  We introduce a notion of refinements in the context of patching, in order to
obtain new results about local-global principles and field invariants in the
context of quadratic forms and central simple algebras. The fields we consider
are finite extensions of the fraction fields of two-dimensional complete
domains that need not be local. Our results in particular give the u-invariant
and period-index bound for these fields, as consequences of more general
abstract results.
