Carlos Parra, Universidad Austral de Chile, Valdivia

FP$_{n}$ injective objects in Grothendieck categories with a projective generator

Abstract: We aim to generalize the theory of torsion pairs from FP$_{n}$-injective and FP$_{n}$-flat modules over $n$-hereditary rings to the context of Grothendieck categories with enough idempotents. In this setting, we show that the class of FP$_{n}$-injective objects is a torsion class if, and only if, the projective dimension of each FP$_{n}$ object is less than 2.  Moreover,  we show that the class of FP$_{n}$-injective complexes over a ring is never a torsion class, along with some applications of our general approach to the category of functors from a small preadditive category to the category of abelian groups.

This is a joint work-in-progress with D. Bravo, S. Odabasi and M. Perez. This work has been supported by the grant CONICYT/FONDECYT/Iniciacion/11160078.