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Applications of patching to quadratic forms and central simple algebras. (English) Zbl 1259.12003
This paper gives applications of field patching, introduced by the first two authors in [Isr. J. Math. 176, 61–107 (2010; Zbl 1213.14052)], to problems in quadratic form theory (computation of the \(u\)-invariant) and central simple algebras (the period-index problem).
The technical setting of field patching is as follows: For a normal irreducible projective curve \(\hat{X}\) over a complete discrete valuation ring \(T\) with function field \(F\), one defines a set \(\mathcal{P}\) of special points, a set \(\mathcal{U}\) of subsets of \(\hat{X}\), and a set \(\mathcal{B}\) of branches at points in \(\mathcal{P}\), and for each \(\xi\in\mathcal{P}\cup\mathcal{U}\cup\mathcal{B}\) a certain field extension \(F_\xi\) of \(F\). Patching now allows to construct objects (like vector spaces) over \(F\), given such objects over the \(F_\xi\) in a compatible way. A key ingredient in the patching process is a factorization result for invertible matrices. Sections 2 and 3 of the paper generalize the matrix factorization results of [loc. cit.] from \(\text{GL}_n\) to connected linear algebraic groups \(G\) over \(F\) that are rational as \(F\)-varieties. From this the authors deduce the following local-global principle for homogeneous spaces: If such a group \(G\) acts on an \(F\)-variety \(H\) such that for every extension \(E\) of \(H\), \(G(E)\) acts transitively on \(H(E)\), then \(H(F)\neq\emptyset\) iff \(H(F_\xi)\neq\emptyset\) for every \(\xi\in\mathcal{P}\cup\mathcal{U}\).
By applying this local-global principle to the rational linear algebraic group \(\text{SO}(q)\), for a quadratic form \(q\) over \(F\) with \(\text{dim}(q)\neq 2\), they deduce that \(q\) is isotropic over \(F\) iff it is isotropic over each \(F_\xi\), \(\xi\in\mathcal{P}\cup\mathcal{U}\). From this they conclude that if \(T\) is a complete (or more generally, excellent henselian) discrete valuation ring with quotient field \(K\) and residue field \(k\), \(\text{char}(k)\neq2\), then \(u_s(K)=2u_s(k)\), where \(u_s\) denotes the strong \(u\)-invariant, defined by \(u_s(K)\leq n\) iff \(u(E)\leq 2^ln\) for every finitely generated extension \(E/k\) of transcendence degree \(l\leq1\), and \(u\) denotes the \(u\)-invariant (in the sense of Kaplansky) defined as the maximal dimension of an anisotropic quadratic form. As a special case they obtain the result of R. Parimala and V. Suresh [Ann. Math. (2) 172, No. 2, 1391–1405 (2010; Zbl 1208.11053)] that the \(u\)-invariant of a function field of one variable over \(\mathbb{Q}_p\), \(p\neq2\), is \(8\). Further concrete results on the \(u\)-invariant are given for function fields over higher local fields, and quotient fields of power series rings like \(k[[x,t]]\), \(k[x][[t]]\), and \(\mathbb{Z}_p[[x]]\).
The applications to central simple algebras concern the period-index problem. Recall that the period \(\text{per}(A)\) of a central simple \(k\)-algebra \(A\) is the order of \(A\) in the Brauer group \(\text{Br}(k)\), and the index \(\text{ind}(A)\) is the degree of the division algebra in the Brauer class of \(A\). The authors define the Brauer dimension of \(k\) away from \(p\) to be at most \(d\) iff \(\text{ind}(A)|\text{per}(A)^{d+l-1}\) for every finitely generated extension \(E/k\) of transcendence degree \(l\leq 1\) and every central simple \(E\)-algebra \(A\) with \(p{\not|}\text{per}(A)\). The technical main result here is that, in the setting described above, for a central simple \(F\)-algebra \(A\), \(\text{ind}(A)=\text{lcm}_{\xi\in\mathcal{P}\cup\mathcal{U}}\text{ind}(A\otimes_F{F_\xi})\).
