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We develop an extension theory for analytic valuation rings in order toestablish Ax-Kochen-Ersov type results for these structures. New is that we canadd in salient cases lifts of the residue field and the value group and showthat the induced structure on the lifted residue field is just its fieldstructure, and on the lifted value group is just its ordered abelian groupstructure. This restores an analogy with the non-analytic AKE-setting that wasmissing in earlier treatments of analytic AKE-theory.

Analytic Ax-Kochen-Ersov theory with lifts of the residue field and value group