Isomorphisms Between Local Cohomology Modules As Truncations of Taylor Series

TL;DR

This paper explores the local cohomology modules of thickenings of a determinantal ideal in a polynomial ring, revealing they can be described via Taylor series of natural logarithm, providing concrete constructions and new insights.

Contribution

It provides explicit constructions for local cohomology modules of determinantal ideal thickenings, linking them to Taylor series of natural logarithm in characteristic zero.

Findings

Local cohomology modules are describable using Taylor series of natural log.

Concrete constructions for local cohomology modules of determinantal ideals.

Connection between algebraic structures and analytic series like Taylor expansion.

Abstract

Let $R$ be a standard graded polynomial ring that is finitely generated over a field, and let $I$ be a homogenous prime ideal of $R$. Bhatt, Blickle, Lyubeznik, Singh, and Zhang examined the local cohomology of $R/I^t$, as $t$ grows arbitrarily large. Such rings are known as thickenings of $R/I$. We consider $R = \mathbb{F}[X]$ where $\mathbb{F}$ is a field of characteristic 0, $X$ is a $2 \times m$ matrix, and $I$ is the ideal generated by size two minors. We give concrete constructions for the local cohomology modules of thickenings of $R/I$. Bizarrely, these local cohomology modules can be described using the Taylor series of natural log.