nLab rigid analytic geometry

Idea

Rigid analytic geometry (often just “rigid geometry” for short) is a form of analytic geometry over a nonarchimedean field $K$ which considers spaces glued from polydiscs, hence from maximal spectra of Tate algebras (quotients of a $K$-algebra of converging power series). This is in contrast to some modern approaches to non-Archimedean analytic geometry such as Berkovich spaces which are glued from Berkovich’s analytic spectra and more recent Huber’s adic spaces.

The issue is that while the p-adic numbers are complete in the p-adic norm, that topology is exotic: $\mathbb{Q}_p$ is a Stone space, hence in particular a totally disconnected topological space.

For that reason the naive idea of formulating p-adic analytic geometry in analogy to complex analytic geometry as modeled on domains in $\mathbb{Q}_p^n$, regarded with their subspace topology, fails, as also all these domains are totally disconnected.

Instead there is (Tate 71) a suitable Grothendieck topology on such affinoid domains – the G-topology – with respect to which there is a good theory of non-archimedean analytic geometry (“rigid analytic geometry”) and hence in particular of p-adic geometry. Moreover, one may sensibly assign to a $p$-adic domain a topological space which is well behaved (in particular locally connected and even locally contractible), this is the analytic spectrum construction. The resulting topological spaces equipped with covers by affinoid domain under the analytic spectrum are called Berkovich spaces.

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.

Cohomology

The related type of cohomology is called rigid cohomology.

Relation to perfectoid spaces

Any smooth rigid-analytic variety $X$ admits a cover $U_{i}\to X$ where the $U_{i}$‘s are affinoid perfectoid spaces (corollary 4.7 of Scholze12). Together with the fact that $H^{i}(U_{et},\mathcal{O}_{X}^{+})$ is almost zero for $i\geq 1$ when $U$ is an affinoid perfectoid space, this allows one to compute the cohomology $H_{et}^{i}(X,\mathcal{O}_{X}^{+})$. The Artin-Schreier sequence can then be used to compute the cohomology $H^{i}(X_{et},\mathbb{Z}/p\mathbb{Z})$ (see the discussion in section 5.7 of Weinstein15).

Applications

• 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.

Related concepts

• complex analytic space

• analytification

• global analytic geometry

• overconvergent global analytic geometry

References

An original article is

• John Tate, Rigid analytic spaces, Invent. Math. 12:257–289, 1971.

and for the construction of the generic fiber of formal schemes over the ring of integers of $K$

• Michel Raynaud, Géométrie analytique rigide d’après Tate, Kiehl,⋯, Table Ronde d’Analyse non archimédienne (Paris, 1972), pp. 319–327. Bull. Soc. Math. France, Mem. No. 39–40, Soc. Math. France, Paris, 1974, MR470254

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

A comprehensive textbook account is in

• Siegfried Bosch, Ulrich Güntzer, Reinhold Remmert, Non-Archimedean Analysis – A systematic approach to rigid analytic geometry, Grundlehren der Mathem. Wissen. 261 (1984) [ ISBN:9783540125464, pdf]

Comparison of various spectra and topologies is in

• M. van der Put, P. Schneider, Points and topologies in rigid geometry, Math. Ann. 302 (1995), no. 1, 81–103, MR96k:32070, doi

Other accounts include

• Ahmed Abbes, Éléments de Géométrie Rigide, vol. I. Construction et étude géométrique des espaces rigides, Progress in Mathematics 286, Birkhäuser 2011, 496 p.book page

• Siegfried Bosch, Lectures on formal and rigid geometry, Preprints of SFB Geom. Struk. Math. Heft 378, pdf (revised 2008)

• J. Fresnel, M. van der Put, Rigid geometry and applications, Birkhäuser (2004) MR2014891

• F. Denef, L. van den Dries, $p$-adic and real subanalytic sets, Ann. of Math. 128 (1988) no. 1, 79–138 MR951508, doi

• Yan Soibelman, On non-commutative analytic spaces over non-archimedean fields, preprint IHES, pdf

• Hans Grauert, Reinhold Remmert, Coherent analytic sheaves, Springer 1984

• R. Cluckers, L. Lipshitz, Fields with analytic structure, J. Eur. Math. Soc. 13, 1147–1223, pdf

  and several articles (in various formalisms) in collection

• R. Cluckers, J. Nicaise, J. Sebag (Editors), Motivic integration and its interactions with model theory and non-archimedean Geometry, 2 vols. London Mathematical Society Lecture Note Series 383, 384

• Peter Schneider, Points of rigid analytic varieties, J. Reine Angew. Math. 434 (1993), 127–157, MR94b:14017, doi

See also

• Kiran Kedlaya, $p$-Adic differential equations (pdf)

The relation to perfectoid spaces can be found in

• Peter Scholze, $p$-adic Hodge theory for rigid-analytic varieties (arXiv:1205.3463)

There is also a survey of the above in chapter 5 of

• Jared Weinstein, Reciprocity laws and Galois representations: recent breakthroughs (pdf)

category: geometry