Non-Archimedean analytic geometry as relative algebraic geometry
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 1, pp. 49-126.

Nous montrons que la géométrie analytique non-archimédienne peut être considérée comme la géométrie algébrique relative, au sens de Toën-Vaquié, au-dessus de la catégorie des espaces de Banach non-archimédiens. Pour toute catégorie symétrique monoïdale fermée quasi-abélienne nous définissons une topologie sur certaines sous-catégories de la catégorie des schémas affines (relatifs). Dans le cas où la catégorie monoïdale est celle des groupes abéliens, la topologie coïncide avec la topologie de Zariski usuelle. En examinant cette topologie pour la catégorie des espaces de Banach, nous retrouvons la G-topologie faible ou encore la topologie des sous-ensembles admissibles sur un affinoïde utilisée en géométrie rigide. Cela donne une approche de type foncteur des points à la géométrie analytique non-archimédienne. Nous démontrons que la catégorie des espaces analytiques de Berkovich (et aussi des espaces analytiques rigides) se plonge de manière pleinement fidèle dans la catégorie des schémas relatifs. Nous définissons une notion de faisceau quasi-cohérent sur les espaces analytiques que nous utilisons pour caractériser les familles couvrantes. En chemin nous utilisons l’algèbre homologique dans les catégories quasi-abéliennes développée par Schneiders.

We show that non-Archimedean analytic geometry can be viewed as relative algebraic geometry in the sense of Toën–Vaquié–Vezzosi over the category of non-Archimedean Banach spaces. For any closed symmetric monoidal quasi-abelian category we define a topology on certain subcategories of the category of (relative) affine schemes. In the case that the monoidal category is the category of abelian groups, the topology reduces to the ordinary Zariski topology. By examining this topology in the case that the monoidal category is the category of Banach spaces we recover the G-topology or the topology of admissible subsets on affinoids which is used in rigid or Berkovich analytic geometry. This gives a functor of points approach to non-Archimedean analytic geometry. We demonstrate that the category of Berkovich analytic spaces (and also rigid analytic spaces) embeds fully faithfully into the category of (relative) schemes in our version of relative algebraic geometry. We define a notion of quasi-coherent sheaf on analytic spaces which we use to characterize surjectivity of covers. Along the way, we use heavily the homological algebra in quasi-abelian categories developed by Schneiders.

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DOI : 10.5802/afst.1526
Ben-Bassat, Oren 1 ; Kremnizer, Kobi 2

1 Department of Mathematics, Faculty of Natural Sciences, University of Haifa, Mount Carmel, Haifa, 31905, Israel
2 Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
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Ben-Bassat, Oren; Kremnizer, Kobi. Non-Archimedean analytic geometry as relative algebraic geometry. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 1, pp. 49-126. doi : 10.5802/afst.1526. http://www.numdam.org/articles/10.5802/afst.1526/

[1] The Stacks Project (http://stacks.math.columbia.edu/tag/00QL)

[2] The Stacks Project (http://stacks.math.columbia.edu/tag/00WX)

[3] The Stacks Project (http://stacks.math.columbia.edu/tag/00WW)

[4] The Stacks Project (http://stacks.math.columbia.edu/tag/00XT)

[5] Adámek, Jiří; Rosický, Jiří Locally Presentable and Accessible Categories, London Mathematical Society Lecture Note Series, 189, Cambridge University Press, 1994, xiv+316 pages

[6] Ardakov, Konstantin; Wadsley, Simon On irreducible representations of compact p-adic analytic groups, Ann. Math., Volume 178 (2013) no. 2, pp. 453-557 | DOI

[7] Artin, Michael; Grothendieck, Alexander; Verdier, Jean-Louis Théorie des topos et cohomologie étale des schémas, Lecture Notes in Mathematics., 269, Springer, 1972, xix+525 pages

[8] Bambozzi, Federico On a generalization of affinoid varieties, University of Padova (Italy) (2013) (Ph. D. Thesis https://arxiv.org/abs/1401.5702)

[9] Bambozzi, Federico; Ben-Bassat, Oren Dagger geometry as Banach algebraic geometry, J. Number Theory, Volume 162 (2016), pp. 391-462 | DOI

[10] Bambozzi, Federico; Ben-Bassat, Oren; Kremnizer, Kobi Stein Domains in Banach Algebraic Geometry (https://arxiv.org/abs/1511.09045)

[11] Ben-Bassat, Oren; Kremnizer, Kobi A perspective on the foundations of derived analytic geometry (preprint)

