Purity for the Brauer group. (English) Zbl 1430.14047

Česnavičius, Kęstutis

Duke Math. J. 168, No. 8, 1461-1486 (2019).

The purity conjecture is a famous conjecture due to A. Grothendieck [Adv. Stud. Pure Math. 3, 88–188 (1968; Zbl 0198.25901)]. It predicts that the Brauer group of a regular scheme \(X\) is not changed when removing a closed subscheme \(Z \subset X\) of codimension at least two. Combining several results due to Grothendieck and Gabber, the conjecture was proved except in some cases. In the present paper, the author settles the these remaining cases. They concern \(p\)-torsion Brauer classes in mixed characteristic \((0,p)\). This finishes the proof of the following theorem.
Theorem [K. Česnavičius, Duke Math. J. 168, No. 8, 1461–1486 (2019; Zbl 07080116)]. For each locally noetherian scheme \(X\) and a closed subscheme \(Z\subset X\), such that for every \(z \in Z\) the local ring \(\mathcal{O}_{X,z}\) of \(X\) at \(z\) is regular of dimension \(\geq 2\), we have \[ H^2_{\text{et}} \left(X, \mathbb{G}_m\right) \overset{\sim}{\longrightarrow} H^2_{\text{et}} \left(X - Z, \mathbb{G}_m\right) \] and \[ H^3_{\text{et}} \left(X, \mathbb{G}_m\right)\hookrightarrow H^3_{\text{et}} \left(X - Z, \mathbb{G}_m\right). \]

Reviewer: Charlotte Ure (Charlottesville)

MSC:

14F22	Brauer groups of schemes
14F20	Étale and other Grothendieck topologies and (co)homologies
14G22	Rigid analytic geometry
16K50	Brauer groups (algebraic aspects)

Keywords:

Brauer group; étale cohomology; perfectoid ring; punctured spectrum; purity

Citations:

Zbl 0198.25901

References:

