×

Wild Pfister forms over Henselian fields, \(K\)-theory, and conic division algebras. (English) Zbl 1222.17009

The present paper constitutes a thorough and comprehensive study of conic algebras, in particular composition algebras, and of pointed quadratic spaces, in particular quadratic spaces given by Pfister forms, over fields with a \(2\)-Henselian valuation, with an emphasis on the case of residue characteristic \(2\). The relation between these objects as well as between them and Milnor \(K\)-theory modulo \(2\) is explored in quite some detail. An extensive bibliography complements this well-written article.
Part I of the paper gives a detailed general study of what the authors call conic algebras over a field \(k\), in particular in the case where \(\text{char(k)}=2\). Here, conic means that there exists a quadratic form \(n:C\to k\) (the norm of \(C\)) with associated polar bilinear form \(t(\,.\,,\,.\,)\) such that each element \(x\in C\) satisfies \(x^2-t(1_C,x)x+n(x)1_C=0\) (some authors call such algebras quadratic). \(C\) is said to be nondegenerate if the bilinear form \(t\) is nondegenerate. Composition algebras are examples of conic algebras. A standard way to construct conic algebras is to take a conic algebra \(B\), a scalar \(\mu\in k^*\) and to use the Cayley-Dickson construction to obtain a conic algebra \(C=\text{Cay}(B,\mu)\) of twice the dimension. The authors explain their version of a non-orthogonal Cayley-Dickson construction in characteristic \(2\). Starting with a purely inseparable field extensions \(K/k\) of exponent \(1\) (i.e. \(K^2\subseteq k\)) together with a scalar \(\mu\in k\) and a unital linear form \(s: K\to k\), they define a unital non-associative \(k\)-algebra \(C:=\text{Cay}(K;\mu,s)\) on the direct sum \(K\oplus Kj\) that is uniquely determined by certain relations that are given explicitly. \(C\) is shown to be nondegenerate, flexible and conic, and if \([K:k]=2^n\), then the norm of \(C\) is given by an \((n+1)\)-fold quadratic Pfister form \(\langle\!\langle a_1,\ldots,a_n,\mu ]]\) for certain \(a_i\) that satisfy \(K=k(\sqrt{a_1},\ldots,\sqrt{a_n})\). Various properties are shown. For example, it is shown that \(C\) is division iff its norm form is anisotropic. Thus, one can easily construct many new examples of conic division algebras. Furthermore, it is shown that \(C\) is a composition algebra iff \([K:k]\leq 4\), in which case \(C= \text{Cay}(K;\mu,s)\) is isomorphic to \(\text{Cay}(K;\mu',s)\) iff their respective norm forms are isometric. These algebras are also used in a proof of a certain version of the Skolem-Noether theorem for composition algebras in characteristic \(2\). It is shown that if \(C\) is such a composition algebra, then every isomorphism between inseparable subfields of \(C\) can be extended to an automorphism of \(C\). The authors also include a short section on Pfister forms that contains a result that is of interest in its own right. Namely, if \(q\) is a quadratic Pfister form containing quadratic Pfister forms \(q_1\) and \(q_2\) with \(\dim q_1\leq \dim q_2\), then there are bilinear Pfister forms \(b_1\), \(b_2\) with \(b_2\otimes b_1\otimes q_1\cong q\cong b_2\otimes q_2\).
