Formal Schemes of Rational Degree

TL;DR

This paper constructs non-Noetherian formal schemes of perfectoid type with rational degree, introduces eka^d rings, and extends key properties like Gabber's Lemma and admissible blow-ups without relying on traditional perfectoid tools.

Contribution

It introduces a new class of non-Noetherian formal schemes with rational degree and the novel eka^d rings, avoiding reliance on Huber's framework or Witt vectors.

Findings

Constructed non-Noetherian formal schemes of perfectoid type with rational degree.

Defined topologically finite presentation and proved Gabber's Lemma in this context.

Introduced eka^d rings that encompass most perfectoid affinoid algebras without traditional constructions.

Abstract

Non notherian Formal schemes of perfectoid type (for example $\mathbb{Z}_p[p^{1/p^\infty}]\langle X^{1/p^\infty} \rangle$ along with its multivariate version) with rational degree are constructed and are shown to be admissible. These formal schemes are a rational degree avatar of Tate affinoid algebras and come equipped with non Notherian rings. The corresponding notion of topologically finite presentation are defined and Gabber's Lemma, admissible blow ups (Raynaud's approach) are shown to hold under certain assumptions. A new notion of rings called eka$^d$ are introduced, which recover most examples of perfectoid affinoid algebras, without resorting to Huber's construction, Witt vectors or Frobenius. This version fixes some errors in the last version