The classical period-index problem studies the relationship between the period and the index of a Brauer class. One may also consider an analogue of this question by working with higher degree cohomology classes instead of the Brauer classes. Recently, this has been studied by Harbater, Hartmann and Krashen by using the patching technique. For some algebraic objects, the patching technique can be carried to a function field of a curve over an excellent henselian discretely valued field. In this talk, I will introduce my work about the generalization of the patching technique, the local-global principle for higher degree cohomology classes and the application to the generalized period-index problem.