Let be a -tilting module with tilting torsion pair in The following conditions are proved to be equivalent: is pure projective; is a definable subcategory of with enough pure projectives; (3) both classes and are finitely axiomatizable; and (4) the heart of the corresponding HRS -structure (in the derived category is Grothendieck. This article explores in this context the question raised by Saorín if the Grothendieck condition on the heart of an HRS -structure implies that it is equivalent to a module category. This amounts to asking if is tilting equivalent to a finitely presented module. This is resolved in the positive for a Krull-Schmidt ring, and for a commutative ring, a positive answer follows from a proof that every pure projective -tilting module is projective. However, a general criterion is found that yields a negative answer to Saorín's Question and this criterion is satisfied by the universal enveloping algebra of a semisimple Lie algebra, a left and right noetherian domain.
Cite this article
Silvana Bazzoni, Ivo Herzog, Pavel Příhoda, Jan Šaroch, Jan Trlifaj, Pure Projective Tilting Modules. Doc. Math. 25 (2020), pp. 401–424DOI 10.4171/DM/752