The period-index problem for fields of transcendence degree 2. (English) Zbl 1337.14020

Lieblich, Max

Ann. Math. (2) 182, No. 2, 391-427 (2015).

The author proves a result on the Brauer group for functions fields \(K=k(X)\) of algebraic surfaces \(X\) over finite fields \(k\). Namely for each Brauer class \(\alpha\in\mathrm{Br}(K)\), the index divides the square of the period, that is \(\mathrm{ind}(\alpha)|\mathrm{per}(\alpha)^2\). The result actually holds if \(K\) is an extension of transcendence degree two over a perfect field \(k\) for which the maximal prime-to-\(l\) extensions are pseudo-algebraically closed fields, with Galois group the \(l\)-adic integers, for all primes \(l\). In general, it is conjectured that \(\mathrm{ind}(\alpha)|\mathrm{per}(\alpha)^{d-1}\) holds for \(C_d\) fields \(K\). Thus Lieblich’s result supports this conjecture, by treating a whole class of \(C_3\) fields of geometric origin.
The proof reduces the assertion to the existence of coherent sheaves of certain ranks on certain stacks, which is then established by geometric methods. The problem is easily reduced to the case that the Brauer class \(\alpha\) has prime period \(l\), and that the ground field \(k\) is pseudo-algebraically closed, which means that every geometrically integral quasiprojective scheme contains a rational point. Lieblich’s approach is to extend the cohomology class \(\alpha\) from the function field \(K=k(X)\) of the algebraic surface to a certain smooth 2-dimensional Deligne–Mumford stack \(\mathcal{X}\rightarrow X\), obtained as the \(l\)th root stack with respect to the ramification divisor \(D\subset X\) of the Brauer class \(\alpha\). There the class is represented by a suitable \(\mu_l\)-gerbe \(\mathcal{X}\rightarrow\mathcal{X}\). The main step is to show that there is an invertible sheaf \(\mathcal{N}\) on \(\mathcal{X}\) and a geometrically integral, in particular nonempty, open substack \(\mathcal{S}\) of the Artin stack of coherent \(\mathcal{X}\)-twisted sheaves on \(\mathcal{X}\), of rank \(l^2\) and determinant \(\mathcal{N}\). For this stack, there is a chart \(S\rightarrow\mathcal{S}\) with \(S\) a geometrically integral quasiprojective scheme. The assumptions on the ground field \(k\) then ensure that \(S\) has a \(k\)-rational point.
The bulk of the paper is a careful construction of the stack \(\mathcal{S}\), starting with sheaves on the preimage \(\mathcal{D}\subset \mathcal{X}\) of the ramification divisor \(D\subset X\), and extending them \(\mathcal{X}\), by using elementary transformations, deformation theory and ideas from [M. Lieblich, Duke Math. J. 138, No. 1, 23–118 (2007; Zbl 1122.14012); K. G. O’Grady, Invent. Math. 123, No. 1, 141–207 (1996; Zbl 0869.14005); D. J. Saltman, J. Algebra 314, No. 2, 817–843 (2007; Zbl 1129.16014); A. J. de Jong et al., Publ. Math., Inst. Hautes Étud. Sci. 114, 1–85 (2011; Zbl 1285.14053)].

Reviewer: Stefan Schröer (Düsseldorf)

MSC:

14F22	Brauer groups of schemes
12G05	Galois cohomology
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
14D20	Algebraic moduli problems, moduli of vector bundles

Keywords:

Brauer group; gerbe; moduli of sheaves; period-index problem; stack; twisted sheaves

Citations:

Zbl 1122.14012; Zbl 0869.14005; Zbl 1129.16014; Zbl 1285.14053

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