# Pure Projective Tilting Modules

### Silvana Bazzoni

Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121 Padova, Italy### Ivo Herzog

The Ohio State University at Lima, 4240 Campus Drive, Lima, OH 45804, USA### Pavel Příhoda

Faculty of Mathematics and Physics, Department of Algebra, Sokolovská 83, 186 75 Praha 8, Czech Republic### Jan Šaroch

Faculty of Mathematics and Physics, Department of Algebra, Sokolovská 83, 186 75 Praha 8, Czech Republic### Jan Trlifaj

Faculty of Mathematics and Physics, Department of Algebra, Sokolovská 83, 186 75 Praha 8, Czech Republic

## Abstract

Let $T_R$ be a $1$-tilting module with tilting torsion pair $(\operatorname{Gen} T, \mathcal{F})$ in $\text{Mod}\text{-}R.$ The following conditions are proved to be equivalent: $(1) T$ is pure projective; $(2) \operatorname{Gen} T$ is a definable subcategory of $\text{Mod}\text{-}R$ 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 $\mathcal{D}^b (\text{Mod}\text{-}R))$ 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.

## Cite this article

Silvana Bazzoni, Ivo Herzog, Pavel Příhoda, Jan Šaroch, Jan Trlifaj, Pure Projective Tilting Modules. Doc. Math. 25 (2020), pp. 401–424

DOI 10.4171/DM/752