Pure Projective Tilting Modules

摘要

Let (T_R) be a (1)-tilting module with tilting torsion pair ((operatorname{Gen} T, mathcal{F})) in (ext{Mod}ext{-}R.) The following conditions are proved to be equivalent: ((1) T) is pure projective; ((2) operatorname{Gen} T) is a definable subcategory of (ext{Mod}ext{-}R) with enough pure projectives; (3) both classes (operatorname{Gen} T) and (mathcal{F}) are finitely axiomatizable; and (4) the heart of the corresponding HRS (t)-structure (in the derived category (mathcal{D}^b (ext{Mod}ext{-}R))) is Grothendieck. This article explores in this context the question raised by Saorín 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 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 (1)-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.