In Part II, the authors study quadratic spaces and composition algebras over \(2\)-Henselian fields where the case of the residue characteristic \(2\) is included. They first recall the definition of a pointed quadratic space \(Q=(V_Q,n_Q,1_Q)\), i.e. a triple consisting of a quadratic form \(n_Q\) on a vector space \(V_Q\) with an element \(1_Q\in V\) with \(n_Q(1_Q)=1\). For \(x\in V_Q\), one defines \(t_Q(x)=\partial n_Q(1_Q,x)\) (where \(\partial n_Q\) denotes the polar bilinear form associated to \(n_Q\)), and \(V_Q^\times = \{ x\in V_Q|n_Q(x)\neq 0\}\). In general, \(n_Q\) is assumed to be round, i.e. the nonzero values represented by \(n_Q\) coincide with the group of similarity factors of \(n_Q\). Pfister forms are examples of round forms. The base field \(F\) is assumed to carry a \(2\)-Henselian discrete valuation \(\lambda: F\to\mathbb{Z}\cup\infty\) with ring of integers \(\mathfrak{o}\), maximal ideal \(\mathfrak{p}\) and residue field \(\overline{F}\). One has a norm \(\lambda_Q:V_Q\to \mathbb{Q}\cup\infty\) with \(\lambda_Q(x)=\frac{1}{2}\lambda (n_Q(x))\). With \(\Gamma_Q=\lambda_Q(V_Q^\times)\) and \(n_Q\) being round, one has \(\Gamma_Q=\frac{1}{e_{Q/F}}\mathbb{Z}\) where \(e_{Q/F}=[\Gamma_Q:\mathbb{Z}]\in \{ 1,2\}\) is the ramification index of \(Q\). To get pointed quadratic residue spaces, one first defines the \(\mathfrak{o}\)-lattices \(\mathfrak{o}_Q\) resp. \(\mathfrak{p}_Q\) consisting of those \(x\in V_Q\) with \(\lambda_Q(x)\geq 0\) resp. \(>0\). Thus, one obtains \(\overline{Q}=(V_{\overline{Q}},n_{\overline{Q}},1_{\overline{Q}})\) where \(V_{\overline{Q}}=\mathfrak{o}_Q/\mathfrak{p}_Q\), \(1_{\overline{Q}}=\overline{1_Q}\), and \(n_{\overline{Q}}(\overline{x}):=\overline{n_Q(x)}\), the residue form. \(f_{Q/F}:=\dim_{\overline{F}}(V_{\overline{Q}})\) is called the residue degree of \(Q\). It is shown that \(e_{Q/F}f_{Q/F}=\dim_F(V_Q)\). \(Q\) is said to be tame if \(t_{\overline{Q}}\neq 0\), wild otherwise. Tame pointed spaces \(Q\) are further subdivided into unramified ones where \(e_{Q/F}=1\), and ramified ones where \(e_{Q/F}=2\). The authors introduce various invariants such as the trace exponent \(\text{texp}_Q\), the norm exponent \(\text{nexp}_Q\), and what they call Tignol’s invariant \(\omega(Q)\). These invariants encode valuation theoretic information about the values \(t_Q\) and \(n_Q\) can take. Various relations between these invariants are shown. The authors then consider a pointed quadratic space \(P\) (generally assumed to be nonsingular, round, anisotropic with ramification index \(e_{P/F}=1\)) and study the behavior of the ramification index, the pointed residue space and the trace exponent when passing to \(Q=\langle\!\langle \mu\rangle\!\rangle\otimes P\), and under which conditions a pointed quadratic \(n\)-Pfister space \(P\) embeds into a a pointed quadratic \((n+1)\)-Pfister space \(Q\).
In analogy to Albert’s absolute-valued algebras the authors then introduce the concept of \(\lambda\)-normed resp. \(\lambda\)-valued conic algebras over a \(2\)-Henselian field \((F,\lambda)\), i.e. conic algebras that are round and anisotropic (considered as quadratic spaces) and that satisfy \(\lambda_C(xy)\geq\lambda_C(x)+\lambda_C(y)\) resp. \(\lambda_C(xy)=\lambda_C(x)+\lambda_C(y)\). \(\lambda\)-valued conic algebras are always division, and composition algebras are \(\lambda\)-valued and conic iff they are division. The authors then interpret their earlier results on pointed quadratic spaces over \((F,\lambda)\) in the context of algebras obtained by the Cayley-Dickson construction. For example, if \(B\) is a \(\lambda\)-normed conic algebra of ramification index \(1\), then criteria on \(B\) and \(\mu\in\mathfrak{o}\) are given for \(\text{Cay}(B,\mu)\) to be \(\lambda\)-valued resp. anisotropic. Applications to composition algebras are also given, such as criteria for a quaternion or octonion division algebra to have a prescribed trace exponent. This part of the paper concludes with extending Tignol’s notion of height (which agrees with Saltman’s notion of level) from the context of central associative division algebras of degree \(p\) over Henselian fields with residue characteristic \(p\) to the setting of composition algebras, and relating the new notions of height to the earlier invariant \(\omega\).
