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For automorphisms of a polynomial ring in two variables over a domain R, we show that local tameness implies global tameness provided that every 2-generated locally free R-module of rank 1 is free. We give examples illustrating this property. (J. Berson), Adrien.Dubouloz@u-bourgogne.fr (A. Dubouloz), email@example.com (J.-P. Furter), firstname.lastname@example.org (S. Maubach).doi:10.1016/j.jpaa.2011.05.013 fatcat:5vvpphfn2zhgliormi7qw5pvua