Skip to main content

Problems About Torsors over Regular Rings

With an Appendix by Yifei Zhao

Abstract

We overview a web of conjectures about torsors under reductive groups over regular rings and survey some techniques that have been used for making progress on such problems.

This is a preview of subscription content, access via your institution.

Notes

  1. We expect them to stay true for regular semilocal rings, but for the sake of focus we chose to neglect this aspect below. Many of their known special cases are established in this generality in the indicated references.

  2. A form of G is an S-group sheaf isomorphic to G locally on S, so it corresponds to an element of \(H^{1}(S, \underline {\text {Aut}}_{\text {gp}}(G))\). A form is inner (resp., pure inner) if this element lifts to H1(S, G/ZG) (resp., even to H1(S, G)), where ZGG is the center and the map \(G/Z_{G} \rightarrow \underline {\text {Aut}}_{\text {gp}}(G)\) is induced by G acting on itself by conjugation.

  3. We recall from [123, Definition 0AP6] that a scheme is ind-quasi-affine if all of its quasi-compact opens are quasi-affine, and that a morphism is ind-quasi-affine if the preimage of every affine open is ind-quasi-affine. By [123, Lemma 0AP8], ind-quasi-affineness of a morphism is fpqc local on the target. Useful examples of ind-quasi-affine but not quasi-affine (that is, not quasi-compact) schemes are character groups of tori or automorphism groups of reductive groups, see Section 1.3.7 below.

  4. Note that the case of loc. cit. that allows a separated G to be merely locally of finite type over a Noetherian S is false, as is pointed out in [20, Theorem 5.3.5 and below]: the Néron lft model of \(\mathbb {G}_{m}\) gives a counterexample because its relative identity component is an open but not closed subgroup identified with \(\mathbb {G}_{m}\).

  5. The representability of E/P is quite remarkable because no general result about quotients ensures it, see Section 1.2.3.

  6. Beyond semilocal S, quasi-splitness is a slightly more delicate notion, see [121, Exposé XXIV, Section 3.9].

  7. An algebra C over a field k is geometrically regular if \(C \otimes _{k} k^{\prime }\) is a regular ring for every finite field extension \(k^{\prime } /k\).

  8. We recall that the Grothendieck ring K0(A) of a commutative ring A is the quotient of the free abelian group on the set of isomorphism classes of finite projective A-modules P by the relations \([P] = [P^{\prime }] + [P^{\prime \prime }]\) for finite projective A-modules P, \(P^{\prime }\), \(P^{\prime \prime }\) with \(P \simeq P^{\prime } \oplus P^{\prime \prime }\), and that the multiplication in K0(A) is induced by the tensor product ⊗A.

  9. We justify the assertion about K-points as follows. Since \(LG \rightarrow \text {Gr}_{G}\) is surjective on K-points and a bijection on sets of connected components, by [105, Theorem 5.1 and the end of the proof of Lemma 17 on p. 198 (with G(L)1 defined after Remark 2 on p. 189)] (their G(L)1 is our (LG)0(K)), we may replace G by a z-extension (see Theorem A.4.1) to reduce to Gder being simply connected. For such G, however, the surjectivity of

    $$ \text{Gr}_{G^{\text{der}}}(K) \rightarrow \text{Gr}_{G}^{0}(K) $$

    follows from [105, last line on p. 197 and proof of Lemma 5 on p. 191] (by the latter, T(L)1 there is our T(Kt)).

  10. CNRS, Université Paris-Saclay, Laboratoire de mathématiques d’Orsay, F-91405, Orsay, France. E-mail address: yifei.zhao@universite-paris-saclay.fr

  11. This definition agrees with [20, Definition B.1.1], but it differs from [120, Exposé IX, Définition 1.1], where G is only required to be fpqc locally isomorphic to \(\mathbb D_{S}(M)\) for an abelian group M (which is not necessarily finitely generated).

  12. We refer to the original paper of Colliot-Thélène–Sansuc [30, Section 0.5] for other equivalent characterizations of flasque lattices, including the one involving Tate cohomology, which often appears in the literature.

  13. Information about \(T_{2} := \text {corad}(G^{\prime })\) is not needed for this proof.

References

  1. Auslander, M, Goldman, O: The Brauer group of a commutative ring. Trans. Amer. Math. Soc. 97, 367–409 (1960). MR0121392 (22 #12130)

    MathSciNet  MATH  Google Scholar 

  2. Asok, A, Hoyois, M, Wendt, M: Affine representability results in A1-homotopy theory, II: Principal bundles and homogeneous spaces. Geom. Topol. 22 (2), 1181–1225 (2018). MR3748687, https://doi.org/10.2140/gt.2018.22.1181

    MathSciNet  MATH  Google Scholar 

  3. Asok, A, Hoyois, M, Wendt, M: Affine representability results in A1-homotopy theory III: Finite fields and complements. Algebr. Geom. 7 (5), 634–644 (2020). MR4156421, https://doi.org/10.14231/ag-2020-023

    MathSciNet  MATH  Google Scholar 

  4. Alper, J: Adequate moduli spaces and geometrically reductive group schemes. Algebr. Geom. 1 (4), 489–531 (2014). MR3272912, https://doi.org/10.14231/AG-2014-022, See also https://arxiv.org/abs/1005.2398v2 for an updated post-publication version

    MathSciNet  MATH  Google Scholar 

  5. Antieau, B, Williams, B: Topology and purity for torsors. Doc. Math. 20, 333–355 (2015). MR3398715

    MathSciNet  MATH  Google Scholar 

  6. Baeza, R: Quadratic Forms over Semilocal Rings. Lecture Notes in Mathematics, vol. 655, p vi+ 199. Springer, Berlin (1978). MR0491773

    Google Scholar 

  7. Bass, H: Some problems in “classical” algebraic K-theory, pp 3–73. Springer, Berlin (1973). Lecture Notes in Math., Vol. 342, 0409606

    MATH  Google Scholar 

  8. Bass, H., Connell, E. H., Wright, D. L.: Locally polynomial algebras are symmetric algebras. Invent. Math. 38(3), 279–299 (1976/77). MR432626, https://doi.org/10.1007/BF01403135

    MathSciNet  MATH  Google Scholar 

  9. Bouthier, A, Česnavičius, K: Torsors on loop groups and the Hitchin fibration. Available at 1908.07480v4 (2021)

  10. Bayer-Fluckiger, E, First, UA., Parimala, R: On the Grothendieck–Serre conjecture for classical groups. Available at 1911.07666v2 (2020)

  11. Bhatt, B: Algebraization and Tannaka duality. Camb. J. Math. 4 (4), 403–461 (2016). MR3572635, https://doi.org/10.4310/CJM.2016.v4.n4.a1

    MathSciNet  MATH  Google Scholar 

  12. Bhatt, B, Halpern-Leistner, D: Tannaka duality revisited. Adv. Math. 316, 576–612 (2017). MR3672914, https://doi.org/10.1016/j.aim.2016.08.040

    MathSciNet  MATH  Google Scholar 

  13. Borovoi, M, Kunyavskii, B: Formulas for the unramified Brauer group of a principal homogeneous space of a linear algebraic group. J. Algebra 225 (2), 804–821 (2000). MR1741563, https://doi.org/10.1006/jabr.1999.8153

    MathSciNet  MATH  Google Scholar 

  14. Beauville, A, Laszlo, Y: Un lemme de descente. C. R. Acad. Sci. Paris Sér. I Math. 320(3), 335–340 (1995). MR1320381

    MathSciNet  MATH  Google Scholar 

  15. Bosch, S, Lütkebohmert, W, Raynaud, M: Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, p x + 325. Springer, Berlin (1990). MR1045822 (91i:14034)

    MATH  Google Scholar 

  16. Bhatwadekar, S.M., Rao, R.A.: On a question of Quillen. Trans. Amer. Math. Soc. 279(2), 801–810 (1983). MR709584

    MathSciNet  MATH  Google Scholar 

  17. Balwe, C, Sawant, A: A1-connectedness in reductive algebraic groups. Trans. Amer. Math. Soc. 369 (8), 5999–6015 (2017). MR3646787, https://doi.org/10.1090/tran/7090

    MathSciNet  MATH  Google Scholar 

  18. Conrad, B, Gabber, O, Prasad, G: Pseudo-Reductive Groups. New Mathematical Monographs, 2nd edn., vol. 26, p xxiv+ 665. Cambridge University Press, Cambridge (2015). MR3362817

    MATH  Google Scholar 

  19. Chambert-Loir, A, Nicaise, J, Sebag, J: Motivic integration. Progress in Mathematics, vol. 325, p xx+ 526. Birkhäuser/Springer, New York (2018). MR3838446, https://doi.org/10.1007/978-1-4939-7887-8

    MATH  Google Scholar 

  20. Conrad, B: Reductive group schemes. In: Autour des schémas en groupes. Vol. I. Panor. Synthèses. MR3362641, vol. 42/43, pp 93–444. Soc. Math., France (2014)

