Pure Projective Tilting Modules

Abstract

Let $T_R$ be a $1$-tilting module with tilting torsion pair $(\mathrm{Gen}T,\mathcal{F})$ in $\text{Mod}\text{-}R.$ The following conditions are proved to be equivalent: $(1)T$ is pure projective; $(2)\mathrm{Gen}T$ is a definable subcategory of $\text{Mod}\text{-}R$ with enough pure projectives; (3) both classes $\mathrm{Gen}T$ and $\mathcal{F}$ are finitely axiomatizable; and (4) the heart of the corresponding HRS $t$-structure (in the derived category ${\mathcal{D}}^b(\text{Mod}\text{-}R))$ 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.