The final Part III deals with Milnor \(K\)-theory modulo \(p\) over Henselian fields \((F,\lambda)\) of characteristic \(0\) with residue characteristic \(p>0\). The authors first recall the relevant classical results in this setting due to Kato and others. Their aim is then to express the results from Part II in terms of valuation-theoretic properties of symbols in Milnor \(K\)-theory modulo \(p=2\), and also in the case of arbitrary \(p>0\), to relate traditional valuation-theoretic terms on associative division algebras of degree \(p\) to Milnor \(K\)-theory modulo \(p\). In this context, the authors prove what they call the Gathering Lemma, which is a nice result of independent interest on the structure of symbols and which says that the slots of a symbol can be chosen in a convenient way in terms of the valuation.

MSC:

17A75 Composition algebras
11E04 Quadratic forms over general fields
11E08 Quadratic forms over local rings and fields
12F15 Inseparable field extensions
12J10 Valued fields
16K20 Finite-dimensional division rings
17A20 Flexible algebras
17A45 Quadratic algebras (but not quadratic Jordan algebras)
17A80 Valued algebras
19D45 Higher symbols, Milnor \(K\)-theory
19F15 Symbols and arithmetic (\(K\)-theoretic aspects)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Albert, A. A., Absolute valued real algebras, Ann. of Math. (2), 48, 495-501 (1947) · Zbl 0029.01001
[2] Albert, A. A., On nonassociative division algebras, Trans. Amer. Math. Soc., 72, 296-309 (1952) · Zbl 0046.03601
[3] Becker, E., Über eine Klasse flexibler quadratischer Divisionsalgebren, J. Reine Angew. Math., 256, 25-57 (1972) · Zbl 0222.17001
[4] Bloch, S.; Kato, K., \(p\)-adic étale cohomology, Publ. Math. Inst. Hautes Etudes Sci., 63, 107-152 (1986) · Zbl 0613.14017
[5] Bourbaki, N., Algebra I: Chapters 1-3 (1989), Springer-Verlag · Zbl 0673.00001
[6] Bourbaki, N., Algebra II: Chapters 4-7 (1990), Springer-Verlag · Zbl 0719.12001
[7] Brown, R. B., On generalized Cayley-Dickson algebras, Pacific J. Math., 20, 415-422 (1967) · Zbl 0168.28501
[8] Bruhat, F.; Tits, J., Groupes réductifs sur un corps local, (Inst. Hautes Études Sci. Publ. Math., vol. 41 (1972)), 5-251 · Zbl 0254.14017
[9] Bruhat, F.; Tits, J., Schémas en groupes et immeubles des groupes classiques sur un corps local, Bull. Soc. Math. France, 112, 2, 259-301 (1984) · Zbl 0565.14028
[10] Colliot-Thélène, J.-L., Cohomologie galoisienne des corps valués discrets henseliens, d’après K. Kato et S. Bloch, (Bass, H.; Kuku, A. O.; Pedrini, C., Algebraic \(K\)-Theory and Its Applications. Algebraic \(K\)-Theory and Its Applications, Trieste, 1997 (1999)), 120-163 · Zbl 0976.11053
[11] Draxl, P. K., Skew Fields, London Math. Soc. Lecture Note, vol. 81 (1983), Cambridge University Press: Cambridge University Press Cambridge, New York · Zbl 0498.16015
[12] Dress, A., Metrische Ebenen über quadratisch perfekten Körpern, Math. Z., 92, 19-29 (1966) · Zbl 0136.15002
[13] Ebbinghaus, H.-D.; Hermes, H.; Hirzebruch, F.; Koecher, M.; Mainzer, K.; Neukirch, J.; Prestel, A.; Remmert, R., Numbers, Grad. Texts in Math., vol. 123 (1991), Springer-Verlag · Zbl 0705.00001
[14] Efrat, I., Valuations, Orderings, and Milnor \(K\)-Theory, Math. Surveys Monogr., vol. 124 (2006), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1103.12002
[15] Elman, R.; Karpenko, N.; Merkurjev, A., The Algebraic and Geometric Theory of Quadratic Forms, Amer. Math. Soc. Colloq. Publ., vol. 56 (2008), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1165.11042
[16] Engler, A. J.; Prestel, A., Valued Fields, Springer Monogr. Math. (2005), Springer-Verlag: Springer-Verlag Berlin · Zbl 1128.12009
[17] Flaut, C., Division algebras with dimension \(2^t, t \in N\), An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat., 13, 2, 31-38 (2005)
[18] Garibaldi, S.; Merkurjev, A.; Serre, J.-P., Cohomological Invariants in Galois Cohomology, Univ. Lecture Ser., vol. 28 (2003), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1159.12311
