2017/03/14 by Bazzoni, Silvana, Herzog, Ivo, Příhoda, Pavel +2 · 1 citation
#16B70 #16D60 #16D90 #18E15 #18E30 #18G10 #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1703.04745
Let T be a 1-tilting module whose tilting torsion pair (\mathcal T, \mathcal F) has the property that the heart \mathcal Ht of the induced t-structure (in the derived category \mathcal D(\rm Mod - 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 - 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 → B, but no almost split sequence beginning with A.