  21. Chernousov, V, Panin, I: Purity of G2-torsors. C. R. Math. Acad. Sci. Paris 345(6), 307–312 (2007). MR2359087, https://doi.org/10.1016/j.crma.2007.07.018

    MathSciNet  MATH  Google Scholar 

  22. Chernousov, V, Panin, I: Purity for Pfister forms and F4-torsors with trivial g3 invariant. J. Reine Angew. Math. 685, 99–104 (2013). MR3181565, https://doi.org/10.1515/crelle-2012-0018

    MathSciNet  MATH  Google Scholar 

  23. Colliot-Thélène, J-L: Formes quadratiques sur les anneaux semi-locaux réguliers. Bull. Soc. Math. France Mém. 59, 13–31 (1979). Colloque sur les Formes Quadratiques, 2 (Montpellier, 1977), 532002

    MATH  Google Scholar 

  24. Colliot-Thélène, J-L: Résolutions flasques des groupes réductifs connexes. C. R. Math. Acad. Sci. Paris 339(5), 331–334 (2004). MR2092058, https://doi.org/10.1016/j.crma.2004.06.012

    MathSciNet  MATH  Google Scholar 

  25. Colliot-Thélène, J-L, Hoobler, RT., Kahn, B: The Bloch-Ogus-Gabber theorem. In: Algebraic K-theory (Toronto, ON 1996) Fields Inst. Commun. MR1466971, vol. 16, pp 31–94. Amer. Math. Soc., Providence (1997)

  26. Colliot-Thélène, J.-L., Harari, D., Skorobogatov, A. N.: Compactification équivariante d’un tore (d’après Brylinski et Künnemann). Expo. Math. 23(2), 161–170 (2005). MR2155008, https://doi.org/10.1016/j.exmath.2005.01.016

    MathSciNet  MATH  Google Scholar 

  27. Colliot-Thélène, J-L, Ojanguren, M: Espaces principaux homogènes localement triviaux. Inst. Hautes Études Sci. Publ. Math. 75, 97–122 (1992). MR1179077

    MATH  Google Scholar 

  28. Colliot-Thélène, J-L, Sansuc, J-J: Cohomologie des groupes de type multiplicatif sur les schémas réguliers. C. R. Acad. Sci. Paris Sér. A-B 287(6), A449–A452 (1978). MR510771

    MATH  Google Scholar 

  29. Colliot-Thélène, J.-L., Sansuc, J.-J.: Fibrés quadratiques et composantes connexes réelles. Math. Ann. 244(2), 105–134 (1979). MR550842, https://doi.org/10.1007/BF01420486

    MathSciNet  MATH  Google Scholar 

  30. Colliot-Thélène, J-L, Sansuc, J-J: Principal homogeneous spaces under flasque tori: applications. J. Algebra 106(1), 148–205 (1987). MR878473, https://doi.org/10.1016/0021-8693(87)90026-3

    MathSciNet  MATH  Google Scholar 

  31. Česnavičius, K: Purity for the Brauer group. Duke Math. J. 168 (8), 1461–1486 (2019). MR3959863, https://doi.org/10.1215/00127094-2018-0057

    MathSciNet  MATH  Google Scholar 

  32. Česnavičius, K: Grothendieck–Serre in the quasi-split unramified case. Forum Math. Pi 10, e9, 30 pp (2022). https://doi.org/10.1017/fmp.2022.5

  33. Česnavičius, K, Scholze, P: Purity for flat cohomology. Available at 1912.10932v2 (2021)

  34. Danilov, V. I.: The geometry of toric varieties. Uspekhi Mat. Nauk 33(2(200)), 85–134, 247 (1978). MR495499

    MathSciNet  Google Scholar 

  35. Deligne, P, Milne, JS., Ogus, A, Shih, K-y: Hodge Cycles, Motives, and Shimura Varieties. Lecture Notes in Mathematics, vol. 900, p ii+ 414. Springer, Berlin (1982). MR654325

    Google Scholar 

  36. Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique. I. Le langage des schémas. Inst. Hautes Études Sci. Publ. Math. 4, 228 (1960). MR0217083 (36 #177a)

    MATH  Google Scholar 

  37. Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Inst. Hautes Études Sci. Publ. Math. 8, 222 (1961). MR0163909 (29 #1208)

    Google Scholar 

  38. Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I. Inst. Hautes Études Sci. Publ. Math. 11, 167 (1961). MR0217085 (36 #177c)

    Google Scholar 

  39. Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II. Inst. Hautes Études Sci. Publ. Math. 24, 231 (1965). MR0199181 (33 #7330)

    MATH  Google Scholar 

  40. Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III. Inst. Hautes Études Sci. Publ. Math. 28, 255 (1966). MR0217086 (36 #178)

    MATH  Google Scholar 

  41. Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV. Inst. Hautes Études Sci. Publ. Math. 32, 361 (1967). MR0238860 (39 #220)

    MATH  Google Scholar 

  42. Elmanto, E, Kulkarni, G, Wendt, M: \(\mathbb {A}^{1}\)-connected components of classifying spaces and purity for torsors. Available at https://arxiv.org/abs/2104.06273v1 (2021)

  43. Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73 (3), 349–366 (1983). MR718935 (85g:11026a), https://doi.org/10.1007/BF01388432

    MathSciNet  MATH  Google Scholar 

  44. Faltings, G, Chai, C-L: Degeneration of Abelian Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22, p xii+ 316. Springer, Berlin (1990). MR1083353 (92d:14036)

    Google Scholar 

  45. Fedorov, R: Affine Grassmannians of group schemes and exotic principal bundles over A1. Amer. J. Math. 138 (4), 879–906 (2016). MR3538146, https://doi.org/10.1353/ajm.2016.0036

    MathSciNet  MATH  Google Scholar 

  46. Fedorov, R: On the Grothendieck–Serre Conjecture about principal bundles and its generalizations. Algebra Number Theory, to appear. Available at 1810.11844v2 (2021)

  47. Fedorov, R: On the Grothendieck–Serre conjecture on principal bundles in mixed characteristic. Trans. Amer. Math. Soc. 375(01), 559–586 (2021). MR4358676, https://doi.org/10.1090/tran/8490

    MathSciNet  MATH  Google Scholar 

  48. Fedorov, R: On the purity conjecture of Nisnevich for torsors under reductive group schemes. Available at 2109.10332 (2021)

  49. First, UA.: An 8-Periodic Exact Sequence of Witt Groups of Azumaya Algebras with Involution. Available at 1910.03232v2 (2021)

  50. Fedorov, R, Panin, I: A proof of the Grothendieck-Serre conjecture on principal bundles over regular local rings containing infinite fields. Publ. Math. Inst. Hautes Études Sci. 122, 169–193 (2015). MR3415067, https://doi.org/10.1007/s10240-015-0075-z

    MathSciNet  MATH  Google Scholar 

  51. Ferrand, D, Raynaud, M: Fibres formelles d’un anneau local noethérien. Ann. Sci. École Norm. Sup. (4) 3, 295–311 (1970). MR0272779

    MathSciNet  MATH  Google Scholar 

  52. González-Avilés, CD.: Flasque resolutions of reductive group schemes. Cent. Eur. J. Math. 11(7), 1159–1176 (2013). MR3085137, https://doi.org/10.2478/s11533-013-0235-7

    MathSciNet  MATH  Google Scholar 

  53. Gabber, O: Some theorems on Azumaya algebras. In: The Brauer group. Sem., Les Plans-sur-Bex (1980). Lecture Notes in Math. MR611868 (83d:13004), vol. 844, pp 129–209. Springer, Berlin (1981)

  54. Gabber, O: Gersten’s conjecture for some complexes of vanishing cycles. Manuscripta Math. 85(3–4), 323–343 (1994). MR1305746, https://doi.org/10.1007/BF02568202

    MathSciNet  MATH  Google Scholar 

  55. Gabber, O.: On space filling curves and Albanese varieties. Geom. Funct. Anal. 11(6), 1192–1200 (2001). MR1878318, https://doi.org/10.1007/s00039-001-8228-2

    MathSciNet  MATH  Google Scholar 

  56. Gabber, O: On purity theorems for vector bundles. Int. Math. Res. Not. 15, 783–788 (2002). MR1891173, https://doi.org/10.1155/S1073792802110087

    MathSciNet  MATH  Google Scholar 

  57. Gille, P.: Torseurs sur la droite affine. Transform. Groups 7(3), 231–245 (2002). MR1923972, https://doi.org/10.1007/s00031-002-0012-3

    MathSciNet  MATH  Google Scholar 

  58. Gille, P.: Errata: “Torsors on the affine line” (French) [Transform. Groups 7 (2002), no. 3, 231–245; MR1923972]. Transform. Groups 10(2), 267–269 (2005). MR2195603, https://doi.org/10.1007/s00031-005-1010-z

    MathSciNet  Google Scholar 

  59. Gille, P: Le problème de Kneser-Tits. Astérisque 326, Exp. No. 983, vii, 39–81 (2010) (2009). Séminaire Bourbaki. Vol. 2007/2008. 2605318

    MATH  Google Scholar 

  60. Gille, P: When is a reductive group scheme linear? Michigan Math. J., to appear. Available at https://arxiv.org/abs/2103.07305v3 (2021)

