Ymir

Introduction

This is intended to be a Stacks Project for complex analytic spaces and non-Archimedean analytic spaces.

Ymir is a giant from Nordic mythology, who gave birth to a male and a female from his armpits. For us, Ymir contains two subclasses: a complex analytic part and a rigid analytic/Berkovich analytic part.

We will freely make use of all results from Stacks Project.

Current structure

Part 1 Preliminaries

• Topology and Bornology

• Banach rings

To do: include the Archimedean theory

• Commutative algebra

To do: include further details.

Part 2 Complex analytic spaces

• Complex analytic local algebras

• The notion of complex analytic spaces

To do: remove the Hausdorff assumptions.

• Constructions of complex analytic spaces

• Local properties of complex analytic spaces

To do, include Gorenstein, local complete intersection. Products.

• Morphisms of complex analytic spaces

To do, include Grauert’s direct image theorem. Include separated, proper, smooth morphisms, etc.

• Analytic sets

• Global properties of complex analytic spaces

To do, reproduce Kiehl’s proof that the flat locus is co-analytic.

Part 3 Non-Archimedean analytic spaces

• Affinoid algebras

To do: include Ducros’ proof of the excellence of affinoid algebras

• Berkovich analytic spaces

To do: reproduce the first half of Berkovich93 PIHES. Include the theory of rigid analytic spaces.

Unwritten parts

• GAGA (complex and non-Archimedean)

• morphisms (Berkovich)

• properties (Berkovich)

• étale cohomology (Berkovich)

• descent (complex and Berkovich)

• differential calculus (complex)

• Resolution of singularities (complex)

• birational maps (complex)

• vanishing theorems (complex)