Pure Projective Tilting Modules

Bazzoni, Silvana ; Herzog, Ivo ; Příhoda, Pavel ; Šaroch, Jan ; Trlifaj, Jan

Documenta mathematica, Tome 25 (2020), pp. 401-424.

Résumé

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.

Détail

Classification : 18E45, 18E10, 16B70, 16D90, 18G10, 18G80
Keywords: tilting module, pure projective module, \(t\)-structure, Grothendieck category, definable subcategory