  61. Giraud, J: Cohomologie non abélienne, p ix+ 467. Springer, Berlin (1971). MR0344253 (49 #8992)

    MATH  Google Scholar 

  62. Gabber, O, Ramero, L: Foundations for almost ring theory. preprint, arXiv version 13. Available at https://arxiv.org/abs/math/0409584v13 (2018)

  63. Grothendieck, A: Torsion homologique et sections rationnelles. Seminaire Claude Chevalley 3(5), 1–29 (1958)

    Google Scholar 

  64. Grothendieck, A: Le groupe de Brauer. II. Théorie cohomologique. In: Dix Exposés sur la Cohomologie des Schémas. MR0244270 (39 #5586b), pp 67–87, North-Holland (1968)

  65. Gros, M, Suwa, N: La conjecture de Gersten pour les faisceaux de Hodge-Witt logarithmique. Duke Math. J. 57(2), 615–628 (1988). MR962522, https://doi.org/10.1215/S0012-7094-88-05727-4

    MathSciNet  MATH  Google Scholar 

  66. Guo, N: The Grothendieck–Serre conjecture over semilocal Dedekind rings. Transform. Groups, to appear. Available at https://arxiv.org/abs/1902.02315v3 (2020)

  67. Guo, N: The Grothendieck–Serre conjecture over valuation rings. Available at https://arxiv.org/abs/2008.02767v4 (2021)

  68. Görtz, U, Wedhorn, T: Algebraic geometry I. Schemes—with examples and exercises. Springer Studium Mathematik—Master, 2nd edn., p vii+ 625. Springer, Spektrum (2020). MR4225278, https://doi.org/10.1007/978-3-658-30733-2

    MATH  Google Scholar 

  69. de Jong, A. J., He, X, Starr, JM: Families of rationally simply connected varieties over surfaces and torsors for semisimple groups. Publ. Math. Inst. Hautes Études Sci. 114, 1–85 (2011). MR2854858, https://doi.org/10.1007/s10240-011-0035-1

    MathSciNet  MATH  Google Scholar 

  70. Haines, TJ., Lourenço, J, Richarz, T: On the normality of Schubert varieties: remaining cases in positive characteristic. Available at https://arxiv.org/abs/1806.11001v4 (2020)

  71. Horrocks, G.: Projective modules over an extension of a local ring. Proc. London Math. Soc. (3) 14, 714–718 (1964). MR0169878

    MathSciNet  MATH  Google Scholar 

  72. Hall, J, Rydh, D: Coherent Tannaka duality and algebraicity of Hom-stacks. Algebra Number Theory 13(7), 1633–1675 (2019). MR4009673, https://doi.org/10.2140/ant.2019.13.1633

    MathSciNet  MATH  Google Scholar 

  73. Haines, TJ., Richarz, T: Normality and Cohen–Macaulayness of parahoric local models. J. Eur. Math. Soc. (JEMS), to appear. Available at https://arxiv.org/abs/1903.10585v4 (2021)

  74. Kottwitz, RE.: Stable trace formula: elliptic singular terms. Math. Ann. 275(3), 365–399 (1986). MR858284, https://doi.org/10.1007/BF01458611

    MathSciNet  MATH  Google Scholar 

  75. Lam, T. Y.: Serre’s Problem on Projective Modules. Springer Monographs in Mathematics, p xxii+ 401. Springer, Berlin (2006). MR2235330

    Google Scholar 

  76. Li, S: The Bass–Quillen conjecture for general groups in equal characteristic Mémoire de Master 2, Sorbonne Université (2021)

  77. Lindel, H: On the Bass-Quillen conjecture concerning projective modules over polynomial rings. Invent. Math. 65(2), 319–323 (1981/82). MR641133, https://doi.org/10.1007/BF01389017

    MathSciNet  MATH  Google Scholar 

  78. Lequain, Y, Simis, A: Projective modules over R[X1,⋯,Xn], R a Prüfer domain. J. Pure Appl. Algebra 18(2), 165–171 (1980). MR585221

    MathSciNet  MATH  Google Scholar 

  79. Matsumura, H: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, 2nd edn., vol. 8, p xiv+ 320. Cambridge University Press, Cambridge (1989). MR1011461 (90i:13001)

    Google Scholar 

  80. Moret-Bailly, L: Un problème de descente. Bull. Soc. Math. France 124(4), 559–585 (1996). MR1432058

    MathSciNet  MATH  Google Scholar 

  81. Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 3rd edn., vol. 34, p xiv+ 292. Springer, Berlin (1994). MR1304906 (95m:14012), https://doi.org/10.1007/978-3-642-57916-5

    Google Scholar 

  82. Moser, L-F: Rational triviale Torseure und die Serre-Grothendiecksche Vermutung. Diplomarbeit, Mathematisches Institut der Ludwig-Maximilians Universität München (2008)

  83. Nisnevich, Y.A.: Etale cohomology and arithmetic of semisimple groups. Thesis (Ph.D.)–Harvard University. ProQuest LLC, Ann Arbor, MI. MR2632405 (1982)

  84. Nisnevich, YA.: Espaces homogènes principaux rationnellement triviaux et arithmétique des schémas en groupes réductifs sur les anneaux de Dedekind. C. R. Acad. Sci. Paris Sér. I Math. 299(1), 5–8 (1984). MR756297

    MathSciNet  MATH  Google Scholar 

  85. Nisnevich, Y: Rationally trivial principal homogeneous spaces, purity and arithmetic of reductive group schemes over extensions of two-dimensional regular local rings. C. R. Acad. Sci. Paris Sér. I Math. 309(10), 651–655 (1989). MR1054270

    MathSciNet  MATH  Google Scholar 

  86. Nisnevich, Y: Stratified canonical forms of matrix valued functions in a neighborhood of a transition point. Internat. Math. Res. Notices 10, 513–527 (1998). MR1634912, https://doi.org/10.1155/S1073792898000336

    MathSciNet  MATH  Google Scholar 

  87. Oda, T: Convex Bodies and Algebraic Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, p viii+ 212. Springer, Berlin (1988). An introduction to the theory of toric varieties; Translated from the Japanese. 922894

    Google Scholar 

  88. Ojanguren, M: Unités représentées par des formes quadratiques ou par des normes réduites. In: Algebraic K-theory, Part II (Oberwolfach, 1980). Lecture Notes in Math. MR689397, vol. 967, pp 291–299. Springer, Berlin (1982)

  89. Ojanguren, M, Panin, I: A purity theorem for the Witt group. Ann. Sci. École Norm. Sup. (4) 32(1), 71–86 (1999). MR1670591, https://doi.org/10.1016/S0012-9593(99)80009-3

    MathSciNet  MATH  Google Scholar 

  90. Ojanguren, M, Panin, I: Rationally trivial Hermitian spaces are locally trivial. Math. Z. 237(1), 181–198 (2001). MR1836777, https://doi.org/10.1007/PL00004859

    MathSciNet  MATH  Google Scholar 

  91. Panin, I: Rationally isotropic quadratic spaces are locally isotropic. Invent. Math. 176(2), 397–403 (2009). MR2495767, https://doi.org/10.1007/s00222-008-0168-0

    MathSciNet  MATH  Google Scholar 

  92. Panin, I. A.: Purity conjecture for reductive groups. Vestnik St. Petersburg Univ. Math. 43(1), 44–48 (2010). MR2662409, https://doi.org/10.3103/S1063454110010085

    MathSciNet  MATH  Google Scholar 

  93. Panin, I: On Grothendieck-Serre conjecture concerning principal bundles. In: Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures. MR3966763, pp 201–221. World Sci. Publ., Hackensack (2018)

  94. Panin, I. A.: Nice triples and moving lemmas for motivic spaces. Izv. Math. 83(4), 796–829 (2019). MR3985694, https://doi.org/10.1070/IM8819

    MathSciNet  MATH  Google Scholar 

  95. Panin, I. A.: Proof of the Grothendieck–Serre conjecture on principal bundles over regular local rings containing a field. Izv. Ross. Akad. Nauk Ser. Mat. 84 (4), 169–186 (2020). (Russian); English transl., Izv. Math. 84 (2020), no. 4, 780–795. 4133391, https://doi.org/10.4213/im8982

    MathSciNet  MATH  Google Scholar 

  96. Panin, I. A.: Two purity theorems and the Grothendieck–Serre’s conjecture concerning principal G-bundles. Mat. Sb. 211(12), 123–142 (2020). MR4181078, https://doi.org/10.4213/sm9393

    MathSciNet  Google Scholar 

  97. Parimala, S.: Failure of a quadratic analogue of Serre’s conjecture. Amer. J. Math. 100(5), 913–924 (1978). MR517136, https://doi.org/10.2307/2373953

    MathSciNet  MATH  Google Scholar 

  98. Poonen, B: Bertini theorems over finite fields. Ann. of Math. (2) 160(3), 1099–1127 (2004). MR2144974, https://doi.org/10.4007/annals.2004.160.1099

