Field of iterated Laurent series and its Brauer group. (English) Zbl 1452.16020

Let \(F\) be an infinite field, \(\mathrm{Br}(F)\) its Brauer group, \(p\) a prime number, \(_p\mathrm{Br}(F) = \{\beta \in\mathrm{Br}(F): p\beta = 0\}\), and \(\mathrm{Brd}_p(F)\) the Brauer \(p\)-dimension of \(F\), i.e., the supremum \(d \le \infty\) of those integers \(n \ge 0\), for which there exists a central division \(F\)-algebra \(\Delta\) of \(p\)-primary exponent exp\((\Delta)\) and degree deg\((\Delta) =\exp(\Delta)^n\). It is well known that if \(F\) contains a primitive \(p\)-th root of unity \(\rho\) and \(A\) is a cyclic \(F\)-algebra of degree \(p\), then \(A\) can be presented as \(F[x, y: x^p = \alpha, y^p = \beta, yx = \rho x, y]\), for some \(\alpha, \beta \in F^{\ast}\); we denote this presentation by \((\alpha, \beta)_{p,F}\). When \(\mathrm{char}(F) = p\), every cyclic \(F\)-algebra of degree \(p\) takes the form \([\alpha, \beta)_{p,F} = F \langle x, y: x^- x = \alpha, y^p = \beta, yxy^{-1} = x + 1 \rangle\), for some \(\alpha \in F\), \(\beta \in F^{\ast}\). These forms are called (Hilbert) symbol presentations of the algebras, and the algebras are also called symbol algebras. It is known that \(_p\mathrm{Br}(F)\) is generated by the Brauer equivalence classes of cyclic \(F\)-algebras of degree \(p\) in the following two cases: if \(F\) contains a primitive \(p\)-th root of unity (see [A. S. Merkur’ev and A. A. Suslin, Math. USSR, Izv. 21, 307–340 (1983; Zbl 0525.18008); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 5, 1011–1046 (1982)]); if \(\mathrm{char}(F) = p\) (see Ch. VII, Theorem 28, in: [A. A. Albert, Structure of algebras. American Mathematical Society (AMS), Providence, RI (1939; JFM 65.0094.02)]). The symbol length \(\mathrm{Sym}(b_p)\) of an element of \(_p\mathrm{Br}(F)\) is the minimal number of symbol algebras needed to express it, and the symbol length of \(_p\mathrm{Br}(F)\) is the supremum \(\mathrm{Sym}_p(F)\) of \(\mathrm{Sym}(b_p): b_p \in{}_{p}\mathrm{Br}(F)\).
The main results of the paper under review are stated and proved in its Sections 3 and 4. They concern the special case where \(F = k_n := k((\alpha_1)) \dots ((\alpha_n))\) is the iterated Laurent formal power series field in \(n\) variables over a field \(k\). Theorem 3.5, the main result of Section 3, states that if \(k\) is a perfect field with \(\mathrm{char}(k) = p\), \(k_{sep}\) is a separable closure of \(k\), and \(m\) is the rank (as a pro-\(p\)-group) of the Galois group \(\mathcal{G}(k(p)/k)\) of the maximal \(p\)-extension \(k(p)\) of \(k\) in \(k_{\mathrm{sep}}\), then \(\mathrm{Sym}_p(F) = n - 1\), provided that \(m < n\), and \(\mathrm{Sym}_p(F) = n\) if \(m \ge n\). As noted by the author, this complements the known result (presented by Proposition 3.1 of the paper) that \(\mathrm{Sym}_{p'}(F) = [n/2]\) in case \(k\) is algebraically closed, \(p'\) is prime,and \(p' \neq\mathrm{char}(k)\). When \(n = 3\), these results indicate that \(\mathrm{Brd}_2(F) = 1\) if and only if \(\mathrm{char}(k) \neq 2\). This fact is generalized in Section 4 as follows: (i) If \(F = k_{n+1}\), where \(k\) is an algebraically closed field with \(\mathrm{char}(k) \neq 2\), then \(I^n F\) is linked, i.e., every two anisotropic bilinear \(n\)-fold Pfister forms over \(F\) share an \((n-1)\)-fold Pfister factor; (ii) when \(k\) is a field of characteristic \(2\) and \(F = k_{n+1}\), \(I_q^n F\) is not linked, i.e., there exists a pair of quadratic \(n\)-fold Pfister forms which do not share an \((n - 1)\)-fold Pfister factor; (iii) for any field \(F\) with \(\mathrm{char}(F) = 2\) and degree \([F: F^2] > 2^n\), \(I^n F\) is not linked, i.e., there exists a pair of anisotropic bilinear Pfister forms over \(F\), which do not share an \((n - 1)\)-fold factor.
Reviewer’s remark. Let \((F, v)\) be a Henselian valued field with a residue field \(\widehat F\), and let \(\mathrm{Br}(\widehat F)_{p}\) be the \(p\)-component of \(\mathrm{Br}(\widehat F)\). Then \(\mathrm{Brd}_p(F) = \mathrm{Sym}_p(F)\) in the following two cases: (i) if \(\mathrm{Br}(\widehat F)_p = \{0\}\) and \(\widehat F\) contains a primitive \(p\)-th root of unity (see (4.7), Theorem 2.3, and Corollary 5.6 of the reviewer’s paper in [J. Pure Appl. Algebra 223, No. 1, 10–29 (2019; Zbl 1456.16015)]; (ii) if \((F, v)\) is maximally complete, \([F: F^p] = p^n\), for some \(n \in \mathbb{N}\), and \(\widehat F\) is perfect (see Proposition 3.5 in the reviewer’s paper in: [Serdica Math. J. 44, 303–328 (2018)]). This recovers the proof of Proposition 3.1 and generalizes Theorem 3.5 of the paper under review to the case where \((F, v)\) is maximally complete with \(\mathrm{char}(F) = p\), \(\widehat F\) perfect and \([F: F^n] = p^n\).


16K20 Finite-dimensional division rings
16S35 Twisted and skew group rings, crossed products
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
11E81 Algebraic theory of quadratic forms; Witt groups and rings
11E04 Quadratic forms over general fields
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