Formal Schemes of Rational Degree

Formal schemes in algebraic geometry consist of power series with integer degree, this idea can be naturally carried over to power series with rational degree. In this paper formal schemes with rational degree are constructed. These schemes are non noetherian and thus require slight modification of the standard approach. The first part of the paper constructs non noetherian continuous valuation rings from discrete valuation rings, and these rings are reffered to as ‘eka’ (one in Hindi) rings. These new rings are designed to carry the properties of discrete valuation to continuous valuation faithfully. The second part of the paper constructs non noetherian formal schemes with rational degree and shows their admissibility. The corresponding flatness and coherence is proved. Finally line bundles of rational degree are constructed and their Cech cohomology computed.