    MathSciNet  MATH  Google Scholar 

  99. Popescu, D: Polynomial rings and their projective modules. Nagoya Math. J. 113, 121–128 (1989). MR986438

    MathSciNet  MATH  Google Scholar 

  100. Popescu, D: Letter to the editor: “General Néron desingularization and approximation” [Nagoya Math. J. 104 (1986), 85–115; MR0868439 (88a:14007)]. Nagoya Math. J. 118, 45–53 (1990). MR1060701, https://doi.org/10.1017/S0027763000002981

    MathSciNet  Google Scholar 

  101. Popescu, D: Around general Neron desingularization. J. Algebra Appl. 16(4), 1750072, 10 (2017). MR3635121

    MathSciNet  MATH  Google Scholar 

  102. Popescu, D: On a question of Swan. Algebr. Geom. 6(6), 716–729 (2019). MR4009178

    MathSciNet  MATH  Google Scholar 

  103. Panin, I, Pimenov, K: Rationally isotropic quadratic spaces are locally isotropic: II. Doc. Math. Extra vol.: Andrei A. Suslin sixtieth birthday. MR2804263, 515–523 (2010)

    MathSciNet  MATH  Google Scholar 

  104. Panin, I., Pimenov, K.: Rationally isotropic quadratic spaces are locally isotropic. III. Algebra i Analiz 27(6), 234–241 (2015). English transl., St. Petersburg Math. J. 27 (2016), no. 6, 1029–1034. MR3589229, https://doi.org/10.1090/spmj/1433

    MathSciNet  MATH  Google Scholar 

  105. Pappas, G., Rapoport, M.: Twisted loop groups and their affine flag varieties. Adv. Math. 219(1), 118–198 (2008). With an appendix by T. Haines and Rapoport. MR2435422, https://doi.org/10.1016/j.aim.2008.04.006

    MathSciNet  MATH  Google Scholar 

  106. Panin, I., Stavrova, A., Vavilov, N.: On Grothendieck-Serre’s conjecture concerning principal G-bundles over reductive group schemes: I. Compos. Math. 151 (3), 535–567 (2015). MR3320571, https://doi.org/10.1112/S0010437X14007635

    MathSciNet  MATH  Google Scholar 

  107. Quillen, D: Higher algebraic K-theory. I. In: Algebraic K-theory, I: Higher K-theories. Proc. Conf., Battelle Memorial Inst., Seattle, Wash. (1972). MR0338129, pp 85–147. Lecture Notes in Math., Vol. 341. Springer, Berlin (1973)

  108. Quillen, D: Projective modules over polynomial rings. Invent. Math. 36, 167–171 (1976). MR0427303, https://doi.org/10.1007/BF01390008

    MathSciNet  MATH  Google Scholar 

  109. Raghunathan, M. S.: Principal bundles on affine space and bundles on the projective line. Math. Ann. 285(2), 309–332 (1989). MR1016097, https://doi.org/10.1007/BF01443521

    MathSciNet  MATH  Google Scholar 

  110. Rao, Ravi A.: On projective \(R_{f_{1}{\cdots } f_{t}}\)-modules. Amer. J. Math. 107(2), 387–406 (1985). MR784289

    MathSciNet  Google Scholar 

  111. Rao, R.A.: The Bass-Quillen conjecture in dimension three but characteristic ≠ 2, 3 via a question of A. Suslin. Invent. Math. 93(3), 609–618 (1988). MR952284

    MathSciNet  MATH  Google Scholar 

  112. Raynaud, M: Faisceaux amples sur les schémas en groupes et les espaces homogènes. Lecture Notes in Mathematics, vol. 119, p ii+ 218. Springer, Berlin (1970). MR0260758 (41 #5381)

    Google Scholar 

  113. Roitman, M: On projective modules over polynomial rings. J. Algebra 58(1), 51–63 (1979). MR535842, https://doi.org/10.1016/0021-8693(79)90188-1

    MathSciNet  MATH  Google Scholar 

  114. Raghunathan, M. S., Ramanathan, A.: Principal bundles on the affine line. Proc. Indian Acad. Sci. Math. Sci. 93(2–3), 137–145 (1984). MR813075, https://doi.org/10.1007/BF02840656

    MathSciNet  MATH  Google Scholar 

  115. Scully, S: The Artin-Springer theorem for quadratic forms over semi-local rings with finite residue fields. Proc. Amer. Math. Soc. 146(1), 1–13 (2018). MR3723116, https://doi.org/10.1090/proc/13744

    MathSciNet  MATH  Google Scholar 

  116. Serre, J.-P.: Espaces fibrés algébriques. Seminaire Claude Chevalley 3(1), 1–37 (1958)

    Google Scholar 

  117. Serre, J-P: Galois cohomology. Springer Monographs in Mathematics. Corrected reprint of the 1997 English edition, p x + 210. Springer, Berlin (2002). Translated from the French by Patrick Ion and revised by the author. MR1867431 (2002i:12004)

    Google Scholar 

  118. Grothendieck, A: Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2). Documents Mathématiques (Paris) [Mathematical Documents (Paris)], vol. 4. Séminaire de Géométrie Algébrique du Bois Marie, 1962; Augmenté d’un exposé de Michèle Raynaud. [With an exposé by Michèle Raynaud]; With a preface and edited by Yves Laszlo; Revised reprint of the 1968 French original/ Société Mathématique de France, Paris, x + 208. MR2171939 (2005)

  119. Gille, P, Polo, P (eds.): Schémas en groupes (SGA 3). Tome I. Propriétés générales des schémas en groupes/ Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 7. Société Mathématique de France, Paris (2011). Séminaire de Géométrie Algébrique du Bois Marie 1962–64. [Algebraic Geometry Seminar of Bois Marie 1962–64]; A seminar directed by M. Demazure and A. Grothendieck with the collaboration of M. Artin, J.-E. Bertin, P. Gabriel, M. Raynaud and J-P. Serre; Revised and annotated edition of the 1970 French original. MR2867621

    Google Scholar 

  120. Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux. Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, vol. 152. Springer, Berlin. ix+ 654, MR0274459 (43 #223b) (1970)

  121. Gille, P, Polo, P (eds.): Schémas en groupes (SGA 3). Tome III. Structure des schémas en groupes réductifs. Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 8. Société Mathématique de France, Paris (2011). Séminaire de Géométrie Algébrique du Bois Marie 1962–64. [Algebraic Geometry Seminar of Bois Marie 1962–64]; A seminar directed by M. Demazure and A. Grothendieck with the collaboration of M. Artin, J.-E. Bertin, P. Gabriel, M. Raynaud and J-P. Serre; Revised and annotated edition of the 1970 French original. MR2867622

    Google Scholar 

  122. Théorie des topos et cohomologie étale des schémas. Tome 3. Lecture Notes in Mathematics, vol. 305. Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat. Springer, Berlin. vi+ 640. MR0354654 (50 #7132) (1973)

  123. de Jong, A.J., et al.: The Stacks Project. Available at http://stacks.math.columbia.edu

  124. Suslin, A. A.: Projective modules over polynomial rings are free. Dokl. Akad. Nauk SSSR 229(5), 1063–1066 (1976). MR0469905

    MathSciNet  Google Scholar 

  125. Swan, R.G.: Projective modules over Laurent polynomial rings. Trans. Amer. Math. Soc. 237, 111–120 (1978). MR0469906, https://doi.org/10.2307/1997613

    MathSciNet  MATH  Google Scholar 

  126. Swan, R.G.: On seminormality. J. Algebra 67(1), 210–229 (1980). MR595029, https://doi.org/10.1016/0021-8693(80)90318-X

    MathSciNet  MATH  Google Scholar 

  127. Swan, R.G.: A simple proof of Gabber’s theorem on projective modules over a localized local ring. Proc. Amer. Math. Soc. 103(4), 1025–1030 (1988). MR954977, https://doi.org/10.2307/2047079

    MathSciNet  MATH  Google Scholar 

  128. Swan, R.G.: Néron-Popescu desingularization. In: Algebra and geometry (Taipei, 1995). Lect. Algebra Geom. MR1697953 (2000h:13006), vol. 2, pp 135–192. Int. Press, Cambridge (1998)

  129. Teissier, B: Résultats récents sur l’approximation des morphismes en algèbre commutative (d’après André, Artin, Popescu et Spivakovsky). Astérisque 227, Exp. No. 784, 4, 259–282 (1995). Séminaire Bourbaki, vol. 1993/94. MR1321650

    MATH  Google Scholar 

  130. Thomason, R. W.: Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes. Adv. in Math. 65(1), 16–34 (1987). MR893468

    MathSciNet  MATH  Google Scholar 

  131. Tsybyshev, A. E.: A step toward the non-equicharacteristic Grothendieck-Serre conjecture. Algebra i Analiz 31(1), 246–254 (2019). (Russian, with Russian summary); English transl., St. Petersburg Math. J. 31 (2020), no. 1, 181–187. MR3932824, https://doi.org/10.1090/spmj/1591

