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Rationality Problem for Algebraic Tori
About this Title
Akinari Hoshi, Department of Mathematics, Niigata University, Niigata 950-2181, Japan and Aiichi Yamasaki, Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
Publication: Memoirs of the American Mathematical Society
Publication Year:
2017; Volume 248, Number 1176
ISBNs: 978-1-4704-2409-1 (print); 978-1-4704-4054-1 (online)
DOI: https://doi.org/10.1090/memo/1176
Published electronically: March 9, 2017
Keywords: Rationality problem,
algebraic tori,
stably rational,
retract rational,
flabby resolution,
Krull-Schmidt theorem,
Bravais group,
Tate cohomology.
MSC: Primary 11E72, 12F20, 13A50, 14E08, 20C10, 20G15.
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries: Tate cohomology and flabby resolutions
- 3. CARAT ID of the $\mathbb {Z}$-classes in dimensions $5$ and $6$
- 4. Krull-Schmidt theorem fails for dimension $5$
- 5. GAP algorithms: the flabby class $[M_G]^{fl}$
- 6. Flabby and coflabby $G$-lattices
- 7. $H^1(G,[M_G]^{fl})=0$ for any Bravais group $G$ of dimension $n\leq 6$
- 8. Norm one tori
- 9. Tate cohomology: GAP computations
- 10. Proof of Theorem
- 11. Proof of Theorem
- 12. Proof of Theorem
- 13. Application of Theorem
- 14. Tables for the stably rational classification of algebraic $k$-tori of dimension $5$
Abstract
We give the complete stably rational classification of algebraic tori of dimensions $4$ and $5$ over a field $k$. In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank $4$ and $5$ is given. We show that there exist exactly $487$ (resp. $7$, resp. $216$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $4$, and there exist exactly $3051$ (resp. $25$, resp. $3003$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $5$. We make a procedure to compute a flabby resolution of a $G$-lattice effectively by using the computer algebra system GAP. Some algorithms may determine whether the flabby class of a $G$-lattice is invertible (resp. zero) or not. Using the algorithms, we determine all the flabby and coflabby $G$-lattices of rank up to $6$ and verify that they are stably permutation. We also show that the Krull-Schmidt theorem for $G$-lattices holds when the rank $\leq 4$, and fails when the rank is $5$. Indeed, there exist exactly $11$ (resp. $131$) $G$-lattices of rank $5$ (resp. $6$) which are decomposable into two different ranks. Moreover, when the rank is $6$, there exist exactly $18$ $G$-lattices which are decomposable into the same ranks but the direct summands are not isomorphic. We confirm that $H^1(G,F)=0$ for any Bravais group $G$ of dimension $n\leq 6$ where $F$ is the flabby class of the corresponding $G$-lattice of rank $n$. In particular, $H^1(G,F)=0$ for any maximal finite subgroup $G\leq \textrm {GL}(n,\mathbb {Z})$ where $n\leq 6$. As an application of the methods developed, some examples of not retract (stably) rational fields over $k$ are given.- James E. Arnold Jr., Groups of permutation projective dimension two, Proc. Amer. Math. Soc. 91 (1984), no. 4, 505–509. MR 746077, DOI 10.1090/S0002-9939-1984-0746077-6
- Gorô Azumaya, Corrections and supplementaries to my paper concerning Krull-Remak-Schmidt’s theorem, Nagoya Math. J. 1 (1950), 117–124. MR 37832
- D. J. Benson, Representations and cohomology. I, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1991. Basic representation theory of finite groups and associative algebras. MR 1110581
- F. A. Bogomolov, The Brauer group of quotient spaces of linear representations, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 3, 485–516, 688 (Russian); English transl., Math. USSR-Izv. 30 (1988), no. 3, 455–485. MR 903621, DOI 10.1070/IM1988v030n03ABEH001024
- F. A. Bogomolov, Brauer groups of the fields of invariants of algebraic groups, Mat. Sb. 180 (1989), no. 2, 279–293 (Russian); English transl., Math. USSR-Sb. 66 (1990), no. 1, 285–299. MR 993459, DOI 10.1070/SM1990v066n01ABEH001173
- Fedor Bogomolov, Jorge Maciel, and Tihomir Petrov, Unramified Brauer groups of finite simple groups of Lie type $A_l$, Amer. J. Math. 126 (2004), no. 4, 935–949. MR 2075729
- Serge Bouc, Burnside rings, Handbook of algebra, Vol. 2, Handb. Algebr., vol. 2, Elsevier/North-Holland, Amsterdam, 2000, pp. 739–804. MR 1759611, DOI 10.1016/S1570-7954(00)80043-1
- Harold Brown, Rolf Bülow, Joachim Neubüser, Hans Wondratschek, and Hans Zassenhaus, Crystallographic groups of four-dimensional space, Wiley-Interscience [John Wiley & Sons], New York-Chichester-Brisbane, 1978. Wiley Monographs in Crystallography. MR 0484179
- K. S. Brown, Cohomology of Groups, Grad. Texts in Math., vol. 87, Springer-Verlag, 1972.
