×

Principal homogeneous spaces under flasque tori; applications. (English) Zbl 0597.14014

The notions of flasque tori and of flasque resolution of tori defined over a field have already played a great rôle in the work V. E. Voskresenskij [”Algebraic tori” (Moscow 1977; Zbl 0499.14013)], of S. Endo and T. Miyata [Nagoya Math. J. 56, 85-104 (1975; Zbl 0301.14008)] and of the authors [Ann. Sci. École Norm. Supér., IV. Sér. 10, 175-229 (1977; Zbl 0356.14007)]. In this paper, flasque tori and flasque resolutions of tori are defined over arbitrary base schemes. If X’/X is a finite Galois cover with group G, an X-torus F split by X’ is called flasque if its character group \(\hat F\) satisfies \(Ext^ 1_ G(\hat F,\hat P)=0\) for any permutation G-module \(\hat P \)(a torus P whose character group is such a \(\hat P\) is called quasi-trivial).
The main cohomological properties of such a torus is that for U an open set of a regular scheme X, the restriction map in flat cohomology \(H^ i(X,F)\to H^ i(U,F)\) is surjective for \(i=1\) (torsors automatically extend) and injective for \(i=2\) (extension of the Auslander-Goldman- Grothendieck result). This is dealt with in §§ 1 and 2. One application is to the finiteness of \(H^ 1(k,F)\) when k is a field finitely generated over the prime field. To any X-torus T one may associate a flasque resolution: \(1\to F\to P\to T\to 1.\)Taking \(X=Spec k\) for k a field, viewing P as a torsor over T under F and using the extension property for torsors leads to an alternate proof (§ 3) of the main results of the quoted paper of the authors.
In § 4 flasque resolutions of the above and of other types over \(X=Spec A\) for A an integral regular semi-local ring with fraction field K together with the properties of flasque tori lead to a proof that the restriction maps \(H^ i(A,M)\to H^ i(K,M)\) for \(i=1, 2\) are injective for any A-group of multiplicative type M. A consequence (§ 5) is that if B/A is a finite étale cover of integral regular semi-local rings with fraction fields L and K, a unit a in A which is a norm for L/K is a norm for B/A. This result still holds if a is not a unit but B/A is Galois. Some applications and extensions to the representation of elements of A by quadratic forms with coefficients in A are then discussed in § 6, which completes the paper by M. D. Choi, T. Y. Lam, B. Reznick and A. Rosenberg in J. Algebra 65, 234-256 (1980; Zbl 0433.10010) in a few points.
§ 7 considers two related problems: given a local ring A with residue field \(\kappa\), an A-torus T and an A-group of multiplicative type M, is the restriction map T(A)\(\to T(\kappa)\) surjective ? Is the restriction map \(H^ 1(A,M)\to H^ 1(\kappa,M)\) surjective ? When \(M=({\mathbb{Z}}/n)_ A\) for n invertible in A, this last question is just the lifting problem for abelian extensions, as studied by D. J. Saltman [Adv. Mat. 43, 250-283 (1982; Zbl 0484.12004); Isr. J. Math. 47, 165-215 (1984; Zbl 0546.14013)]. Using flasque resolutions, we recover and generalize most of Saltman’s results. For instance the above questions have an affirmative answer when \(T_{\kappa}\) resp. \(M_{\kappa}\) are split by a cyclic extension of \(\kappa\), which explains the special rôle played by powers of 2 in the lifting problem for abelian extensions \((({\mathbb{Z}}/n)_{\kappa}\) is split by the field of n-th roots of unity over \(\kappa)\). Sticking to tori over a field k one shows that the first problem has an affirmative answer for arbitrary local k-algebras A if and only if T is a k-birational direct factor of an affine space over k. The proof is inspired by Saltman’s study of retract rational fields (see the 1984-paper cited above). The section closes with a discussion of Saltman’s approach to the failure of the Noether problem (rationality of the field of invariants of a finite group) via Wang’s counterexample to Grunwald’s theorem and with some remarks to the effect that in spite of the failure of Noether’s problem Hilbert’s irreducibility theorem sometimes holds for the field of invariants. In § 8, which parallels § 7, K is a discretely valued field and \(\hat K\) its completion. Given T a K-torus, resp. M a K-group of multiplicative type, the following questions are discussed: is T(K) dense in T(\^K) ? Is the map \(H^ 1(K,M)\to H^ 1(\hat K,M)\) surjective ?
