  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.