- Gregory Butler and John McKay, The transitive groups of degree up to eleven, Comm. Algebra 11 (1983), no. 8, 863–911. MR 695893, DOI 10.1080/00927878308822884
- J. Opgenorth, W. Plesken, T. Schulz, CARAT, GAP 4 package, version 2.1b1, 2008, available at http://wwwb.math.rwth-aachen.de/carat/.
- Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
- H. Chu, S. Hu, M. Kang, B. E. Kunyavskii, Noether’s problem and the unramified Brauer group for groups of order 64, Int. Math. Res. Not. IMRN 2010 2329–2366.
- Huah Chu, Shou-Jen Hu, Ming-chang Kang, and Y. G. Prokhorov, Noether’s problem for groups of order 32, J. Algebra 320 (2008), no. 7, 3022–3035. MR 2442008, DOI 10.1016/j.jalgebra.2008.07.007
- Huah Chu and Ming-chang Kang, Rationality of $p$-group actions, J. Algebra 237 (2001), no. 2, 673–690. MR 1816710, DOI 10.1006/jabr.2000.8615
- Anne Cortella and Boris Kunyavskiĭ, Rationality problem for generic tori in simple groups, J. Algebra 225 (2000), no. 2, 771–793. MR 1741561, DOI 10.1006/jabr.1999.8150
- Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, La $R$-équivalence sur les tores, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 2, 175–229 (French). MR 450280
- Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, Principal homogeneous spaces under flasque tori: applications, J. Algebra 106 (1987), no. 1, 148–205. MR 878473, DOI 10.1016/0021-8693(87)90026-3
- Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders; Pure and Applied Mathematics; A Wiley-Interscience Publication. MR 632548
- Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. II, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. With applications to finite groups and orders; A Wiley-Interscience Publication. MR 892316
- E. C. Dade, The maximal finite groups of $4\times 4$ integral matrices, Illinois J. Math. 9 (1965), 99–122. MR 170958
- Andreas Dress, On the Krull-Schmidt theorem for integral group representations of rank $1$, Michigan Math. J. 17 (1970), 273–277. MR 263933
- Andreas W. M. Dress, Contributions to the theory of induced representations, Algebraic $K$-theory, II: “Classical” algebraic $K$-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 183–240. Lecture Notes in Math., Vol. 342. MR 0384917
- Andreas W. M. Dress, The permutation class group of a finite group, J. Pure Appl. Algebra 6 (1975), 1–12. MR 360773, DOI 10.1016/0022-4049(75)90008-0
- Shizuo Endo, The rationality problem for norm one tori, Nagoya Math. J. 202 (2011), 83–106. MR 2804547, DOI 10.1215/00277630-1260459
- S. Endo, private communications, 2012.
