Period-index problems in WC-groups IV: a local transition theorem

Let K be a complete discretely valued field with perfect residue field k. Assuming upper bounds on the relation between period and index for WC-groups over k, we deduce corresponding upper bounds on the relation between period and index for WC-groups over K. Up to a constant depending only on the dimension of the torsor, we recover theorems of Lichtenbaum and Milne in a "duality free" context. Our techniques include the use of LLR models of torsors under abelian varieties with good reduction and a generalization of the period-index obstruction map to flat cohomology. In an appendix, we consider some related issues of a field-arithmetic nature.