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We introduce a notion of refinements in the context of patching, in order toobtain new results about local-global principles and field invariants in thecontext of quadratic forms and central simple algebras. The fields we considerare finite extensions of the fraction fields of two-dimensional completedomains that need not be local. Our results in particular give the u-invariantand period-index bound for these fields, as consequences of more generalabstract results.

Refinements to patching and applications to field invariants