- Shizuo Endô and Yumiko Hironaka, Finite groups with trivial class groups, J. Math. Soc. Japan 31 (1979), no. 1, 161–174. MR 519042, DOI 10.2969/jmsj/03110161
- Shizuo Endô and Takehiko Miyata, Invariants of finite abelian groups, J. Math. Soc. Japan 25 (1973), 7–26. MR 311754, DOI 10.2969/jmsj/02510007
- Shizuo Endô and Takehiko Miyata, On a classification of the function fields of algebraic tori, Nagoya Math. J. 56 (1975), 85–104. MR 364203
- Shizuo Endô and Takehiko Miyata, Integral representations with trivial first cohomology groups, Nagoya Math. J. 85 (1982), 231–240. MR 648425
- Alberto Facchini, The Krull-Schmidt theorem, Handbook of algebra, Vol. 3, Handb. Algebr., vol. 3, Elsevier/North-Holland, Amsterdam, 2003, pp. 357–397. MR 2035101, DOI 10.1016/S1570-7954(03)80066-9
- M. Florence, Non rationality of some norm-one tori, preprint (2006).
- The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.12; 2008. (http://www.gap-system.org).
- Robert M. Guralnick and Al Weiss, Transitive permutation lattices in the same genus and embeddings of groups, Linear algebraic groups and their representations (Los Angeles, CA, 1992) Contemp. Math., vol. 153, Amer. Math. Soc., Providence, RI, 1993, pp. 21–33. MR 1247496, DOI 10.1090/conm/153/01305
- Mowaffaq Hajja, A note on monomial automorphisms, J. Algebra 85 (1983), no. 1, 243–250. MR 723077, DOI 10.1016/0021-8693(83)90128-X
- Mowaffaq Hajja, Rationality of finite groups of monomial automorphisms of $k(x,y)$, J. Algebra 109 (1987), no. 1, 46–51. MR 898335, DOI 10.1016/0021-8693(87)90162-1
- Mowaffaq Hajja and Ming Chang Kang, Finite group actions on rational function fields, J. Algebra 149 (1992), no. 1, 139–154. MR 1165204, DOI 10.1016/0021-8693(92)90009-B
- Mowaffaq Hajja and Ming Chang Kang, Three-dimensional purely monomial group actions, J. Algebra 170 (1994), no. 3, 805–860. MR 1305266, DOI 10.1006/jabr.1994.1366
- G. Ellis, HAP, GAP 4 package, version 1.10.6, 2012, available at http://hamilton.nuigalway.ie/Hap/www/.
- W. Hürlimann, On algebraic tori of norm type, Comment. Math. Helv. 59 (1984), no. 4, 539–549. MR 780075, DOI 10.1007/BF02566365
- Peter Hindman, Lee Klingler, and Charles J. Odenthal, On the Krull-Schmidt-Azumaya theorem for integral group rings, Comm. Algebra 26 (1998), no. 11, 3743–3758. MR 1647074, DOI 10.1080/00927879808826371
- Akinari Hoshi and Ming-Chang Kang, Twisted symmetric group actions, Pacific J. Math. 248 (2010), no. 2, 285–304. MR 2741249, DOI 10.2140/pjm.2010.248.285
- Akinari Hoshi, Ming-chang Kang, and Hidetaka Kitayama, Quasi-monomial actions and some 4-dimensional rationality problems, J. Algebra 403 (2014), 363–400. MR 3166080, DOI 10.1016/j.jalgebra.2014.01.019
- Akinari Hoshi, Ming-Chang Kang, and Boris E. Kunyavskii, Noether’s problem and unramified Brauer groups, Asian J. Math. 17 (2013), no. 4, 689–713. MR 3152260, DOI 10.4310/AJM.2013.v17.n4.a8
- Akinari Hoshi, Hidetaka Kitayama, and Aiichi Yamasaki, Rationality problem of three-dimensional monomial group actions, J. Algebra 341 (2011), 45–108. MR 2824511, DOI 10.1016/j.jalgebra.2011.06.004
- Akinari Hoshi and Yūichi Rikuna, Rationality problem of three-dimensional purely monomial group actions: the last case, Math. Comp. 77 (2008), no. 263, 1823–1829. MR 2398796, DOI 10.1090/S0025-5718-08-02069-3
- Alfredo Jones, On representations of finite groups over valuation rings, Illinois J. Math. 9 (1965), 297–303. MR 175981
- M. Kang, Retract rationality and Noether’s problem, Int. Math. Res. Not. IMRN 2009 2760–2788.
