Behaviour of the Support of Lyubeznik Functors under Ring Extensions

TL;DR

This paper investigates how the Zariski closedness of the support of Lyubeznik functors behaves under various types of ring extensions, including flat, pure, and cyclically pure extensions, with specific results for direct summands.

Contribution

It provides new insights into the transfer of support properties of Lyubeznik functors across different classes of ring extensions, especially pure and cyclically pure cases.

Findings

Zariski closedness of support descends from extended ring to base ring for certain extensions.

Support sets are compared in the case of pure extensions where the base ring is a direct summand.

Results clarify the behavior of Lyubeznik functors under various ring extension conditions.

Abstract

Let $R\rightarrow S$ be an arbitrary ring extension of Noetherian rings. In this article we study the behaviour of Zariski closedness of the support of Lyubeznik functors $\mathrm{T}$, when the ring extension $R\rightarrow S$ is namely `flat', `faithfully flat', `pure' and lastly `cyclically pure'. We show that the Zariski closedness of the support comes down from extended ring to the base ring for faithfully flat, pure and finally for cyclically pure ring extensions. Lastly, we focus on a special case of pure extension i.e. when $R$ is a direct summand of $S$ and we compare the sets $\mathrm{Supp}_S(\mathrm{T}(R)\otimes_R S)$ and $\mathrm{Supp}_S \mathrm{T}(S)$.