    MathSciNet  Google Scholar 

  132. Temkin, M., Tyomkin, I.: Ferrand pushouts for algebraic spaces. Eur. J. Math. 2(4), 960–983 (2016). MR3572553, https://doi.org/10.1007/s40879-016-0115-3

    MathSciNet  MATH  Google Scholar 

  133. Vasiu, A.: Extension theorems for reductive group schemes. Algebra Number Theory 10(1), 89–115 (2016). MR3463037, https://doi.org/10.2140/ant.2016.10.89

    MathSciNet  MATH  Google Scholar 

  134. Yengui, I.: The Hermite ring conjecture in dimension one. J. Algebra 320(1), 437–441 (2008). MR2417998, https://doi.org/10.1016/j.jalgebra.2008.02.007

    MathSciNet  MATH  Google Scholar 

  135. Zhu, X.: An introduction to affine Grassmannians and the geometric Satake equivalence. In: Geometry of moduli spaces and representation theory. IAS/Park City Math. Ser. MR3752460, vol. 24, pp 59–154. Amer. Math. Soc., Providence, RI (2017)

Download references

Acknowledgements

I thank Viện Toán Học for the invitation to contribute to the special issue of Acta Mathematica Vietnamica. I thank Yifei Zhao for the appendix and for helpful comments on the main body of the text. I thank the referee for helpful comments and suggestions. I thank Alexis Bouthier, Jean-Louis Colliot-Thélène, Sean Cotner, Roman Fedorov, Ofer Gabber, Ning Guo, Shang Li, Ivan Panin, Federico Scavia, Yifei Zhao, and many others for useful conversations and correspondence related to the subject of this article. This project received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation program (grant agreement no. 851146).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kęstutis Česnavičius.

Additional information

Dedicated to Professor Nguyen Tu Cuong on the occasion of his 70th birthday

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix. Resolutions of reductive groups

Appendix. Resolutions of reductive groups

Yifei ZhaoFootnote 10

This appendix is an exposition on the construction of flasque and coflasque resolutions of a reductive group G over a general base scheme S, subject only to the condition that rad(G) be isotrivial.

The notions of flasque and coflasque tori are due to Colliot-Thélène–Sansuc [30]. The existence of a coflasque resolution strengthens that of a z-extension of Langlands and Kottwitz [74, Section 1], which is often stated over a field of characteristic zero ([35, Chapter V, Section 3], [13], for example). When the base is a field of arbitrary characteristic, both resolutions are constructed by Colliot-Thélène in [24], but, as observed by González-Avilés [52], the same proof yields the existence of flasque resolutions over locally Noetherian, geometrically unibranch schemes (e.g., a normal scheme). Our proof follows Colliot-Thélène’s strategy, but we replace the hypotheses on S by the isotriviality of rad(G), which holds whenever S is locally Noetherian and geometrically unibranch but could remain valid in other contexts.

Sections A.2A.3 are preparatory and the main construction appears as Theorem A.4.1. As an application, we explain a simple reduction of the Grothendieck–Serre conjecture 3.1.1 to the case when the derived subgroup is simply connected, see Proposition A.5.1. The author thanks K. Česnavičius for many helpful conversations and comments.

A.1 Group schemes of multiplicative type

In this section, we review group schemes of multiplicative type. The most important notion for us is the isotriviality of such group schemes.

A.1.1

Let S be a scheme. For an fppf sheaf of abelian groups \({\mathscr{F}}\) over S, one may consider the fppf sheaf \(\mathbb D_{S}({\mathscr{F}})\) whose value at an affine S-scheme \(S^{\prime }\) is \(\text {Hom}({\mathscr{F}}_{S^{\prime }}, \mathbb G_{m, S^{\prime }})\). Here, Hom is viewed in the category of fppf sheaves of abelian groups over \(S^{\prime }\). The fppf sheaf \(\mathbb D_{S}({\mathscr{F}})\) again takes values in abelian groups, the group structure being inherited from \(\mathbb G_{m}\).

A.1.2

An S-group scheme G is diagonalizable if there exist a finitely generated abelian group M and an isomorphism between G and the group scheme \(\mathbb D_{S}(M_{S})\), where MS denotes the constant sheaf with values in M. An S-group scheme G is of multiplicative type if it is diagonalizable fppf locally on S.Footnote 11 In fact, every S-group scheme of multiplicative type is diagonalizable étale locally on S ([120, Exposé X, Corollaire 4.5] or [20, Proposition B.3.4]).

If an S-group scheme G of multiplicative type becomes diagonalizable after base change along \(\widetilde S\rightarrow S\), then G is said to be split by \(\widetilde S\). If S is connected, then any S-group scheme of multiplicative type is split by some fppf (equivalently, étale) surjection \(\widetilde S\rightarrow S\) ([120, Exposé IX, Remarque 1.4.1]). By fppf descent, any S-group scheme G of multiplicative type is S-affine.

An S-group G of multiplicative type is a torus if fppf (equivalently, étale) locally on S it is of the form \(\mathbb D_{S}(M_{S})\) for some finitely generated free abelian group M.

A.1.3

Groups of multiplicative type enjoy certain closure properties:

(1) an S-flat, finitely presented closed subgroup of an S-group scheme of multiplicative type is again of multiplicative type ([120, Exposé X, Corollaire 4.7 b)] or [20, Corollary B.3.3]);

(2) a commutative extension of group schemes of multiplicative type is again of multiplicative type ([120, Exposé XVII, Proposition 7.1.1] or [20, Corollary B.4.2]).

Furthermore, an S-group scheme G of multiplicative type is reflexive in the sense that the natural transformation \(G \rightarrow \mathbb D_{S}(\mathbb D_{S}(G))\) is an isomorphism. Indeed, this statement may be verified fppf locally on S, where it follows from [120, Exposé VIII, Théorème 1.2].

A.1.4

An S-group scheme G of multiplicative type is called isotrivial if G is split by a finite étale surjection \(\widetilde S\rightarrow S\). When S is connected, an isotrivial S-group scheme of multiplicative type is split by a finite connected étale Galois cover \(\widetilde S\rightarrow S\). We discuss some ways to obtain isotrivial S-group schemes of multiplicative type.

Lemma A.1.5

Let S be a connected scheme. Then any finite S-group scheme G of multiplicative type is isotrivial.

Proof

Let \({\mathscr{F}}\) denote the fppf sheaf of abelian groups \(\mathbb D_{S}(G)\). Since S is connected, there is an fppf surjection \(\widetilde S\rightarrow S\) such that \({\mathscr{F}}_{\widetilde S}\) is isomorphic to the constant sheaf \(M_{\widetilde S}\) for a finite abelian group M. The descent data of \({\mathscr{F}}_{\widetilde S}\) allow us to construct an Aut(M)-torsor \({\mathscr{P}}\) over S such that \({\mathscr{F}}\) is the fppf sheaf of abelian groups induced from \({\mathscr{P}}\). Since Aut(M) is finite, \({\mathscr{P}}\) is representable by a finite étale surjection \(\widetilde S^{\prime }\rightarrow S\). In particular, \({\mathscr{P}}\) is trivialized by \(\widetilde S^{\prime }\). It follows that G is split by \(\widetilde S^{\prime }\). □

The same argument proves more generally that an S-group scheme of multiplicative type whose maximal torus has rank ≤ 1 is isotrivial. This fact can be compared with Lemma A.1.6 (ii) below.

Lemma A.1.6

Let S be a connected scheme. Given a short exact sequence of S-group schemes of multiplicative type:

$$ 1 \rightarrow G_{1} \rightarrow G \rightarrow G_{2} \rightarrow 1, $$

(i) if G is diagonalizable (resp. isotrivial), then both G1 and G2 are diagonalizable (resp. isotrivial);

(ii) if G1 is isotrivial and G2 is finite, then G is isotrivial.

Proof

Statement (i) is established in [120, Exposé IX, Proposition 2.11]. To prove statement (ii), we may assume that both G1 and G2 are diagonalizable by replacing S with a connected finite étale cover. Since \(\mathbb D_{S}\) is an anti-equivalence on reflexive fppf sheaves of abelian groups [120, Exposé VIII, Proposition 1.0.1], it restricts to an exact functor on the full subcategory of S-group schemes of multiplicative type. In particular, we obtain a short exact sequence of fppf sheaves of abelian groups:

$$ 1 \rightarrow M_{2, S} \rightarrow \mathbb D_{S}(G) \rightarrow M_{1, S} \rightarrow 1. $$

Here, Mi, S (for i = 1,2) denotes the constant sheaf associated to a finitely generated \(\mathbb Z\)-module Mi. The finiteness of G2 allows us to assume that M2 is finite.

