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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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