- Ming-chang Kang, Retract rational fields, J. Algebra 349 (2012), 22–37. MR 2853623, DOI 10.1016/j.jalgebra.2011.10.024
- Ming-chang Kang and Yuri G. Prokhorov, Rationality of three-dimensional quotients by monomial, J. Algebra 324 (2010), no. 9, 2166–2197. MR 2684136, DOI 10.1016/j.jalgebra.2010.07.037
- Donald E. Knuth, The art of computer programming. Vol. 2, Addison-Wesley, Reading, MA, 1998. Seminumerical algorithms; Third edition [of MR0286318]. MR 3077153
- B. E. Kunyavskii, Three-dimensional algebraic tori, Selecta Math. Soviet. 9 (1990) 1–21.
- B. È. Kunyavskiĭ, Three-dimensional algebraic tori, Investigations in number theory (Russian), Saratov. Gos. Univ., Saratov, 1987, pp. 90–111 (Russian). Translated in Selecta Math. Soviet. 9 (1990), no. 1, 1–21. MR 1032541
- Boris Kunyavskiĭ, The Bogomolov multiplier of finite simple groups, Cohomological and geometric approaches to rationality problems, Progr. Math., vol. 282, Birkhäuser Boston, Boston, MA, 2010, pp. 209–217. MR 2605170, DOI 10.1007/978-0-8176-4934-0_{8}
- Nicole Lemire and Martin Lorenz, On certain lattices associated with generic division algebras, J. Group Theory 3 (2000), no. 4, 385–405. MR 1790337, DOI 10.1515/jgth.2000.031
- Nicole Lemire, Vladimir L. Popov, and Zinovy Reichstein, Cayley groups, J. Amer. Math. Soc. 19 (2006), no. 4, 921–967. MR 2219306, DOI 10.1090/S0894-0347-06-00522-4
- Lieven Le Bruyn, Generic norm one tori, Nieuw Arch. Wisk. (4) 13 (1995), no. 3, 401–407. MR 1378805
- H. W. Lenstra Jr., Rational functions invariant under a finite abelian group, Invent. Math. 25 (1974), 299–325. MR 347788, DOI 10.1007/BF01389732
- Martin Lorenz, Multiplicative invariant theory, Encyclopaedia of Mathematical Sciences, vol. 135, Springer-Verlag, Berlin, 2005. Invariant Theory and Algebraic Transformation Groups, VI. MR 2131760
- M. Matsumoto, T. Nishimura, Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator, ACM Trans. on Modeling and Computer Simulation 8 (1998) 3–30.
- Primož Moravec, Unramified Brauer groups of finite and infinite groups, Amer. J. Math. 134 (2012), no. 6, 1679–1704. MR 2999292, DOI 10.1353/ajm.2012.0046
- J. Opgenorth, W. Plesken, and T. Schulz, Crystallographic algorithms and tables, Acta Cryst. Sect. A 54 (1998), no. 5, 517–531. MR 1645546, DOI 10.1107/S010876739701547X
- Wilhelm Plesken, On reducible and decomposable representations of orders, J. Reine Angew. Math. 297 (1978), 188–210. MR 466193, DOI 10.1515/crll.1978.297.188
- Wilhelm Plesken, Bravais groups in low dimensions, Match 10 (1981), 97–119 (loose microfiche). MR 620803
- W. Plesken, Finite unimodular groups of prime degree and circulants, J. Algebra 97 (1985), no. 1, 286–312. MR 812182, DOI 10.1016/0021-8693(85)90086-9
- W. Plesken and W. Hanrath, The lattices of six-dimensional Euclidean space, Math. Comp. 43 (1984), no. 168, 573–587. MR 758205, DOI 10.1090/S0025-5718-1984-0758205-5
- Wilhelm Plesken and Michael Pohst, On maximal finite irreducible subgroups of $\textrm {GL}(n, \textbf {Z})$. I. The five and seven dimensional cases, Math. Comp. 31 (1977), no. 138, 536–551. MR 444789, DOI 10.1090/S0025-5718-1977-0444789-X
- Wilhelm Plesken and Michael Pohst, On maximal finite irreducible subgroups of $\textrm {GL}(n,\,\textbf {Z})$. III. The nine-dimensional case, Math. Comp. 34 (1980), no. 149, 245–258. MR 551303, DOI 10.1090/S0025-5718-1980-0551303-7