It remains to show that any class in \(\text {Ext}^{1}_{\text {fppf}}(M_{1,S}, M_{2,S})\) comes from \(\text {Ext}^{1}_{\mathbb Z}(M_{1}, M_{2})\) after passing to a finite étale cover \(\widetilde S\rightarrow S\). For this statement, it suffices to treat the case where M1 is a cyclic group. For \(M_{1} {=} \mathbb Z\), we have \(\text {Ext}^{1}_{\text {fppf}}(\mathbb Z_{S}, M_{2, S}) {\cong } H^{1}_{\text {fppf}}(S, M_{2}) {\cong } H^{1}_{\acute {\text {e}}\text {t}}(S, M_{2})\), and because M2 is finite, any class in \(H^{1}_{\acute {\text {e}}\text {t}}(S, M_{2})\) vanishes over a finite étale surjection \(\widetilde S\rightarrow S\). For \(M_{1} = \mathbb Z/n\) for an integer n ≥ 1, we have an exact sequence:

$$ \text{Hom}(\mathbb Z, M_{2}) \xrightarrow{n} \text{Hom}(\mathbb Z, M_{2}) \rightarrow \text{Ext}^{1}_{\text{fppf}}((\mathbb Z/n)_{S}, M_{2,S}) \rightarrow \text{Ext}^{1}_{\text{fppf}}(\mathbb Z_{S}, M_{2,S}). $$

By the same argument as above, any class in \(\text {Ext}^{1}_{\text {fppf}}((\mathbb Z/n)_{S}, M_{2,S})\) has zero image in \(\text {Ext}^{1}_{\text {fppf}}(\mathbb Z_{S}, M_{2, S})\) after passing to a connected finite étale cover \(\widetilde S\rightarrow S\). Equivalently, this means that over \(\widetilde S\), the class comes from \(\text {Ext}^{1}_{\mathbb Z}(\mathbb Z/n, M_{2})\). □

Remark A.1.7

In Lemma A.1.6 (ii), the finiteness hypothesis on G2 cannot be dropped. Indeed, whenever \(H^{1}_{\acute {\text {e}}\text {t}}(S, \mathbb Z)\neq 0\), there exist self-extensions of \(\mathbb G_{m}\) which are not isotrivial. To see this, we use the isomorphism \(\text {Ext}^{1}_{\text {fppf}}(\mathbb G_{m}, \mathbb G_{m}) \cong H^{1}_{\acute {\text {e}}\text {t}}(S, \mathbb Z)\) and the fact that any class of \(H^{1}_{\acute {\text {e}}\text {t}}(S, \mathbb Z)\) which vanishes on a finite étale cover of S must already be zero (because \(H^{1}_{\acute {\text {e}}\text {t}}(S, \mathbb {Z}) \hookrightarrow H^{1}_{\acute {\text {e}}\text {t}}(S, \mathbb {Q})\)).

A.1.8

There is a convenient condition on the base scheme S which guarantees that all multiplicative type S-group schemes are isotrivial.

A local ring R is geometrically unibranch if its strict Henselization Rsh has a unique minimal prime, see [123, Definition 0BPZand Lemma 06DM]. A scheme S is geometrically unibranch if so are its local rings. For example, a normal scheme (in the usual sense that its local rings are normal domains) is geometrically unibranch. Every connected component of a locally Noetherian, geometrically unibranch scheme is irreducible (see [68, Exercise 3.16 (a)]).

Let S be a locally Noetherian, geometrically unibranch scheme. By [120, Exposé X, Théorème 5.16], every S-group scheme G of multiplicative type splits over a finite étale surjection \(\widetilde S \rightarrow S\). When S is connected, we may further assume that \(\widetilde S \rightarrow S\) is a connected Galois cover.

A.2 Flasque and coflasque tori

In this section, we focus on isotrivial tori. The study of these objects is equivalent to that of Galois modules with integral coefficients. We discuss several conditions on isotrivial tori (quasi-trivial, flasque, and coflasque) which are “of Galois cohomology nature.”

A.2.1

Suppose that S is a connected scheme and let \(\widetilde S \rightarrow S\) be a connected finite étale Galois cover. By [120, Exposé X, Proposition 1.1], the construction \(\mathbb D_{\widetilde S}\) induces an equivalence of categories between

(1) group schemes \(G\rightarrow S\) of multiplicative type split by \(\widetilde S\);

(2) finitely generated \(\mathbb Z\)-modules M equipped with a \(\text {Gal}(\widetilde S/S)\)-action.

Under this equivalence, tori \(T\rightarrow S\) split by \(\widetilde S\) correspond to finitely generated free \(\mathbb Z\)-modules M equipped with a \(\text {Gal}(\widetilde S/S)\)-action—such an M is called the character lattice of T, and we denote it by \(\check {{{\varLambda }}}_{T, \widetilde S}\). Its \(\mathbb Z\)-linear dual is called the cocharacter lattice of T, which we denote by \({{\varLambda }}_{T, \widetilde S}\).

A.2.2

Let Γ be a finite group. A Γ-lattice Λ, i.e., a finitely generated free \(\mathbb Z\)-module equipped with a Γ-action, is called quasi-trivial if it has a Γ-stable \(\mathbb Z\)-basis. Clearly, Λ is quasi-trivial if and only if its \(\mathbb Z\)-linear dual \(\check {{{\varLambda }}} := \text {Hom}_{\mathbb {Z}}({{\varLambda }}, \mathbb {Z})\), equipped with the contragredient Γ-action, is quasi-trivial.

The following lemma describes the conditions that end up defining flasque tori.

Lemma A.2.3

Let Γ be a finite group and let Λ be a lattice equipped with a Γ-action. The following conditions are equivalent:

(i) \(H^{1}({{\varGamma }}^{\prime }, {{\varLambda }}) = 0\) for any subgroup \({{\varGamma }}^{\prime } \subset {{\varGamma }}\);

(ii) \(\text {Ext}^{1}_{\mathbb Z[{{\varGamma }}]}(P, {{\varLambda }}) = 0\) for any quasi-trivial Γ-lattice P.

Proof

The key observation is as follows. Suppose that P is a quasi-trivial lattice with a basis X that consists of a single Γ-orbit. Fix an xX and let \({{\varGamma }}^{\prime }\subset {{\varGamma }}\) be the stabilizer of x. Then we have an isomorphism \(P \cong \mathbb Z[{{\varGamma }}/{{\varGamma }}^{\prime }]\) of \(\mathbb Z[{{\varGamma }}]\)-modules, and so an isomorphism \(\text {Ext}^{1}_{\mathbb Z[{{\varGamma }}]}(P, {{\varLambda }}) \cong H^{1}({{\varGamma }}^{\prime }, {{\varLambda }})\). □

A.2.4

Let Γ be a finite group. A Γ-lattice Λ is called

(1) coflasque if it satisfies the equivalent conditions of Lemma A.2.3; and

(2) flasque if its \(\mathbb Z\)-linear dual \(\check {{{\varLambda }}}\) satisfies the equivalent conditions of Lemma A.2.3.Footnote 12

By Shapiro lemma, any quasi-trivial Γ-lattice is both flasque and coflasque.

A.2.5

In the setting of Section A.2.1, a torus \(T\rightarrow S\) split by \(\widetilde S\) is called quasi-trivial (resp. flasque; resp., coflasque) if its character lattice \(\check {{{\varLambda }}}_{T, \widetilde S}\) is quasi-trivial (resp. flasque; resp., coflasque). By [30, Lemma 1.1], these notions are independent of the choice of the Galois cover \(\widetilde S\).

For any scheme S, a torus \(T\rightarrow S\) is called quasi-trivial (resp. flasque; resp., coflasque) if every connected component of S admits a connected finite étale Galois cover \(\widetilde S\) such that T is split by \(\widetilde S\) and quasi-trivial (resp. flasque; resp., coflasque) with respect to \(\widetilde S\) (again, these notions do not depend on \(\widetilde {S}\)). If a torus \(T\rightarrow S\) is quasi-trivial (resp. flasque, coflasque), then so is its base change along any morphism \(S^{\prime }\rightarrow S\) with \(S^{\prime }\) still connected.

Quasi-trivial tori are both flasque and coflasque, and they can be made more explicit as follows.

Lemma A.2.6

Let S be a connected scheme. A torus \(T\rightarrow S\) is quasi-trivial if and only if it is a finite product of Weil restrictions of \(\mathbb G_{m}\) along finite étale surjections \(S^{\prime } \rightarrow S\).

Proof

Suppose that \(T\rightarrow S\) is quasi-trivial. Let \(\widetilde S\rightarrow S\) be a connected finite étale Galois cover such that T is split by \(\widetilde S\). Without loss of generality, we may assume that \({{\varLambda }}_{T, \widetilde S}\) has a basis X that consists of a single \(\text {Gal}(\widetilde S/S)\)-orbit. Then the \(\text {Gal}(\widetilde S/S)\)-set X gives rise to a finite étale cover \(S^{\prime } \rightarrow S\), and, by Section A.2.1, we have an isomorphism \(T \simeq \text {Res}_{S^{\prime }/S}(\mathbb G_{m})\). The converse is analogous. □

Flasque and coflasque tori enjoy the following pleasant splitting property.

Lemma A.2.7

In the setting of Section A.2.1, a short exact sequence of S-tori split by \(\widetilde S\)

$$ 1 \rightarrow T_{1} \rightarrow T_{2} \rightarrow T_{3} \rightarrow 1 $$

is split if either of the following conditions holds:

(i) T1 is quasi-trivial and T3 is coflasque;

(ii) T1 is flasque and T3 is quasi-trivial.