[12] Ben-Bassat, Oren; Temkin, Michael Berkovich spaces and tubular descent, Adv. Math., Volume 2343 (2013), pp. 217-238 | DOI

[13] Berkovich, Vladimir G. Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, 1990, ix+169 pages

[14] Berkovich, Vladimir G. Non-Archimedean analytic spaces (2009) (Advanced School on p-adic Analysis and Applications, ICTP, Trieste, www.wisdom.weizmann.ac.il/~vova/Trieste_2009.pdf)

[15] Block, Jonathan Mayer-Vietoris sequences in cyclic homology of topological algebras (1987) (https://www.math.upenn.edu/~blockj/papers/msri.pdf, MSRI 01208-88)

[16] Bosch, S.; Güntzer, Ulrich; Remmert, Reinhold Non-Archimedean analysis. A systematic approach to rigid analytic geometry, Grundlehren der Mathematischen Wissenschaften, 261, Springer, 1984, xii+436 pages

[17] Bourbaki, Nicolas Topological vector spaces, Springer, 1987, vii+364 pages (Transl. from the French by H. G. Eggleston and S. Madan.)

[18] Braverman, Alexander; Kazhdan, David Representations of affine Kac-Moody groups over local and global fields: a survey of some recent results (https://arxiv.org/abs/1205.0870)

[19] Cohn, Lee Differential graded categories are k-linear stable -categories (https://arxiv.org/abs/1308.2587)

[20] Deligne, Pierre Catégories tannakiennes, The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. II (Prog. Math.), Volume 87, Birkhäuser, 1990, pp. 111-195

[21] Dugger, Daniel; Hollander, Sharon; Isaksen, Daniel C. Hypercovers and simplicial presheaves, Math. Proc. Camb. Philos. Soc., Volume 136 (2004) no. 1, pp. 9-51 | DOI

[22] Eschmeier, Jörg; Putinar, Mihai Spectral decompositions and analytic sheaves, London Mathematical Society Monographs. New Series., 10, Oxford Univ. Press., 1996, x+362 pages

[23] Grothendieck, Alexander Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas, Publ. Math., Inst. Hautes Étud. Sci., Volume 32 (1967), pp. 1-361

[24] Gruson, Laurent Théorie de Fredholm p-adique, Bull. Soc. Math. Fr., Volume 94 (1966), pp. 67-95 | DOI

[25] Hakim, Monique Topos anneles et schemas rélatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete., 64, Springer, 1972, vi+158 pages

[26] Helemskii, Alexander Ya. Lectures and exercises on functional analysis, Translations of Mathematical Monographs, 233, American Mathematical Society, 2006, xvii+468 pages

[27] Séminaire Banach (Houzel, Christian, ed.), Lecture Notes in Mathematics., 277, Springer, 1972, v+229 pages

[28] Houzel, Christian Espaces analytiques relatifs et théorème de finitude, Math. Ann., Volume 205 (1973), pp. 13-54 | DOI

[29] Huber, Roland Continuous valuations, Math. Z., Volume 212 (1993) no. 3, pp. 455-477 | DOI

[30] Ingleton, Audrey W. The Hahn-Banach theorem for non-Archimedean-valued fields, Proc. Camb. Philos. Soc., Volume 48 (1952), pp. 41-45 | DOI

[31] Kapranov, Mikhail The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups (https://arxiv.org/abs/math/0001005)

[32] Kontsevich, Maxim; Rosenberg, Alexander Noncommutative smooth spaces (https://arxiv.org/abs/math/9812158)

[33] Lurie, Jacob Higher algebra (www.math.harvard.edu/~lurie/papers/HA.pdf)

[34] Lurie, Jacob Tanaka duality for geometric stacks (https://arxiv.org/abs/math/0412266)

[35] Lurie, Jacob Higher topos theory, Annals of Mathematics Studies, 170, Princeton University Press, 2009, xv+925 pages

[36] Macpherson, Andrew W. Skeleta in non-Archimedean and tropical geometry (https://arxiv.org/abs/1311.0502)

[37] Meyer, Ralf Embeddings of derived categories of bornological modules (https://arxiv.org/abs/math/0410596)

[38] Meyer, Ralf Local and analytic cyclic homology, EMS Tracts in Mathematics, 3, European Mathematical Society, 2007, viii+360 pages

[39] Patnaik, Manish M. Geometry of Loop Eisenstein Series, Yale University (USA) (2008) (Ph. D. Thesis)

[40] Paugam, Frederic Global analytic geometry (http://arxiv.org/pdf/0803.0148v3)