Finally, § 9 discusses the so-called centre \(Z_ n(2)\) of the ring of two n by n generic matrices over a field k. Using Formanek’s presentation [E. Formanek, Linear Multilinear Algebra 7, 203-212 (1979; Zbl 0419.16010)] of this centre as the function field of a certain K-torus T over a purely trancendental extension K of k, we recover two results of Saltman: when n is prime, this centre is ”retract rational” and in general the Brauer group of a smooth proper model of \(Z_ n(2)\) is trivial [D. J. Saltman, J. Algebra 97, 53-67 (1985; Zbl 0586.13005)].

MSC:

14F20 Étale and other Grothendieck topologies and (co)homologies
14M17 Homogeneous spaces and generalizations
13B02 Extension theory of commutative rings
14L30 Group actions on varieties or schemes (quotients)
20G10 Cohomology theory for linear algebraic groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Artin, M.; Grothendieck, A.; Verdier, J.-L, Théorie des Topos et Cohomologie Étale des Schémas, (Lecture Notes in Math. (1972-1973), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York), Nos. 269, 270, 305 · Zbl 0234.00007
[2] Bourbaki, N., Algèbre, (Modules et anneaux semi-simples (1958), Hermann: Hermann Paris), Chap. 8 · Zbl 0455.18010
[3] Bourbaki, N., Algèbre commutative (1961-1965), Hermann: Hermann Paris, Chaps. 1-7 · Zbl 0141.03501
[4] Brylinski, J.-L, Décomposition simpliciale d’un réseau, invariante par un groupe fini d’automorphismes, C. R. Acad. Sci. Paris Sér. A, 288, 137-139 (1979) · Zbl 0406.14022
[5] Cassels, J. W.S, Rational Quadratic Forms (1978), Academic Press: Academic Press London/New York/San Francisco · Zbl 0395.10029
[6] Choi, M. D.; Lam, T. Y.; Reznick, B.; Rosenberg, A., Sums of squares in some integral domains, J. Algebra, 65, 234-256 (1980) · Zbl 0433.10010
[7] Colliot-Thélène, J.-L; Sansuc, J.-J, Torseurs sous des groupes de type multiplicatif; applications à l’étude des points rationnels de certaines variétés algébriques, C. R. Acad. Sci. Paris Sér. A, 282, 1113-1116 (1976) · Zbl 0337.14014
[8] Colliot-Thélène, J.-L; Sansuc, J.-J, \(La R\)-équivalence sur les tores, Ann. Sci. École Norm. Sup. 4ème Sér., 10, 175-229 (1977) · Zbl 0356.14007
[9] Colliot-Thélène, J.-L; Sansuc, J.-J, Cohomologie des groupes de type multiplicatif sur les schémas réguliers, C. R. Acad. Sci. Paris Sér. A, 287, 449-452 (1978) · Zbl 0399.14011
[10] Colliot-Thélène, J.-L; Sansuc, J.-J, Thèse (1978), Orsay
[11] Colliot-Thélène, J.-L, Formes quadratiques sur les anneaux semi-locaux réguliers, (Colloque sur les formes quadratiques. Colloque sur les formes quadratiques, Montpellier 1977. Colloque sur les formes quadratiques. Colloque sur les formes quadratiques, Montpellier 1977, Bull. Soc. Math. France, Mém., 59 (1979)), 13-31 · Zbl 0407.10018
[12] Colliot-Thélène, J.-L; Sansuc, J.-J, La descente sur les variétés rationnelles, (Beauville, A., Journées de géométrie algébrique d’Angers (1980), Sijthoff and Noordhoof: Sijthoff and Noordhoof Alphen aan Rijn), 223-237, 1979 · Zbl 0451.14018
[13] Demazure, M.; Grothendieck, A., Schémas en Groupes, (Lecture Notes in Math. (1970), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York), Nos. 151, 152, 153 · Zbl 0212.52808
