[Paper Review] Formal and rigid geometry: an intuitive introduction, and some applications

Johannes Nicaise|arXiv (Cornell University)|Jan 19, 2007

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper provides an intuitive introduction to formal and rigid geometry, emphasizing their roles in connecting algebraic geometry over non-archimedean fields with arithmetic and singularity theory. It explains how formal schemes capture infinitesimal structures and how rigid geometry enables analytic techniques in non-archimedean settings, with key applications to the Milnor fibration and motivic zeta functions via the work of Sebag and the author.

ABSTRACT

We give an informal introduction to formal and rigid geometry over complete discrete valuation rings, and we discuss some applications in algebraic and arithmetic geometry and singularity theory, with special emphasis on recent applications to the Milnor fibration and the motivic zeta function by J. Sebag and the author.

Motivation & Objective

• To provide a conceptual and accessible overview of formal and rigid geometry for researchers in algebraic and arithmetic geometry.
• To clarify the relationship between formal schemes, their generic fibers (rigid varieties), and special fibers over residue fields.
• To motivate the use of these tools in studying singularities, particularly through the Milnor fibration and motivic zeta functions.
• To illustrate how rigid geometry enables topological techniques in non-archimedean settings, overcoming limitations of classical Grothendieck topologies.
• To connect abstract formal and rigid structures with concrete applications in Galois theory and semi-stable reduction of curves.

Proposed method

• Uses formal schemes over a complete discrete valuation ring $ R $ to encode both the special fiber $ X_0 $ over $ k $ and its infinitesimal neighborhood via $ \widehat{X} $.
• Constructs the generic fiber $ X_\eta $ of a formal scheme $ \widehat{X} $ by inverting the uniformizer $ \pi $, yielding a rigid analytic space over $ K $.
• Applies Berkovich’s spectral theory to rigid geometry, introducing a true topology on analytic spaces, enabling classical algebraic topology techniques.
• Employs the reduction map from rigid spaces to their canonical reductions over $ \widetilde{L} $, identifying formal fibers as key tools for detecting singularities.
• Uses patching techniques on affinoid covers of rigid analytic curves to construct semi-stable models via gluing of canonical reductions.
• Applies GAGA-type theorems to algebraize rigid analytic covers into algebraic étale covers, enabling construction of Galois extensions over non-archimedean fields.

Experimental results

Research questions

• RQ1 How do formal schemes over $ R $ unify the special fiber and its infinitesimal neighborhood in arithmetic geometry?
• RQ2 In what way does rigid geometry provide a more geometric framework than classical Grothendieck topologies for non-archimedean analytic geometry?
• RQ3 How can the formal fiber of a rigid space over a closed point in the canonical reduction detect geometric properties like smoothness or ordinary double points?
• RQ4 What role do $ R' $-sections (for finite extensions $ R' $ of $ R $) play in connecting the generic and special fibers of a formal scheme?
• RQ5 How can patching techniques in rigid geometry be used to construct semi-stable models of curves and Galois covers?

Key findings

• The generic fiber $ X_\eta $ of a formal scheme $ \widehat{X} $ is a rigid analytic space over $ K $, and its $ K' $-points correspond to $ R' $-sections of $ X $, establishing a canonical link between the generic and special fibers.
• A closed point $ x $ on the canonical reduction $ \widetilde{U} $ of an affinoid domain $ U $ is smooth if and only if its formal fiber is isomorphic to an open disc of radius 1.
• The construction of a semi-stable model for a curve $ C $ over $ L $ is achieved by finding a finite admissible cover $ \mathfrak{U} $ of $ C^{an} $ such that the canonical reductions $ \widetilde{U} $ are semi-stable and can be glued together.
• Using rigid geometry, one can construct étale covers of rigid analytic curves and algebraize them via GAGA theorems, leading to algebraic étale covers over the base field.
• Abhyankar’s conjecture for the projective line is realized via rigid geometry: a finite group $ G $ is realizable as a Galois group of a cover of $ \mathbb{P}^1_k $, unramified outside $ \infty $, if and only if $ G $ is generated by elements of $ p $-power order.
• Harbater’s theorem is realized in the rigid setting: for any finite group $ G $, there exists a Galois cover $ X \to \mathbb{P}^1_L $ with $ X $ smooth, projective, and absolutely irreducible, and unramified at some $ L $-rational point.