[41] Pirkovskij, A. Yu. On certain homological properties of Stein algebras, J. Math. Sci., New York, Volume 95 (1999) no. 6, pp. 2690-2702 | DOI

[42] Poineau, Jérôme Les espaces de Berkovich sont angéliques, Bull. Soc. Math. Fr., Volume 141 (2013) no. 2, pp. 267-297 | DOI

[43] Porta, Mauro Derived complex analytic geometry I: GAGA theorems (https://arxiv.org/abs/1506.09042)

[44] Porta, Mauro Derived complex analytic geometry II: square-zero extensions (https://arxiv.org/abs/1507.06602)

[45] Porta, Mauro; Yu, Tony Yue Higher analytic stacks and GAGA theorems (https://arxiv.org/abs/1412.5166)

[46] Porta, Mauro; Yue Yu, Tony Derived non-Archimedean analytic spaces (https://arxiv.org/abs/1601.00859)

[47] Prosmans, Fabienne Algèbre homologique quasi-abélienne, Université Paris 13 (France) (1995) (Ph. D. Thesis)

[48] Ramis, Jean-Pierre; Ruget, Gabriel Résidus et dualité, Invent. Math., Volume 26 (1974), pp. 89-131 | DOI

[49] Robert, Alain M. A course in p-adic analysis, Graduate Texts in Mathematics, 1998, Springer, 2000, xv+437 pages

[50] Ryan, Raymond A. Introduction to tensor products of Banach spaces, Springer Monographs in Mathematics., Springer, 2002, xiv+225 pages

[51] Schneider, Peter Nonarchimedean functional analysis, Springer Monographs in Mathematics, Springer, 2002, 156 pages (Revised course notes from Winter 1997/1998 course at the University of Münster)

[52] Schneiders, Jean-Pierre Quasi-abelian categories and sheaves, Mém. Soc. Math. Fr., Nouv. Sér., Volume 76 (1998), pp. 1-140

[53] Soibelman, Yan S. On non-commutative analytic spaces over non-Archimedean fields, Homological mirror symmetry. New developments and perspectives (Lecture Notes in Physics), Volume 757, Springer, 2009, pp. 221-247

[54] Taylor, Joseph L. A general framwork for a multi-operator functional calculus, Adv. Math., Volume 9 (1972), p. 1833-252 | DOI

[55] Temkin, Michael Introduction to Berkovich analytic spaces (people.math.gatech.edu/~mbaker/pdf/aws07mb_v4.pdf)

[56] Temkin, Michael A new proof of the Gerritzen-Grauert theorem, Math. Ann., Volume 333 (2005) no. 2, pp. 261-269 | DOI

[57] Thuillier, Amaury Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels, Manuscr. Math., Volume 123 (2007) no. 4, pp. 381-451 | DOI

[58] Toën, Bertrand Simplicial presheaves and derived algebraic geometry (https://hal.archives-ouvertes.fr/hal-00772850)

[59] Toën, Bertrand; Vaquié, Michel Algébrisation des variétés analytiques complexes et catégories dérivées, Math. Ann., Volume 342 (2008) no. 4, pp. 789-831 | DOI

[60] Toën, Bertrand; Vaquié, Michel Under Spec , J. K-Theory, Volume 3 (2009) no. 3, pp. 437-500 | DOI

[61] Toën, Bertrand; Vezzosi, Gabriele From HAG to DAG: derived moduli stacks, Axiomatic, enriched and motivic homotopy theory. Proceedings of the NATO Advanced Study Institute, Cambridge, UK, September 9–20, 2002 (NATO Science Series II: Mathematics, Physics and Chemistry), Volume 131, Kluwer Academic Publishers (2004), pp. 173-216

[62] Toën, Bertrand; Vezzosi, Gabriele Homotopical algebraic geometry I: Topos theory, Adv. Math., Volume 193 (2005) no. 2, pp. 257-372 | DOI

[63] Toën, Bertrand; Vezzosi, Gabriele Brave new algebraic geometry and global derived moduli spaces of ring spectra, Elliptic cohomology. Geometry, applications, and higher chromatic analogues. Selected papers of the workshop, Cambridge, UK, December 9–20, 2002 (London Mathematical Society Lecture Note Series), Volume 342, Cambridge University Press (2007), p. 325-259

[64] Toën, Bertrand; Vezzosi, Gabriele Homotopical algebraic geometry II: Geometric stacks and applications, Mem. Am. Math. Soc., 902, American Mathematical Society, 2008, 224 pages

[65] Yu, Tony Yue Gromov compactness in tropical geometry and in non-Archimedean analytic geometry (https://arxiv.org/abs/1401.6452)

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