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 $A.$
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2017
- DOI:
- 10.48550/arXiv.1703.04745
- arXiv:
- arXiv:1703.04745
- Bibcode:
- 2017arXiv170304745B
- Keywords:
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- Mathematics - Representation Theory;
- Mathematics - Rings and Algebras;
- 18E30;
- 18E15;
- 16D90;
- 18G10;
- 16B70;
- 16D60