
## 2. The Brauer group

1. #### 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 $\C$?
• #### 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:

#### Problem 2.3.

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