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Homogeneous spaces, algebraic \(K\)-theory and cohomological dimension of fields. (English) Zbl 1509.12002

Let \(K\) be a field of characteristic \(\text{char}(K)\), \(K_{\mathrm{sep}}\) a separable closure of \(K\), \(\mathcal{G}_K\) the absolute Galois group of \(K\) (defined to be the Galois group \(\mathcal{G}(K_{\mathrm{sep}}/K)\)), and for each prime number \(\ell\), let \(\text{cd}_{\ell }(\mathcal{G}_K)\) be the \(\ell\)-cohomological dimension of \(\mathcal{G}_K\). By definition, the \(\ell\)-cohomological dimension \(\text{cd}_{\ell }(K)\) of \(F\) is equal to \(\text{cd}_{\ell }(\mathcal{G}_K)\), for every prime \(\ell \neq char(K)\). It is well-known that \(\text{cd}_p(\mathcal{G}_F) \le 1\) if \(\text{char}(F) = p > 0\) (see [J.-P. Serre, Cohomologie Galoisienne. 5ème éd., rév. et complété. Berlin: Springer-Verlag (1994; Zbl 0812.12002)]). Then \(\text{cd}_p(K)\) is defined to be the smallest integer \(i\) such that \([K: K^p] \le p^i\) and the Kato-Milne cohomology group \(H_p^{i+1}(L)\) (for its definition, see [J. S. Milne, Ann. Sci. Éc. Norm. Supér. (4) 9, 171–201 (1976; Zbl 0334.14010)]) is trivial, for every finite extension \(L\) of \(K\); \(\text{cd}_p(K)\) is infinity if \(i\) does not exist. The cohomological dimension \(\text{cd}(K)\) of \(K\) is the supremum of all the \(\text{cd}_{\ell }(K)\) when \(\ell\) runs across the set of prime numbers. It follows from the Bloch-Kato conjecture (proved by V. Voevodsky [Ann. Math. (2) 174, No. 1, 401–438 (2011; Zbl 1236.14026)], J. Riou [Astérisque 361, 421–463, Exp. No. 1073 (2014; Zbl 1366.19001)], and further references there) that \(\text{cd}(K) \le n\) if and only if \(K\) satisfies condition \(C _{0} ^{n}\) introduced in: [K. Kato and T. Kuzumaki, J. Number Theory 24, 229–244 (1986; Zbl 0608.12029)].
The research presented in the paper under review is motivated by the influence of some diophantine properties of fields on their cohomological dimension. Perhaps the earliest result of this kind states that any finitely-generated extension \(F\) of an algebraically closed field \(F _{0}\) of transcendence degree \(d\) is an \(C_d\)-field with \(\text{cd}(\mathcal{G}_{F}) = d\) (see [S. Lang, Ann. Math. (2) 55, 373–390 (1952; Zbl 0046.26202)], and Ch. II, Proposition 11 of [Serre, loc. cit.]). When \(K\) is a field with \(\text{char}(K) = 0\), the condition that \(\text{cd}(K) \le q \le 2\) is characterized by the surjectivity of: norm mappings of finite separable extensions \(L'/L\) of \(K\), for \(q = 1\); reduced norm mappings of finite-dimensional central simple algebras over finite extensions of \(K\), for \(q = 2\). This ensures that \(\text{cd}(K) \le q\) in case \(K\) is a \(C _{q}\)-field (for \(q = 1\) and \(q = 2\), see Ch. II, 3.1, of [Serre, loc. cit.], and Theorem 24.8, Corollary 24.9 in: [A. A. Suslin, J. Sov. Math. 30, 2556–2611 (1985; Zbl 0566.12016); A. A. Suslin, in: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 25, 115–207 (1984; Zbl 0558.12013)], respectively). The latter result has been generalized to the case where \(\text{char} (K) \neq 0\) (see Theorem 7 in: [P. Gille, \(K\)-Theory 21, No. 1, 57–100 (2000; Zbl 0993.20031)]). It has also been proved that \(\text{cd}(\mathcal{G}_{E}) \le q\) if \(E\) is a \(C _{q}\)-field and \(3 \ge q \le 4\) [D. Krashen and E. Matzri, Proc. Am. Math. Soc. 143, No. 7, 2779–2788 (2015; Zbl 1329.12003)]. On the other hand, a number of results obtained in the course of time (including [J. Ax, Proc. Am. Math. Soc. 16, 1214–1221 (1965; Zbl 0142.30001); A. S. Merkur’ev, Math. USSR, Izv. 38, No. 1, 215–221 (1991; Zbl 0763.12003); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 55, No. 1, 218–224 (1991); J.-L. Colliot-Thélène and D. A. Madore, J. Inst. Math. Jussieu 3, No. 1, 1–16 (2004; Zbl 1056.14030)]) show that neither the classical \(C _{m}\)-property nor its variant \(C _{q} ^{0}\) introduced by Kato and Kuzumaki [loc. cit.] characterize the cohomological dimension of fields.
