Behaviour of the Support of Lyubeznik Functors under Ring Extensions

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 $\mathrm{Supp}_S \mathrm{T}(S)$.