Pure Projective Tilting Modules

Abstract

Let $T$ be a $1$-tilting module whose tilting torsion pair $(\mathcal{T},\mathcal{F})$ has the property that the heart ${\mathcal{H}}_t$ of the induced $t$-structure (in the derived category ${\mathcal D}({\rm Mod} \mbox{-} R)$ is Grothendieck. It is proved that such tilting torsion pairs are characterized in several ways: (1) the $1$-tilting module $T$ is pure projective; (2) $\mathcal{T}$ is a definable subcategory of ${\rm Mod} \mbox{-} R$ with enough pure projectives, and (3) both classes $\mathcal{T}$ and $\mathcal{F}$ are finitely axiomatizable. This study addresses the question of Saorín that asks whether the heart is equivalent to a module category, i.e., whether the pure projective $1$-tilting module is tilting equivalent to a finitely presented module. The answer is positive for a Krull-Schmidt ring and for a commutative ring, every pure projective $1$-tilting module is projective. A criterion is found that yields a negative answer to Saorín's Question for a left and right noetherian ring. A negative answer is also obtained for a Dubrovin-Puninski ring, whose theory is covered in the Appendix. Dubrovin-Puninski rings also provide examples of (1) a pure projective $2$-tilting module that is not classical; (2) a finendo quasi-tilting module that is not silting; and (3) a noninjective module $A$ for which there exists a left almost split morphism $m:A\to B,$ but no almost split sequence beginning with $A.$