A note on prismatic sites for p-quasisyntomic rings

Abstract

Let $p$ be a fixed prime number and let $R$ be a $p$-quasisyntomic ring. In this note, we provide conditions for objects in the absolute prismatic site $R_\Prism$ to cover the final object in $\Shv(R_\Prism)$. More precisely, we introduce in $R_\Prism$ the so-called transversal objects, with which coproducts exist in $R_\Prism$. Immediately generalizing this, we introduce the so-called relatively quasiregular semiperfectoid covers of $R$, whose prismatic cohomology (of $δ$-pairs, in the sense of Antieau-Krause-Nikolaus) would produce in $R_\Prism$ objects with which coproducts exist.