The -structure induced by an -tilting module

We study the t t -structure induced by an n n -tilting module T T in the derived category D ( R ) \mathcal {D}(R) of a ring R R . Our main objective is to determine when the heart of the t t -structure is a Grothendieck category. We obtain characterizations in terms of properties of the module category over the endomorphism ring of T T and, as a main result, we prove that the heart is a Grothendieck category if and only if T T is a pure projective R R -module.