From this the authors deduce that if \(T\) is a complete (or more generally, excellent henselian) discrete valuation ring with quotient field \(K\) and residue field \(k\), and \(k\) has Brauer dimension \(d\geq0\) away from \(\text{char}(k)\), then \(K\) has Brauer dimension at most \(d+1\) away from \(\text{char}(k)\). As a special case they obtain the result of D. J. Saltman [J. Ramanujan Math. Soc. 12, No. 1, 25–47 (1997; Zbl 0902.16021)] that \(\text{ind}(A)|\text{per}(A)^2\) for any central simple algebra \(A\) over a function field of one variable over a finite extension of \(\mathbb{Q}_p\). Further results on the period-index problem are given for function fields of one variable over higher local fields, and for quotient fields of rings like \(k[[x,t]]\), \(k[x][[t]]\) and \(\mathbb{Z}_p^{\text{ur}}[[x]]\).

MSC:
12E30 Field arithmetic
11E08 Quadratic forms over local rings and fields
16K20 Finite-dimensional division rings
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References:
[1] Abhankar, S.S.: Resolution of singularities of algebraic surfaces. In: Algebraic Geometry, Proc. of Internat. Colloq., Bombay, 1968. Tata Inst. Fund. Res., pp. 1–11. Oxford University Press, London (1969)
[2] Artin, M.: Algebraic approximation of structures over complete local rings. Publ. Math. Inst. Hautes Études Sci. 36, 23–58 (1969) · Zbl 0181.48802
[3] Borel, A.: Linear Algebraic Groups, 2nd edn. Graduate Texts in Mathematics, vol. 126. Springer, Berlin (1991) · Zbl 0726.20030
[4] Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron Models. Ergebnisse der Mathematik, vol. 21. Springer, Berlin/Heidelberg (1990) · Zbl 0705.14001
[5] Colliot-Thélène, J.-L., Ojanguren, M., Parimala, R.: Quadratic forms over fraction fields of two-dimensional Henselian rings and Brauer groups of related schemes. In: Proceedings of the International Colloquium on Algebra, Arithmetic and Geometry. Tata Inst. Fund. Res. Stud. Math., vol. 16, pp. 185–217. Narosa, New Delhi (2002) · Zbl 1055.14019
[6] de Jong, A.J.: The period-index problem for the Brauer group of an algebraic surface. Duke Math. J. 123, 71–94 (2004) · Zbl 1060.14025
[7] Eisenbud, D.: Commutative Algebra. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995) · Zbl 0819.13001
[8] Ford, T.J.: The Brauer group of a curve over a strictly local discrete valuation ring. Israel J. Math. 96, 259–266 (1996) · Zbl 0873.13004
[9] Greenberg, M.J.: Rational points in Henselian discrete valuation rings. Publ. Math. Inst. Hautes Études Sci. 31, 59–64 (1966) · Zbl 0146.42201
[10] Grothendieck, A.: Éléments de géométrie algébrique III. Publ. Math. Inst. Hautes Études Sci. 11 (1961)
[11] Grothendieck, A.: Éléments de géométrie algébrique IV, 2 e partie. Publ. Math. Inst. Hautes Études Sci. 24 (1965)
[12] Grothendieck, A.: Éléments de géométrie algébrique, IV, 4 e partie. Publ. Math. Inst. Hautes Études Sci. 32 (1967)
[13] Grothendieck, A.: Le groupe de Brauer. Parts I–III. In: Dix Exposés sur la Cohomologie des Schémas, pp. 46–188. North-Holland, Amsterdam (1968)
[14] Grothendieck, A.: Séminaire de Géométrie Algébrique 1. Lecture Notes in Mathematics, vol. 224. Springer, New York (1971)
[15] Grove, L.: Classical Groups and Geometric Algebra. Graduate Studies in Mathematics, vol. 39. American Mathematical Society, Providence (2002)
