Toward an efficient algorithm for deciding the vanishing of local cohomology modules in prime characteristic

TL;DR

This paper presents a modified algorithm for deciding the vanishing of local cohomology modules in polynomial rings over fields of prime characteristic, aiming to improve practicality by reducing memory consumption.

Contribution

It introduces a memory-efficient modification of Lyubeznik's algorithm for local cohomology vanishing, making it more feasible for computational use.

Findings

Reduced memory usage in the algorithm

Potential for practical implementation in algebraic computations

Improved efficiency over the original Lyubeznik algorithm

Abstract

Let $R=k[x_1,\dots,x_n]$ be a ring of polynomials over a field $k$ of characteristic $p>0$. There is an algorithm due to Lyubeznik for deciding the vanishing of local cohomology modules $H^i_I(R)$ where $I\subset R$ is an ideal. This algorithm has not been implemented because its complexity grows very rapidly with the growth of $p$ which makes it impractical. In this paper we produce a modification of this algorithm that consumes a modest amount of memory.