Pure projective tilting modules

Let TR be a 1-tilting module with tilting torsion pair (Gen T , F ) in Mod-R. The following conditions are proved to be equiv- alent: (1) T is pure projective; (2) Gen T is a definable subcategory of Mod-R with enough pure projectives; (3) both classes GenT and F are finitely axiomatizable; and (4) the heart of the corresponding HRS t-structure (in the derived category Db(Mod-R)) is Grothendieck. This article explores in this context the question raised by Saor ́ın if the Grothendieck condition on the heart of an HRS t-structure implies that it is equivalent to a module category. This amounts to asking if T is tilting equivalent to a finitely presented module. This is re- solved in the positive for a Krull-Schmidt ring, and for a commutative ring, a positive answer follows from a proof that every pure projective 1-tilting module is projective. However, a general criterion is found that yields a negative answer to Saor ́ın’s Question and this criterion is satisfied by the universal enveloping algebra of a semisimple Lie algebra, a left and right noetherian domain.