higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
Rigid analytic geometry is a form of analytic geometry over a nonarchimedean field $K$ which considers spaces glued from maximal spectra of so-called Tate algebras (quotients of a $K$-algebra of converging power series). This is in contrast to some modern approaches to non-Archimedean analytic geometry. In contrast to this are the Berkovich analytic spaces which are glued from Berkovich analytic spectra and more recent Huber’s adic spaces.
The idea goes back to John Tate. According to Kedlaya, p. 18, the terminology “rigid” is because
… one develops everything “rigidly” by imitating the theory of schemes in algebraic geometry, but using rings of convergent power series instead of polynomials.
See also global analytic geometry.
The related type of cohomology is called rigid cohomology.
The solution by Raynaud and Harbater of Abyhankar’s conjecture concerning fundamental groups of curves in positive characteristic uses the rigid analytic GAGA theorems (whose proofs are very similar to Serre’s proofs in the complex-analytic case).
Work of Kisin on modularity of Galois representations makes creative use of rigid-analytic spaces associated to Galois deformation rings.
An original article is
and for the construction of the generic fiber of formal schemes over the ring of integers of $K$
Introductions are in
Johannes Nicaise, Formal and rigid geometry: an intuitive introduction, and some applications (pdf)
Brian Conrad, Several approaches to non-Archimedean geometry, pdf
Peter Schneider, Basic notions of rigid analytic geometry, in: Galois representations in arithmetic algebraic geometry (Durham, 1996), 369–378, London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press 1998, doi
Comparison of various spectra and topologies is in
Other accounts include
See also