Proof

By considering character lattices, we translate the problem to splitting the exact sequence

$$ 0 \rightarrow \check{{{\varLambda}}}_{T_{3}, \widetilde S} \rightarrow \check{{{\varLambda}}}_{T_{2}, \widetilde S} \rightarrow \check{{{\varLambda}}}_{T_{1}, \widetilde S} \rightarrow 0 $$
(A.2.7.1)

of \(\text {Gal}(\widetilde S/S)\)-lattices. Suppose that T1 is quasi-trivial and T3 is coflasque. Then, by definition,

$$ \text{Ext}^{1}_{\mathbb Z[\text{Gal}(\widetilde S/S)]}(\check{{{\varLambda}}}_{T_{1}, \widetilde S}, \check{{{\varLambda}}}_{T_{3}, \widetilde S}) = 0, $$

so (A.2.7.1) splits. Suppose that T1 is flasque and T3 is quasi-trivial. Then the dual of (A.2.7.1) splits for the same reason, so, by dualizing again, (A.2.7.1) splits as well. □

Next, we shall construct “resolutions” of S-group schemes of multiplicative type split by \(\widetilde S\) in terms of flasque and coflasque tori. The following Lemma of Colliot-Thélène–Sansuc [30] will be the basis of our construction of resolutions of reductive S-group schemes.

Lemma A.2.8

In the setting of Section A.2.1, let \(G\rightarrow S\) be a group scheme of multiplicative type split by \(\widetilde S\). There exist S-tori T1 and T2 split by \(\widetilde S\) that fit into a short exact sequence of S-group schemes

$$ 1 \rightarrow G \rightarrow T_{1} \rightarrow T_{2} \rightarrow 1. $$
(A.2.8.1)

Furthermore, we may arrange (A.2.8.1) so that either of the following conditions is satisfied:

(i) T1 is flasque and T2 is quasi-trivial;

(ii) T1 is quasi-trivial and T2 is coflasque.

Proof

By Section A.2.1, the problem translates into one concerning finitely generated \(\mathbb Z\)-modules equipped with a \(\text {Gal}(\widetilde S/S)\)-action, which is addressed in [30, Lemma 0.6]. □

Remark A.2.9

In the setting of Section A.2.1, let \(T \rightarrow S\) be a torus split by \(\widetilde S\). In the same vein as Lemma A.2.8, [30, Lemma 0.6] implies the existence of resolutions by tori split by \(\widetilde S\):

$$ 1 \rightarrow T_{1} \rightarrow T_{2} \rightarrow T \rightarrow 1, $$

such that either

(i) T1 is flasque and T2 is quasi-trivial; or

(ii) T1 is quasi-trivial and T2 is coflasque.

These resolutions are the flasque, respectively coflasque resolutions of the torus T. The main result we shall prove (Theorem A.4.1) can be viewed as its generalization where T is replaced by a reductive S-group with isotrivial radical. In its proof, however, we will only need a special case of the result for T: the existence of a surjection \(P \twoheadrightarrow T\) from a quasi-trivial torus P split by \(\widetilde S\).

A.3 Central isogenies and the simply connected cover

Before proceeding to construct the promised resolutions of reductive groups in Section 7, we review the notion of central isogenies that plays an important role there. Recall the notion of the center ZG of a reductive group scheme \(G\rightarrow S\) as defined in Section 1.3.3.

A.3.1

For a scheme S, a morphism \(f \colon G^{\prime } \rightarrow G\) of reductive S-group schemes is called a central isogeny if

(1) f is finite, flat, and surjective;

(2) \(\ker (f)\) lies in the center of \(G^{\prime }\).

We only define the notion of central isogenies for reductive S-group schemes, as is done in [121, Exposé XXII, Définition 4.2.9]. One may generalize this notion to other S-group schemes, but it may become pathological: for example, the composition of two central isogenies may fail to be central, see [20, Exercise 3.4.4 (ii)]. We now show that such phenomena do not occur for reductive group schemes and then we use central isogenies to define the simply connected cover of a semisimple group scheme in Proposition A.3.4.

Lemma A.3.2

Let \(f \colon G^{\prime } \rightarrow G\) be a central isogeny of reductive S-group schemes.

(i) The induced map \(Z_{G^{\prime }} \rightarrow f^{-1}(Z_{G})\) is an isomorphism.

(ii) For any other central isogeny \(g \colon G^{\prime \prime } \rightarrow G^{\prime }\) of reductive group schemes over S, the composition \(f \circ g \colon G^{\prime \prime } \rightarrow G\) is also a central isogeny.

Proof

In (i), the problem is étale local on S, so we may assume that G contains a split maximal torus T, whose inverse image \(T^{\prime } := f^{-1}(T)\) is then a split maximal torus of \(G^{\prime }\). Since f is a central isogeny, the induced map on character lattices \(\check {{{\varLambda }}}_{T} \rightarrow \check {{{\varLambda }}}_{T^{\prime }}\) restricts to a bijection between the roots of (G, T) and \((G^{\prime }, T^{\prime })\), see [20, Example 6.1.9]. The result then follows from the characterization of ZG as the kernel of the adjoint action \(T \rightarrow \text {GL}(\text {Lie}(G))\) (see Section 1.3.3), that is, as the intersection of the \(\ker (\alpha )\) over all the roots \(\alpha \colon T \rightarrow \mathbb G_{m}\) of (G, T).

In (ii), fg is finite, flat, and surjective, so we need to verify that \(\ker (f \circ g) \subset Z_{G^{\prime \prime }}\). Indeed, we have

$$ \ker(f \circ g) \cong g^{-1}(\ker(f)) \subset g^{-1}(Z_{G^{\prime}}) \cong Z_{G^{\prime\prime}}, $$

where the last isomorphism comes from (i). □

Remark A.3.3

Suppose that \(f : G^{\prime } \rightarrow G\) is a central isogeny of reductive S-group schemes. Then \(\ker (f)\) is an S-group scheme of multiplicative type. Indeed, \(Z_{G^{\prime }}\) is of multiplicative type (see Section 1.3.3) so this assertion follows from the closure property in Section A.1.3.

Proposition A.3.4

Let S be a scheme and let G be a semisimple S-group scheme. Consider the category of pairs \((G^{\prime }, f)\) consisting of a semisimple S-group scheme \(G^{\prime }\) and a central isogeny \(f \colon G^{\prime } \rightarrow G\), with morphisms \((G^{\prime }_{1}, f_{1}) \rightarrow (G^{\prime }_{2}, f_{2})\) being given by central isogenies \(\alpha \colon G_{1}^{\prime } \rightarrow G_{2}^{\prime }\) such that f1 = f2α. This category has an initial object (Gsc,f), the simply connected cover of G.

Proof

The proof relies on the classification of pinned reductive groups by root data ([121, Exposé XXV, Théorème 1.1] or [20, Theorem 6.1.16]). The universal property allows us to work étale locally on S, so we may assume that G is split with respect to a split maximal torus TG.

The split maximal torus T allows us to extract the root data \(({{\varLambda }}_{T}, {{\varPhi }}, \check {{{\varLambda }}}_{T}, \check {{{\varPhi }}})\). Let \({{{\varLambda }}_{T}^{r}}\subset {{\varLambda }}_{T}\) denote the sublattice generated by the coroots Φ. There is a morphism of root data:

$$ ({{{\varLambda}}_{T}^{r}}, {{\varPhi}}, \check{{{\varLambda}}}_{T}^{r}, \check{{{\varPhi}}}) \rightarrow ({{\varLambda}}_{T}, {{\varPhi}}, \check{{{\varLambda}}}_{T}, \check{{{\varPhi}}}) $$
(A.3.4.1)

which induces the identity maps on Φ and \(\check {{{\varPhi }}}\). The root data \(({{{\varLambda }}_{T}^{r}}, {{\varPhi }}, \check {{{\varLambda }}}_{T}^{r}, \check {{{\varPhi }}})\) define a pinned reductive S-group Gsc with split maximal torus Tsc and (A.3.4.1) comes from a central isogeny \(f \colon G^{\text {sc}} \rightarrow G\) compatible with the splitting (i.e., mapping Tsc to T), but f is only unique up to conjugation by (T/ZG)(S) ([20, Theorem 6.1.16 (1)]). The pair (Gsc,f), however, is canonically defined thanks to the isomorphism \(T^{\text {sc}}/Z_{G^{\text {sc}}} \cong T/Z_{G}\) induced by f. Next, we argue that (Gsc,f) is canonically independent of the choice of the split maximal torus TG. Indeed, conjugation defines an isomorphism between G/NG(T) and the scheme parametrizing maximal tori of G ([20, Theorem 3.1.6]) so the claim follows from the isomorphism \(G^{\text {sc}}/N_{G^{\text {sc}}}(T^{\text {sc}}) \cong G/N_{G}(T)\) induced by f.

