Pure Projective Tilting Modules

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

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

Résumé

Let TR​ be a 1-tilting module with tilting torsion pair (GenT,F) in Mod-R. The following conditions are proved to be equivalent: (1)T is pure projective; (2)GenT is a definable subcategory of Mod-R with enough pure projectives; (3) both classes GenT and F are finitely axiomatizable; and (4) the heart of the corresponding HRS t-structure (in the derived category Db(Mod-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

DOI : 10.4171/dm/752

Classification : 16B70, 16D90, 18E10, 18E45, 18G10, 18G80
Mots-clés : tilting module, t-structure, Grothendieck category, pure projective module, definable subcategory