A correspondence is shown between lim-perfectoid splitting of projective schemes and lim-perfectoid purity of their Gorenstein section rings, supplying new examples of lim-perfectoid pure rings.
Algebraization of absolute perfectoidization via section rings
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abstract
We construct and study a graded version of absolute perfectoidization for $G$-graded adic rings. As a main geometric application, we show that the absolute perfectoidization of the structure sheaf of a projective-type formal scheme admits an algebraization.
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A local-global correspondence for perfectoid purity
A correspondence is shown between lim-perfectoid splitting of projective schemes and lim-perfectoid purity of their Gorenstein section rings, supplying new examples of lim-perfectoid pure rings.