To show that the pair (Gsc,f) satisfies the universal property of an initial object, we suppose being given another central isogeny \(f^{\prime } \colon G^{\prime }\rightarrow G\). For a split maximal torus TG as above, we write \(T^{\prime } \subset G^{\prime }\) for the induced maximal torus. Arguing with root data as above, we find a central isogeny \(\alpha _{1} \colon G^{\text {sc}} \rightarrow G^{\prime }\) such that f and \(f^{\prime }\circ \alpha _{1}\) differ by conjugation by an element of (T/ZG)(S). The isomorphism \(T^{\prime }/Z_{G^{\prime }} \cong T/Z_{G}\) then allows us to construct the unique central isogeny \(\alpha \colon G^{\text {sc}} \rightarrow G^{\prime }\) which satisfies \(f = f^{\prime }\circ \alpha \). □

Remark A.3.5

Another definition of the simply connected cover is given in [20, Exercise 6.5.2 (i)], which characterizes the central isogeny \(f \colon G^{\text {sc}} \rightarrow G\) by the fact that the geometric S-fibers of Gsc are simply connected, i.e., they admit no nontrivial central isogenies from semisimple groups. It is easy to see that the two definitions agree. In particular, the formation of the simply connected cover Gsc of G commutes with arbitrary base change \(S^{\prime } \rightarrow S\).

A.4 Existence of resolutions

In this section, we construct flasque and coflasque resolutions of reductive group schemes with isotrivial radical tori. Recall that to a reductive S-group scheme G, we have associated several other reductive S-group schemes in the main text: the derived subgroup Gder, which is semisimple, and the tori rad(G) and corad(G) := G/Gder (see Section 1.3.3).

Theorem A.4.1

Let S be a connected scheme and let G be a reductive S-group scheme such that rad(G) is isotrivial. Fix a central isogeny \(f \colon G^{\prime {\text {der}}} \rightarrow G^{{\text {der}}}\). There exists a central extension

$$ 1 \rightarrow T_{1} \rightarrow G^{\prime} \rightarrow G \rightarrow 1 $$
(A.4.1.1)

of reductive S-group schemes such that \(\text {rad}(G^{\prime })\) is isotrivial and \(G^{\prime } \rightarrow G\) induces f on derived subgroups. Furthermore, setting \(T_{2} := \text {corad}(G^{\prime })\), we may arrange (A.4.1.1) so that one of the following conditions is satisfied:

(a) T1 is a flasque torus and T2 is a quasi-trivial torus;

(b) T1 is a quasi-trivial torus and T2 is a coflasque torus.

Remark A.4.2

The most typical application of Theorem A.4.1 is with \(f \colon G^{\prime {\text {der}}} \rightarrow G^{{\text {der}}}\) being the simply connected cover reviewed in Lemma A.3.4. In this case, we obtain a resolution (A.4.1.1) where \(G^{\prime }\) has a simply connected derived subgroup and the tori T1, T2 satisfy the conditions above. These are called flasque, respectively coflasque resolutions of G.

Note that if an S-torus T admits a flasque (or coflasque) resolution, then it must be isotrivial (Lemma A.1.6 (i)). Thus the hypothesis that rad(G) be isotrivial cannot be dropped.

Finally, we remark that if S is locally Noetherian and geometrically unibranch (such as a normal scheme), then the isotriviality condition on rad(G) is automatically satisfied (see Section A.1.8).

Proof of Theorem A.4.1

By composing the canonical central isogeny \(G^{{\text {der}}} \times \text {rad}(G) \rightarrow G\) of (1.3.3.1) with f ×idrad(G), we obtain a central isogeny of reductive S-group schemes:

$$ 1 \rightarrow H_{2} \rightarrow G^{\prime{\text{der}}} \times \text{rad}(G) \rightarrow G \rightarrow 1. $$
(A.4.2.1)

In particular, H2 is a finite S-group scheme of multiplicative type (Remark A.3.3).

Let us denote by \(\widetilde S \rightarrow S\) a connected finite étale Galois cover which splits rad(G). By Remark A.2.9, we may choose a short exact sequence of tori split by \(\widetilde S\):

$$ 1 \rightarrow H_{1} \rightarrow P \rightarrow \text{rad}(G) \rightarrow 1, $$
(A.4.2.2)

where P is quasi-trivial. Compose the central isogeny (A.4.2.1) with the surjection \(P\rightarrow \text {rad}(G)\), we obtain a central extension of reductive S-groups:

$$ 1 \rightarrow M \rightarrow G^{\prime{\text{der}}} \times P \rightarrow G \rightarrow 1. $$
(A.4.2.3)

Let us study the commutative S-group scheme M. By construction, it is an extension of H2 by H1. Since H1 is an isotrivial torus and H2 is a finite S-group scheme of multiplicative type, M is of multiplicative type (Section A.1.3) and even isotrivial (Lemma A.1.6 (ii)). Thus, we may take another connected finite étale Galois cover \(\widetilde S^{\prime }\rightarrow \widetilde S\) and assume that M is split by \(\widetilde S^{\prime }\). Using Lemma A.2.8, we find a resolution of M by S-tori which are also split by \(\widetilde S^{\prime }\):

$$ 1 \rightarrow M \rightarrow T_{1} \rightarrow Q \rightarrow 1, $$
(A.4.2.4)

where either

(1) T1 is flasque and Q is quasi-trivial; or

(2) T1 is quasi-trivial and Q is coflasque.

Let us form the pushout of the extension (A.4.2.3) along the map \(M \rightarrow T_{1}\). This gives rise to a central extension of G by T1 that fits into a commutative diagram

By construction, the map α induces an isomorphism on derived subgroups. Hence, the morphism \(G^{\prime } \rightarrow G\) induces the given central isogeny \(f \colon G^{\prime {\text {der}}} \rightarrow G^{{\text {der}}}\) on derived subgroups. Recall that the formation of radicals is preserved under quotient maps. (This statement may be verified over geometric points, where it is [121, Exposé XIX, Section 1.7].) Hence \(\text {rad}(G^{\prime })\) is a quotient of the torus T1 × P. Since the latter is split by \(\widetilde S^{\prime }\), so is \(\text {rad}(G^{\prime })\) (Lemma A.1.6 (i)).

Finally, we show that the two types of resolutions (A.4.2.4) give rise to the two conditions in the statement of Theorem A.4.1. Indeed, write \(T_{2} := \text {corad}(G^{\prime })\). Since T2 is a quotient of \(\text {rad}(G^{\prime })\), it is also split by \(\widetilde S^{\prime }\). We have a short exact sequence of S-tori split by \(\widetilde S^{\prime }\):

$$ 1 \rightarrow P \rightarrow T_{2} \rightarrow Q \rightarrow 1. $$
(A.4.2.5)

Since P is quasi-trivial and Q is at least coflasque, Lemma A.2.7 shows that (A.4.2.5) splits. In particular, T2 is quasi-trivial (resp. coflasque) whenever Q is. □

A.5 An application to the Grothendieck–Serre conjecture

We use Theorem A.4.1 to reduce the Grothendieck–Serre conjecture 3.1.1 to the case when the group G has a simply connected derived subgroup. This argument is suggested to me by K. Česnavičius.

Proposition A.5.1

Let R be a regular local ring, let K := Frac(R), let G be a reductive R-group scheme, and consider the pullback map

$$ H^{1}_{\acute{\text{e}}\text{t}}(R, G) \rightarrow H^{1}_{\acute{\text{e}}\text{t}}(K, G). $$
(A.5.1.1)

If this map has trivial kernel whenever G is replaced by some central extension \(G^{\prime }\) of G whose derived subgroup \(G^{\prime {\text {der}}}\) is simply connected, then it has trivial kernel for G itself.

Proof

By Theorem A.4.1, we may find a central extension of reductive R-group schemes

$$ 1 \rightarrow T_{1} \rightarrow G^{\prime} \rightarrow G \rightarrow 1, $$

such that the map \(G^{\prime } \rightarrow G\) induces the simply connected cover \(G^{\prime {\text {der}}} \cong (G^{{\text {der}}})^{\text {sc}} \rightarrow G^{{\text {der}}}\) on the derived subgroups and T1 is a quasi-trivial torus.Footnote 13 By (1.2.2.1), this extension gives rise to the following map of exact sequences of pointed sets:

Since T1 is a quasi-trivial torus, by Lemma A.2.6, it is isomorphic to a finite product of tori of the form \(\text {Res}_{R^{\prime }/R}(\mathbb G_{m})\) for some finite étale maps \(R \rightarrow R^{\prime }\). In particular, Hilbert 90 implies that α1 is an isomorphism between singletons. Grothendieck’s theorem on the Brauer group [64, Corollaire 1.8] implies that α2 is injective. Therefore, if \(\beta ^{\prime }\) has trivial kernel, then so does β, as desired. □

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Česnavičius, K. Problems About Torsors over Regular Rings. Acta Math Vietnam 47, 39–107 (2022). https://doi.org/10.1007/s40306-022-00477-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40306-022-00477-y

Keywords

  • Bass–Quillen
  • Grothendieck–Serre
  • Quillen patching
  • Reductive group
  • Torsor

Mathematics Subject Classification (2010)

  • Primary: 14M17
  • Secondary: 14L15
  • 20G10