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2. The Brauer group


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      Problem 2.1.

      Fix d \geq 3. Assume that we know:

      \operatorname{ind} (\alpha) | \operatorname{Per}(\alpha)^{d-1}, for all \alpha \in \operatorname{Br}(X), for all smooth, projective X over \overline{\mathbb{F}_p}. (Here d = \dim (X).)

      Can we use this, by a boundedness argument, to show the same thing for \alpha \in \operatorname{Br}(Y), where Y smooth, projective, \dim(Y) = d over \C?


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          Problem 2.2.

          Is every Brauer class over \C(s,t) cyclic?
            1. Remark. This type of question does not reduce to the prime case.

                •     A variant of Problem 2.2:

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                  Problem 2.3.

                  Higher cohomology H^d(F, \pmb{\mu}_n^{\otimes d}), where d = \dim (F).

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                      Problem 2.4.

                      When is \operatorname{Br}(X) = \operatorname{Br'}(X)?
                        1. Remark. If X is quasi-proj, then yes.

                              Cite this as: AimPL: Deformation theory and the Brauer group, available at http://aimpl.org/deformationbrauer.