[19] Gille, P.; Szamuely, T., Central Simple Algebras and Galois Cohomology, Cambridge Stud. Adv. Math., vol. 101 (2006), Cambridge University Press · Zbl 1137.12001
[20] Hoffmann, D. W., Diagonal forms of degree \(p\) in characteristic \(p\), (Contemp. Math., vol. 344 (2004)), 135-183 · Zbl 1074.11023
[21] Hoffmann, D. W.; Laghribi, A., Quadratic forms and Pfister neighbors in characteristic 2, Trans. Amer. Math. Soc., 356, 4019-4053 (2004) · Zbl 1116.11020
[22] Jacob, B., Quadratic forms over dyadic valued fields. I. The graded Witt ring, Pacific J. Math., 126, 1, 21-79 (1987) · Zbl 0606.10013
[23] Kato, K., A generalization of local class field theory by using \(K\)-groups. I, J. Fac. Sci. Univ. Tokyo IA, 26, 303-376 (1979) · Zbl 0428.12013
[24] Kato, K., A generalization of local class field theory by using \(K\)-groups. II, J. Fac. Sci. Univ. Tokyo IA, 27, 603-683 (1980) · Zbl 0463.12006
[25] Kato, K., Galois cohomology of complete discrete valuation fields, (Algebraic \(K\)-Theory, Part II. Algebraic \(K\)-Theory, Part II, Oberwolfach, 1980. Algebraic \(K\)-Theory, Part II. Algebraic \(K\)-Theory, Part II, Oberwolfach, 1980, Lecture Notes in Math., vol. 967 (1982), Springer-Verlag), 215-238
[26] Kato, K., Symmetric bilinear forms, quadratic forms and Milnor \(K\)-theory in characteristic two, Invent. Math., 66, 3, 493-510 (1982) · Zbl 0497.18017
[27] Knebusch, M., Generic splitting of quadratic forms. I, Proc. Lond. Math. Soc. (3), 33, 1, 65-93 (1976) · Zbl 0351.15016
[28] Knebusch, M., Specialization of quadratic and symmetric bilinear forms (2007)
[29] Knus, M.-A.; Merkurjev, A.; Rost, M.; Tignol, J.-P., The Book of Involutions, Amer. Math. Soc. Colloq. Publ., vol. 44 (1998), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI
[30] Lam, T.-Y., Introduction to Quadratic Forms over Fields, Grad. Stud. Math., vol. 67 (2005), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1068.11023
[31] Loos, O., Algebras with scalar involution revisited (2010), preprint
[32] McCarthy, P. J., Algebraic Extensions of Fields (1991), Dover Publications Inc.: Dover Publications Inc. New York · Zbl 0143.05802
[33] McCrimmon, K., Nonassociative algebras with scalar involution, Pacific J. Math., 116, 1, 85-109 (1985) · Zbl 0558.17002
[34] Merkurjev, A. S.; Suslin, A. A., \(K\)-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Math. USSR Izv., 21, 2, 307-340 (1983) · Zbl 0525.18008
[35] O’Meara, O. T., Introduction to Quadratic Forms, Grundlehren Math. Wiss., vol. 117 (1963), Springer-Verlag: Springer-Verlag New York · Zbl 0107.03301
[36] Osborn, J. M., Quadratic division algebras, Trans. Amer. Math. Soc., 105, 202-221 (1962) · Zbl 0136.30303
[37] Rodríguez Palacios, Á., Absolute-valued algebras, and absolute-valuable Banach spaces, (Advanced Courses of Mathematical Analysis I (2004), World Sci. Publ.: World Sci. Publ. Hackensack, NJ), 99-155 · Zbl 1093.46022
[38] Petersson, H. P., Borel subalgebras of alternative and Jordan algebras, J. Algebra, 16, 541-560 (1970) · Zbl 0209.06802
[39] Petersson, H. P., Eine Bemerkung zu quadratischen Divisionsalgebren, Arch. Math. (Basel), 22, 59-61 (1971) · Zbl 0216.07301
[40] Petersson, H. P., Jordan-Divisionsalgebren und Bewertungen, Math. Ann., 202, 215-243 (1973) · Zbl 0351.17015
[41] Petersson, H. P., Composition algebras over a field with a discrete valuation, J. Algebra, 29, 414-426 (1974) · Zbl 0291.17013
[42] Petersson, H. P.; Racine, M. L., Albert algebras, (Kaup, W.; McCrimmon, K.; Petersson, H. P., Jordan Algebras, Proceedings of a conference at Oberwolfach, 1992 (1994), de Gruyter: de Gruyter Berlin), 197-207 · Zbl 0810.17021
[43] Pumplün, S., A non-orthogonal Cayley-Dickson doubling, J. Algebra Appl., 5, 2, 193-199 (2006) · Zbl 1138.17002
[44] F. Rosemeier, Semiquadratische Algebren, PhD thesis, FernUniversität in Hagen, 2002.; F. Rosemeier, Semiquadratische Algebren, PhD thesis, FernUniversität in Hagen, 2002.
