Graded perfectoid rings

Ryo Ishizuka; Shou Yoshikawa

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arXiv · 2511.02322

Graded perfectoid rings

Ryo Ishizuka·Shou Yoshikawa

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Abstract

We introduce and study graded perfectoid rings as graded analogues of Scholze's (integral) perfectoid rings. We establish a categorical equivalence between graded perfectoid rings and graded perfect prisms, extending the Bhatt-Scholze's correspondence to the graded setting. We also construct the initial graded perfectoid cover of any graded semiperfectoid rings and prove a graded version of Andr\'e's flatness lemma. These results lay the foundations for a graded theory of perfectoid rings.

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Ryo Ishizuka, Shou Yoshikawa. 2025-11-04. Graded perfectoid rings. https://arxiv.org/abs/2511.02322

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