Twisted sheaves and the period-index problem.

*(English)*Zbl 1133.14018The author uses the method of twisted sheaves, which was invented by A. J. de Jong [A result of Gabber, preprint, http://www.math.columbia.edu/~dejong/papers/2-gabber.pdf], to prove some results on Brauer groups and period-index problems.

The two main results are as follows: Let \(K\) be field of characteristic \(p>0\) that has transcendence degree two over its prime field, and \(\alpha\) be a class from the Brauer group of \(K\), with period prime to \(p\). If \(\alpha\) is unramified, then the index of \(\alpha \) equals the period of \(\alpha\). In general, the index divides the cube of the period. Here the period is the order of \(\alpha\) in the Brauer group, and the index is the gcd of degrees of field extensions trivializing \(\alpha\). It is conjectured that for \(C_d\)-fields, however, the index should divide the \((d-1)\)-th power of the period.

The central idea is to represent a given Brauer class \(\alpha\) by an algebraic stack, and then to define and investigate certain moduli stacks of sheaves on this stack (= twisted sheaves). Period-index problems then are related to the existence of rational points in certain components of the moduli stack. Along the way, the author also gives interesting new proofs of known results, for example Gabber’s Theorem on the equality of the Brauer group and the cohomological Brauer group for affine schemes.

The two main results are as follows: Let \(K\) be field of characteristic \(p>0\) that has transcendence degree two over its prime field, and \(\alpha\) be a class from the Brauer group of \(K\), with period prime to \(p\). If \(\alpha\) is unramified, then the index of \(\alpha \) equals the period of \(\alpha\). In general, the index divides the cube of the period. Here the period is the order of \(\alpha\) in the Brauer group, and the index is the gcd of degrees of field extensions trivializing \(\alpha\). It is conjectured that for \(C_d\)-fields, however, the index should divide the \((d-1)\)-th power of the period.

The central idea is to represent a given Brauer class \(\alpha\) by an algebraic stack, and then to define and investigate certain moduli stacks of sheaves on this stack (= twisted sheaves). Period-index problems then are related to the existence of rational points in certain components of the moduli stack. Along the way, the author also gives interesting new proofs of known results, for example Gabber’s Theorem on the equality of the Brauer group and the cohomological Brauer group for affine schemes.

Reviewer: Stefan Schröer (Düsseldorf)