[45] Saltman, D. J., Division algebras over discrete valued fields, Comm. Algebra, 8, 18, 1749-1774 (1980) · Zbl 0442.16016
[46] Springer, T. A., Quadratic forms over fields with a discrete valuation. I. Equivalence classes of definite forms, Indag. Math., 17, 352-362 (1955) · Zbl 0067.27605
[47] Springer, T. A.; Veldkamp, F. D., Octonions, Jordan Algebras and Exceptional Groups, Springer Monogr. Math. (2000), Springer-Verlag: Springer-Verlag Berlin · Zbl 1087.17001
[48] Suslin, A.; Joukhovitski, S., Norm varieties, J. Pure Appl. Algebra, 206, 1-2, 245-276 (2006) · Zbl 1091.19002
[49] Teichmüller, O., Diskret bewertete perfekte Körper mit unvollkommenem Restklassenkörper, J. Reine Angew. Math., 176, 141-152 (1937) · JFM 62.1114.01
[50] Tietze, U. P., Zur Theorie quadratischer Formen über Hensel-Korpern, Arch. Math., 17, 352-362 (1955)
[51] Tignol, J.-P., Algèbres à division et extensions de corps sauvagement ramifiées de degré premier, J. Reine Angew. Math., 404, 1-38 (1990) · Zbl 0684.16011
[52] Tignol, J.-P., Classification of wild cyclic field extensions and division algebras of prime degree over a Henselian field, (Proceedings of the International Conference on Algebra, Part 2. Proceedings of the International Conference on Algebra, Part 2, Novosibirsk, 1989. Proceedings of the International Conference on Algebra, Part 2. Proceedings of the International Conference on Algebra, Part 2, Novosibirsk, 1989, Contemp. Math., vol. 131 (1992), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 491-508 · Zbl 0795.16013
[53] van der Blij, F.; Springer, T. A., The arithmetics of octaves and of the group \(G_2\), Indag. Math., 21, 406-418 (1959) · Zbl 0089.25803
[54] Wadsworth, A. R., Valuation theory on finite dimensional division algebras, (Valuation Theory and Its Applications, vol. I. Valuation Theory and Its Applications, vol. I, Saskatoon, SK, 1999. Valuation Theory and Its Applications, vol. I. Valuation Theory and Its Applications, vol. I, Saskatoon, SK, 1999, Fields Inst. Commun., vol. 32 (2002), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 385-449 · Zbl 1017.16011
[55] Weiss, R. M., Quadrangular Algebras, Math. Notes, vol. 46 (2006), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 1129.17001
[56] R.M. Weiss, On the existence of certain affine buildings of type \(E_6E_7\); R.M. Weiss, On the existence of certain affine buildings of type \(E_6E_7\)
[57] Zhevlakov, K. A.; Slin’ko, A. M.; Shestakov, I. P.; Shirshov, A. I., Rings that are Nearly Associative, Pure Appl. Math., vol. 104 (1982), Academic Press: Academic Press New York · Zbl 0487.17001
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.