- Wilhelm Plesken and Tilman Schulz, Counting crystallographic groups in low dimensions, Experiment. Math. 9 (2000), no. 3, 407–411. MR 1795312
- S. Yu. Popov, Galois lattices and their birational invariants, Vestn. Samar. Gos. Univ. Mat. Mekh. Fiz. Khim. Biol. 4 (1998), 71–83 (Russian, with English and Russian summaries). MR 2119726
- S. S. Ryškov, Maximal finite groups of $n\times n$ integral matrices and full integral automorphism groups of positive quadratic forms (Bravais types), Trudy Mat. Inst. Steklov. 128 (1972), 183–211, 261 (Russian). Collection of articles dedicated to Academician Ivan Matveevič Vinogradov on his eightieth birthday, II. MR 0344199
- S. S. Ryškov, The maximal finite groups of integer $n\times n$ matrices, Dokl. Akad. Nauk SSSR 204 (1972), 561–564 (Russian). MR 0304501
- S. S. Ryškov and Z. D. Lomakina, A proof of the theorem on maximal finite groups of integral $5\times 5$ matrices, Trudy Mat. Inst. Steklov. 152 (1980), 204–215, 238 (Russian). Geometry of positive quadratic forms. MR 603825
- David J. Saltman, Retract rational fields and cyclic Galois extensions, Israel J. Math. 47 (1984), no. 2-3, 165–215. MR 738167, DOI 10.1007/BF02760515
- David J. Saltman, Noether’s problem over an algebraically closed field, Invent. Math. 77 (1984), no. 1, 71–84. MR 751131, DOI 10.1007/BF01389135
- David J. Saltman, Multiplicative field invariants, J. Algebra 106 (1987), no. 1, 221–238. MR 878475, DOI 10.1016/0021-8693(87)90028-7
- David J. Saltman, Multiplicative field invariants and the Brauer group, J. Algebra 133 (1990), no. 2, 533–544. MR 1067425, DOI 10.1016/0021-8693(90)90288-Y
- Bernd Souvignier, Irreducible finite integral matrix groups of degree $8$ and $10$, Math. Comp. 63 (1994), no. 207, 335–350. With microfiche supplement. MR 1213836, DOI 10.1090/S0025-5718-1994-1213836-X
- Richard G. Swan, Induced representations and projective modules, Ann. of Math. (2) 71 (1960), 552–578. MR 138688, DOI 10.2307/1969944
- Richard G. Swan, Noether’s problem in Galois theory, Emmy Noether in Bryn Mawr (Bryn Mawr, Pa., 1982) Springer, New York-Berlin, 1983, pp. 21–40. MR 713790
- Richard G. Swan, Torsion free cancellation over orders, Illinois J. Math. 32 (1988), no. 3, 329–360. MR 947032
- Richard G. Swan, The flabby class group of a finite cyclic group, Fourth International Congress of Chinese Mathematicians, AMS/IP Stud. Adv. Math., vol. 48, Amer. Math. Soc., Providence, RI, 2010, pp. 259–269. MR 2744226
- V. E. Voskresenskiĭ, On two-dimensional algebraic tori. II, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 711–716 (Russian). MR 0214597
- V. E. Voskresenskiĭ, Birational properties of linear algebraic groups, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 3–19 (Russian). MR 0262251
- V. E. Voskresenskiĭ, Stable equivalence of algebraic tori, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 3–10 (Russian). MR 0342515
- V. E. Voskresenskiĭ, Projective invariant Demazure models, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 2, 195–210, 431 (Russian). MR 651645
- V. E. Voskresenskiĭ, Algebraic groups and their birational invariants, Translations of Mathematical Monographs, vol. 179, American Mathematical Society, Providence, RI, 1998. Translated from the Russian manuscript by Boris Kunyavski [Boris È. Kunyavskiĭ]. MR 1634406
- Aiichi Yamasaki, Negative solutions to three-dimensional monomial Noether problem, J. Algebra 370 (2012), 46–78. MR 2966827, DOI 10.1016/j.jalgebra.2012.07.018