[14] Endo, S.; Miyata, T., Invariants of finite abelian groups, J. Math. Soc. Japan, 25, 7-26 (1973) · Zbl 0245.20007
[15] Endo, S.; Miyata, T., Corrigendum, 79, 187-190 (1980) · Zbl 0471.14007
[16] Endo, S.; Miyata, T., On integral representations of finite groups, Sugaku, 27, 231-240 (1975), [Japanese]
[17] Formanek, E., The centre of the ring of 3 × 3 generic matrices, Linear Multilinear Algebra, 7, 203-212 (1979) · Zbl 0419.16010
[18] Formanek, E., The centre of the ring of 4 × 4 generic matrices, J. Algebra, 62, 304-319 (1980) · Zbl 0437.16013
[19] Grothendieck, A.; Dieudonné, J., Éléments de Géométrie Algébrique, Publ. Math. I.H.E.S., 32 (1962-1968), Paris · Zbl 0163.03201
[20] Grothendieck, A., III. Exemples et compléments, (Dix exposés sur la Cohomologie des schémas (1968), North-Holland: North-Holland Amsterdam) · Zbl 0198.25901
[21] Kervaire, M., Fractions rationnelles invariantes (d’après H.W. Lenstra), (Séminaire Bourbaki (1974)), 445
[22] Lenstra, H. W., Rational functions invariant under a finite abelian group, Invent. Math., 25, 299-325 (1974) · Zbl 0292.20010
[23] Nisnevich, Ye. A., Espaces homogènes principaux rationnellement triviaux et arithmétique des schémas en groupes réductifs sur les anneaux de Dedekind, C. R. Acad. Sci. Paris Sér. I, 299, 5-8 (1984) · Zbl 0587.14033
[24] Ojanguren, M., A splitting theorem for quadratic forms, Comment. Math. Helv., 57, 145-157 (1982) · Zbl 0487.13005
[25] Ojanguren, M., Quadratic forms over regular rings, J. Indian Math. Soc., 44, 109-116 (1980) · Zbl 0621.10017
[26] Ojanguren, M., (Dennis, K., Lecture Notes in Math. (1982), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York), 291-299, No. 967 · Zbl 0498.10015
[27] W. Pardon; W. Pardon · Zbl 0531.10024
[28] Pardon, W., (Bak, A., Lecture Notes in Math. (1984), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York/Tokyo), 261-328, No. 1046
[29] Procesi, C., Relazioni tra geometria algebrica ed algebra non commutativa. Algebre cicliche e problema di Lüroth, Boll. Un. Mat. Ital. A (5), 18, 1-10 (1981) · Zbl 0493.16010
[30] Saltman, D. J., Generic Galois extensions and problems in field theory, Advan. in Math., 43, 250-283 (1982) · Zbl 0484.12004
[31] Saltman, D. J., Retract rational fields and cyclic Galois extensions, Israël J. Math., 47, 165-215 (1984) · Zbl 0546.14013
[32] Saltman, D. J., The Brauer group and the centre of generic matrices, J. Algebra, 97, 53-67 (1985) · Zbl 0586.13005
[33] Sansuc, J.-J, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math., 327, 12-80 (1981) · Zbl 0468.14007
[34] Swan, R. G., Invariant rational functions and a problem of Steenrod, Invent. Math., 7, 148-158 (1969) · Zbl 0186.07601
[35] Swan, R. G., Expanded version: Noether’s Problem in Galois theory, (Srinivasan, B.; Sally, J., Emmy Noether in Bryn Mawr (1983), Springer-Verlag: Springer-Verlag Berlin), 21-40 · Zbl 0144.22602
[36] Voskresenskiǐ, V. E., Russian Math. Surveys, 28, 79-105 (1973) · Zbl 0289.14006
[37] Voskresenskiǐ, V. E., Algebraic Tori (1977), Nauka: Nauka Moscow · Zbl 0379.14001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.