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The period-index problem for the Brauer group of an algebraic surface. (English) Zbl 1060.14025
Let \(K\) be a field, and \(A\) a central simple \(K\)-algebra. Then one has two numerical invariants: First, the period of \(A\), which is the order of \(A\) in the Brauer group. Second, the index of \(A\), which gives the size of the central divison algebra over which \(A\) is a matrix algebra. The period always divides the index, and both numbers have the same prime factors.
Now assume that \(K\) is the function field of an algebraic surface \(X\) over some separably closed field \(k\). Suppose that the period of \(A\) is prime to the characteristic of the field. The main result of this beautiful paper is that, under these assumptions, the period of \(A\) actually equals the index of \(A\).
In the case that the central simple algebra \(A\) extends to an Azumaya algebra on \(X\), the key ideas of the proof are as follows: The author constructs a suitable surface \(W_0\), which is possibly nonreduced or reducible, having a proper surjection \(W_0\rightarrow X\) so that the Brauer class of \(A\) extends to an Azumaya algebra of very small size on \(W_0\). The surface \(W_0\) appears as a special fiber in a family of surfaces \(W_t\), \(t\in C\), so that the total space has a map \(W\rightarrow X\), and some fiber \(W_\infty\) has an irreducible component birational to \(X\). He then argues that it is possible to modify the Azumaya algebra on \(W_0\) via elementray transformations so that all deformation obstructions vanish. Deforming the Azumaya algebra and specializing at \(W_\infty\), one retains enough control over period and index of the original central simple \(K\)-algebra \(A\) to infer the desired equality.
Along the way, the author analyses Brauer groups on reducible surfaces, deformation obstructions, and elementary transformations for Azumaya algebra. The paper also contains some speculations about the period-index problem for fields of higher transcendence degree.

14F22 Brauer groups of schemes
16K50 Brauer groups (algebraic aspects)
Full Text: DOI
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