Torsion-free abelian groups with optimal Scott families.

We prove that for any computable successor ordinal of the form alpha = delta + 2k (delta limit and k epsilon omega) there exists computable torsion-free abelian group (TFAG) that is relatively Delta(0)(alpha)-categorical and not Delta(0)(alpha-1)-categorical. Equivalently, for any such alpha there exists a computable TFAG whose initial segments are uniformly described by Sigma(c)(alpha) infinitary computable formulae up to automorphism (i.e. it has a c.e. uniformly Sigma(c)(alpha)-Scott family), and t here is no syntactically simpler (c.e.) family of formulae that would capture these orbits. As far as we know, the problem of finding such optimal examples of (relatively) Delta(0)(alpha) -categorical TFAGs for arbitrarily large alpha was first raised by Goncharov at least 10 years ago, but it has resisted solution (see e.g. Problem 7.1 in survey [Computable abelian groups, Bull. Symbolic Logic 20(3) (2014) 315 356]). As a byproduct of the proof, we introduce an effective functor that transforms a 0"-computable worthy labeled free (to be defined) into a computable TFAG. We expect that this technical result will find further applications not necessarily related to categoricity questions.