The reviewed paper introduces a variant of the \(C_1^q\) property and proves that, contrary to the \(C_1^q\) property, they characterize the cohomological dimensions of fields. It proves that \(\text{cd}(K) \le q\) if and only if, for any finite extension \(L/K\) and for any homogeneous space \(Z\) under a smooth linear connected algebraic group defined over \(L\), the \(q\)-th Milnor \(K\)-theory group \(K_q^M(L)\) is spanned by the images of the norms coming from those finite extensions of \(L\) over which \(Z\) has a rational point. The authors also obtain a variant of this result for imperfect fields. As explained in the text, the main theorem of the present paper unifies and significantly generalizes the above-noted results of Suslin and Gille as well as theorems due to Springer and Steinberg (see Ch. II, 2.4, in [Serre, loc. cit]), and Wittenberg (see Corollaries 5.6 and 5.8 in: [O. Wittenberg, Duke Math. J. 164, No. 11, 2185–2211 (2015; Zbl 1348.11037)]).

MSC:

12G10 Cohomological dimension of fields
19D45 Higher symbols, Milnor \(K\)-theory
11E72 Galois cohomology of linear algebraic groups
14M17 Homogeneous spaces and generalizations
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[1] Schémas en groupes (SGA3). Tome I. Propriétés générales des schémas en groupes. Doc. Math. (Paris) 7, Société Mathématique de France, Paris (2011) Zbl 1241.14002 MR 2867621
[2] Colliot-Thélène, J.-L., Madore, D. A.: Surfaces de del Pezzo sans point rationnel sur un corps de dimension cohomologique un. J. Inst. Math. Jussieu 3, 1-16 (2004) Zbl 1056.14030 MR 2036596 · Zbl 1056.14030
[3] Demarche, C., Lucchini Arteche, G.: Le principe de Hasse pour les espaces homogènes: réduction au cas des stabilisateurs finis. Compos. Math. 155, 1568-1593 (2019) Zbl 07077747 MR 3977320 · Zbl 1504.14040
[4] Flicker, Y. Z., Scheiderer, C., Sujatha, R.: Grothendieck’s theorem on non-abelian H 2 and local-global principles. J. Amer. Math. Soc. 11, 731-750 (1998) Zbl 0893.14015 MR 1608617 · Zbl 0893.14015
[5] Gille, P.: Invariants cohomologiques de Rost en caractéristique positive. K-Theory 21, 57-100 (2000) Zbl 0993.20031 MR 1802626 · Zbl 0993.20031
[6] Gille, P., Szamuely, T.: Central Simple Algebras and Galois Cohomology. Cambridge Stud. Adv. Math. 165, Cambridge University Press, Cambridge (2017) Zbl 1373.19001 MR 3727161 · Zbl 1373.19001
[7] Grothendieck, A.: Le groupe de Brauer. III. Exemples et compléments. In: Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math. 3, North-Holland, Amsterdam, 88-188 (1968) Zbl 0198.25901 MR 244271 · Zbl 0198.25901
[8] Harpaz, Y., Wittenberg, O.: Zéro-cycles sur les espaces homogènes et problème de Galois inverse. J. Amer. Math. Soc. 33, 775-805 (2020) Zbl 07225790 MR 4127903 · Zbl 1469.14053