[16] Harbater, D.: Patching and Galois theory. In: Schneps, L. (ed.) Galois Groups and Fundamental Groups. MSRI Publications Series, vol. 41, pp. 313–424. Cambridge University Press, Cambridge (2003) · Zbl 1071.14029
[17] Harbater, D., Hartmann, J.: Patching over fields. Israel J. Math. (2007, to appear) (also available at arXiv:0710.1392 ) · Zbl 1213.14052
[18] Kneser, M.: Schwache Approximation in algebraischen Gruppen. In: Centre Belge Rech. Math. Colloque Théorie des Groupes Algébriques, Bruxelles 1962-41-52 (1962)
[19] Knus, M.-A., Merkurjev, A.S., Rost, M., Tignol, J.-P.: The Book of Involutions. American Mathematical Society, Providence (1998) · Zbl 0955.16001
[20] Lam, T.-Y.: Introduction to Quadratic Forms over Fields. American Mathematical Society, Providence (2005) · Zbl 1068.11023
[21] Lieblich, M.: Period and index in the Brauer group of an arithmetic surface, with an appendix by D. Krashen. Preprint arXiv:math/0702240v3 (2008)
[22] Lipman, J.: Introduction to resolution of singularities. In: Algebraic Geometry, Humboldt State University, Arcata, CA, 1974. Proceedings of Symposia in Pure Mathematics, vol. 29, pp. 187–230. American Mathematical Society, Providence (1975)
[23] Merkurjev, A.S., Suslin, A.A.: K-cohomology of Severi-Brauer varieties and the norm residue homomorphism. Izv. Akad. Nauk SSSR Ser. Mat. 46, 1011–1046, 1135–136 (1982) (in Russian; English translation: Math. USSR-Izv. 21, 307–340 (1983))
[24] Milne, J.S.: On the arithmetic of Abelian varieties. Invent. Math. 17, 177–190 (1972) · Zbl 0249.14012
[25] Parimala, R., Suresh, V.: The u-invariant of the function fields of p-adic curves. Preprint arXiv:0708.3128 (2007) · Zbl 1208.11053
[26] Pfister, A.: Quadratic Forms with Applications to Algebraic Geometry and Topology. LMS Lecture Notes Series, vol. 217. Cambridge University Press, Cambridge (1995) · Zbl 0847.11014
[27] Pierce, R.S.: Associative Algebras. Springer, New York (1982) · Zbl 0497.16001
[28] Reiner, I.: Maximal Orders. Academic Press, New York (1975) · Zbl 0305.16001
[29] Saltman, D.J.: Division algebras over p-adic curves. J. Ramanujan Math. Soc. 12(1), 25–47 (1997) · Zbl 0902.16021
[30] Saltman, D.J.: Correction to: Division algebras over p-adic curves. J. Ramanujan Math. Soc. 13(2), 125–129 (1998) · Zbl 0920.16008
[31] Saltman, D.J.: Lectures on Division Algebras. CBMS Regional Conference Series in Mathematics, vol. 94. American Mathematical Society, Providence (1999) · Zbl 0934.16013
[32] Saltman, D.J.: Cyclic algebras over p-adic curves. J. Algebra 314, 817–843 (2007) · Zbl 1129.16014
[33] Seelinger, G.F.: Brauer-Severi schemes of finitely generated algebras. Israel J. Math. 111, 321–337 (1999) · Zbl 0964.16026
[34] Serre, J.-P.: Cohomologie Galoisienne, 4th edn. Lecture Notes in Mathematics, vol. 5. Springer, New York (1973) · Zbl 0259.12011
[35] Shatz, S.S.: Profinite Groups, Arithmetic, and Geometry. Annals of Mathematics Studies, vol. 67. Princeton University Press, Princeton (1972) · Zbl 0236.12002
[36] Van den Bergh, M.: The Brauer-Severi scheme of the trace ring of generic matrices. In: Perspectives in Ring Theory, Antwerp, 1987. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 233, pp. 333–338. Kluwer Academic, Dordrecht (1988) · Zbl 0761.13003
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