Formal Schemes of Rational Degree

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