[9] Izquierdo, D.: On a conjecture of Kato and Kuzumaki. Algebra Number Theory 12, 429-454 (2018) Zbl 1442.11070 MR 3803709 · Zbl 1442.11070
[10] Kato, K.: A generalization of local class field theory by using K-groups. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, 603-683 (1980) Zbl 0463.12006 MR 603953 · Zbl 0463.12006
[11] Kato, K.: Galois cohomology of complete discrete valuation fields. In: Algebraic K-Theory, Part II (Oberwolfach, 1980), Lecture Notes in Math. 967, Springer, Berlin, 215-238 (1982) Zbl 0506.12022 MR 689394 · Zbl 0506.12022
[12] Kato, K., Kuzumaki, T.: The dimension of fields and algebraic K-theory. J. Number Theory 24, 229-244 (1986) Zbl 0608.12029 MR 863657 · Zbl 0608.12029
[13] Matsumura, H.: Commutative Ring Theory. Cambridge Stud. Adv. Math. 8, Cambridge University Press, Cambridge (1986) Zbl 0603.13001 MR 879273 · Zbl 0603.13001
[14] Merkur’ev, A. S.: Simple algebras and quadratic forms. Izv. Akad. Nauk SSSR Ser. Mat. 55, 218-224 (1991) Zbl 0733.12008 MR 1130036 · Zbl 0733.12008
[15] Ono, T.: Arithmetic of algebraic tori. Ann. of Math. (2) 74, 101-139 (1961) Zbl 0119.27801 MR 124326 · Zbl 0119.27801
[16] Riou, J.: La conjecture de Bloch-Kato (d’après M. Rost et V. Voevodsky). Astérisque 361, Exp. No. 1073, x, 421-463 (2014) Zbl 1366.19001 MR 3289290 · Zbl 1366.19001
[17] Serre, J.-P.: Local Fields. Grad. Texts in Math. 67, Springer, New York (1979) Zbl 0423.12016 MR 554237 · Zbl 0423.12016
[18] Serre, J.-P.: Cohomologie Galoisienne. Lecture Notes in Math. 5, Springer, Berlin, 5th ed. (1994) Zbl 0812.12002 MR 1324577 · Zbl 0812.12002
[19] Serre, J.-P.: Cohomologie galoisienne: progrès et problèmes. Astérisque 227, Exp. No. 783, 4, 229-257 (1995) Zbl 0837.12003 MR 1321649 · Zbl 0837.12003
[20] Springer, T. A.: Nonabelian H 2 in Galois cohomology. In: Algebraic Groups and Discontin-uous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), American Mathematical Society, Providence, 164-182 (1966) Zbl 0193.48902 MR 0209297 · Zbl 0193.48902
[21] Springer, T. A.: Linear Algebraic Groups. Mod. Birkhäuser Class., Birkhäuser, Boston, 2nd ed. (2009) Zbl 1202.20048 MR 2458469 · Zbl 1202.20048
[22] Steinberg, R.: Regular elements of semisimple algebraic groups. Publ. Math. Inst. Hautes Études Sci. 25, 49-80 (1965) MR 180554 · Zbl 0136.30002
[23] Suslin, A. A.: Algebraic K-theory and the norm-residue homomorphism, J. Sov. Math. 30, 2556-2611 (1985) Zbl 0566.12016 · Zbl 0566.12016
[24] Suslin, A., Joukhovitski, S.: Norm varieties. J. Pure Appl. Algebra 206, 245-276 (2006) Zbl 1091.19002 MR 2220090 · Zbl 1091.19002
[25] Wittenberg, O.: Sur une conjecture de Kato et Kuzumaki concernant les hypersurfaces de Fano. Duke Math. J. 164, 2185-2211 (2015) Zbl 1348.11037 MR 3385132 · Zbl 1348.11037
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