The presented splitting lemma extends the techniques of Gromov and Forstnerič to
glue local sections of a given analytic sheaf, a key step in the proof of all Oka
principles. The novelty on which the proof depends is a lifting lemma for transition
maps of coherent sheaves, which yields a reduction of the proof to the work of
Forstnerič. As applications we get shortcuts in the proofs of Forster and Ramspott’s
Oka principle for admissible pairs and of the interpolation property of sections of
elliptic submersions, an extension of Gromov’s results obtained by Forstnerič and
Prezelj.
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