[1]	[AG] M. Auslander and O. Goldman, The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960), 367-409. · Zbl 0100.26304
[2]	[Alp] J. Alper, Adequate moduli spaces and geometrically reductive group schemes, Algebr. Geom. 1 (2014), no. 4, 489-531. See also https://arxiv.org/abs/1005.2398 for an updated postpublication version. · Zbl 1322.14026
[3]	[And] Y. André, Le lemme d’Abhyankar perfectoide, Publ. Math. Inst. Hautes Études Sci. 127 (2018), 1-70. · Zbl 1419.14030
[4]	[BLR] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Ergeb. Math. Grenzgeb. (3) 21, Springer, Berlin, 1990.
[5]	[BMS] B. Bhatt, M. Morrow, and P. Scholze, Integral \(p\)-adic Hodge theory, Publ. Math. Inst. Hautes Études Sci. 128 (2018), 219-397. · Zbl 1446.14011
[6]	[CGP] B. Conrad, O. Gabber, and G. Prasad, Pseudo-reductive groups, 2nd ed., New Math. Monogr. 26, Cambridge University Press, Cambridge, 2015. · Zbl 1314.20037
[7]	[CTS1] J.-L. Colliot-Thélène and J.-J. Sansuc, Fibrés quadratiques et composantes connexes réelles, Math. Ann. 244 (1979), no. 2, 105-134. · Zbl 0418.14016
[8]	[CTS2] J.-L. Colliot-Thélène and J.-J. Sansuc, Principal homogeneous spaces under flasque tori: Applications, J. Algebra 106 (1987), no. 1, 148-205. · Zbl 0597.14014
[9]	[DF] F. R. DeMeyer and T. J. Ford, On the Brauer group of surfaces, J. Algebra 86 (1984), no. 1, 259-271. · Zbl 0491.14022
[10]	[EGA_2] A. Grothendieck, Éléments de géométrie algébrique, II: Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961).
[11]	[EGA_4] A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, II, Inst. Hautes Études Sci. Publ. Math. 24 (1965); IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967).
[12]	[Elk] R. Elkik, Solutions d’équations à coefficients dans un anneau hensélien, Ann. Sci. École Norm. Sup. (4) 6 (1973), 553-603, 1974. · Zbl 0327.14001
[13]	[Fal] G. Faltings, “Almost étale extensions” in Cohomologies p-adiques et applications arithmétiques, II, Astérisque 279, Soc. Math. France, Paris, 2002, 185-270.
[14]	[Fuj] K. Fujiwara, “A proof of the absolute purity conjecture (after Gabber)” in Algebraic Geometry 2000, Azumino (Hotaka, Japan, 2000), Adv. Stud. Pure Math. 36, Math. Soc. Japan, Tokyo, 2002, 153-183.
[15]	[Ga81] O. Gabber, “Some theorems on Azumaya algebras” in The Brauer group (Les Plans-sur-Bex, Switzerland, 1980), Lecture Notes in Math. 844, Springer, Berlin, 1981, 129-209.
[16]	[Ga93] O. Gabber, An injectivity property for étale cohomology, Compositio Math. 86 (1993), no. 1, 1-14. · Zbl 0828.14011
[17]	[Ga04] O. Gabber, “On purity for the Brauer group” in Arithmetic Algebraic Geometry (Oberwolfach, Germany, 2004), Oberwolfach Rep. 1, Eur. Math. Soc., Zurich, 2004, no. 3, 1971-2013.
[18]	[Gir] J. Giraud, Cohomologie non abélienne, Grundlehren Math. Wiss. 179, Springer, Berlin, 1971.
[19]	[GR03] O. Gabber and L. Ramero, Almost Ring Theory, Lecture Notes in Math. 1800, Springer, Berlin, 2003. · Zbl 1045.13002
[20]	[GR18] O. Gabber and L. Ramero, Foundations for almost ring theory, preprint, arXiv:math/0201175 [math.AG].
[21]	[Gro1] A. Grothendieck, “Le groupe de Brauer, II: Théorie cohomologique” in Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math. 3, North-Holland, Amsterdam, 1968, 67-87.
[22]	[Gro2] A. Grothendieck, “Le groupe de Brauer, III: Exemples et complements” in Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math. 3, North-Holland, Amsterdam, 1968, 88-188.
[23]	[Hoo] R. T. Hoobler, A cohomological interpretation of Brauer groups of rings, Pacific J. Math. 86 (1980), no. 1, 89-92. · Zbl 0459.13002
[24]	[Hub1] R. Huber, Continuous valuations, Math. Z. 212 (1993), no. 3, 455-477. · Zbl 0788.13010
[25]	[Hub2] R. Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects Math. E30, Friedr. Vieweg & Sohn, Braunschweig, 1996.
[26]	[ILO] L. Illusie, Y. Laszlo, and F. Orgogozo (eds.), Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents, Astérisque 363-364, Soc. Math. France, Paris, 2014.
[27]	[KL] K. S. Kedlaya and R. Liu, Relative p-adic Hodge theory: Foundations, Astérisque 371, Soc. Math. France, Paris, 2015.
[28]	[Mat] H. Matsumura, Commutative Ring Theory, 2nd ed., Cambridge Stud. Adv. Math. 8, Cambridge Univ. Press, Cambridge, 1989.
[29]	[P] B. Poonen, Rational Points on Varieties, Grad. Stud. Math. 186, American Mathematical Society, Providence, 2017. · Zbl 1387.14004
[30]	[Sch1] P. Scholze, Perfectoid spaces, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 245-313. · Zbl 1263.14022
[31]	[Sch2] P. Scholze, Étale cohomology of diamonds, preprint, arXiv:1709.07343v1 [math.AG].
[32]	[Ser] J.-P. Serre, Galois Cohomology, corrected reprint of 1997English edition, Springer Monogr. Math., Springer, Berlin, 2002.
[33]	[SGA_2] A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, Seminaire de Géométrie Algébrique du Bois Marie 1962(SGA 2), Doc. Math. (Paris) 4, Soc. Math. France, Paris, 2005.
[34]	[SGA_3] M. Demazure and A. Grothendieck, Schémas en groupes, I: Propriétés générales des schémas en groupes, Séminaire de Géométrie Algebrique du Bois Marie 1962-1964 (SGA 3), Lecture Notes in Math. 151, Springer, Berlin, 1970; II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Séminaire de Géométrie Algébrique du Bois Marie 1962-1964 (SGA 3), Lecture Notes in Math. 152, Springer, Berlin, 1970. · Zbl 0209.24201
[35]	[SGA_4] M. Artin, A. Grothendieck, and J.-L. Verdier, Théorie des topos et cohomologie étale des schémas, Tome 2, Séminaire de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4), Lecture Notes in Math. 270, Springer, Berlin, 1972; Tome 3, Séminaire de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4), Lecture Notes in Math. 305, Springer, Berlin, 1973.
[36]	[SGA_5] A. Grothendieck, Cohomologie l-adique et fonctions L, Séminaire de Géometrie Algébrique du Bois-Marie (SGA 5), Lecture Notes in Math. 589, Springer, Berlin, 1977.
[37]	[Shi] K. Shimomoto, An application of the almost purity theorem to the homological conjectures, J. Pure Appl. Algebra 220 (2016), no. 2, 621-632. · Zbl 1327.13025
[38]	[SP] The Stacks Project, http://stacks.math.columbia.edu, accessed March 21, 2019.
[39]	[Tho1] R. W. Thomason, Absolute cohomological purity, Bull. Soc. Math. France 112 (1984), no. 3, 397-406. · Zbl 0584.14007
[40]	[Tho2] R. W. Thomason, Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. in Math. 65 (1987), no. 1, 16-34. · Zbl 0624.14025