Division algebras over surfaces

AbstractWe consider here a number of topics concerning the theory of division algebras over the function field of a surface. One result relates the obstruction for ramification data to be from a division algebra and third etale cohomology. Another result shows this obstruction is always zero when the surface is Spec of a regular local ring (with some mild assumptions). At the same time we study the Brauer group of this function field as it relates to the Brauer group of the function field of the henselization. Finally we prove a result which says that Brauer group elements which “look like” they are of prime index q (unequal to any characteristic) must have all their ramification split by a cyclic Galois extension of the same degree. This last result requires a primitive q root of one