqul Islam, Tony J. Puthenpurakal
TL;DR
This work investigates upper bounds for Bass numbers μ_i(P, M) of Lyubeznik functors M = T(R) acting on polynomial rings R = K[x_1,...,x_m] over an uncountable algebraically closed field K of characteristic 0. It constructs a stratification of primes via sets S_g(t) and H_g(t) and derives monotone bound functions φ^g_i depending on the multiplicity e(M) to bound μ_i(P, M) for primes in these strata. The results cover both graded and non-graded scenarios, proving explicit bounds for maximal and height m−1 primes (Theorems A–D) and culminating in a general bound (Theorem F) for primes in S_g(t). By extending previous work and removing the need for K to be algebraically closed, the paper advances finiteness properties of Bass numbers for local cohomology and iterated Lyubeznik functors, with bounds expressed in terms of Weyl-algebra multiplicities and simple polynomial factors. The findings have implications for the structure of local cohomology modules and their associated primes in characteristic zero.
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 $μ_j(P,M)$ 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)$.