Lorenzo Ramero

Effective estimates for unirationality

In this work we refine some estimates that where set up by Morin and Predonzan in [1] and [2]. While the language has been updated, most of the ideas go back to these authors and we have essentially added some observations in the main stream of the proofs.

7. Analytic geometry

7.1 Derived completion functor 7.2 Cotangent complex for formal schemes and adic spaces 7.3 Deformations of formal schemes and adic spaces 7.4 Analytic geometry over a deeply ramified base 7.5 Semicontinuity of the discriminant

6. Valuation theory

6.1 Ordered groups and valuations 6.2 Basic ramification theory 6.3 Algebraic extensions 6.4 Logarithmic differentials 6.5 Transcendental extensions 6.6 Deeply ramified extensions

5. Henselization and completion of almost algebras

5.1 Henselian pairs 5.2 Criteria for unramified morphisms 5.3 Topological algebras and modules 5.4 Henselian approximation of structures over adically complete rings 5.5 Lifting theorems for henselian pairs 5.6 Smooth locus of an affine almost scheme 5.7 Quasi-projective almost schemes 5.8 Lifting and descent of torsors

2. Homological theory

2.1 Some ring-theoretic preliminaries 2.2 Categories of almost modules and algebras 2.3 Uniform spaces of almost modules 2.4 Almost homological algebra 2.5 Almost homotopical algebra