Wide coreflective subcategories and torsion pairs

A subcategory X of the module category Mod A over a ring A is said to be reflective, respectively coreflective, if the inclusion functor X → Mod A admits a left, respectively right, adjoint. A result of Gabriel and de la Peña characterizes the subcategories which are both reflective and coreflective as those which arise as module categories X = Mod B from some ring epimorphism A → B. Much less is known when only one of the two conditions is satisfied, even when restricting to wide, i.e. exact abelian, subcategories of Mod A.

In my talk I will review a construction going back to work of Ingalls and Thomas which assigns to a torsion pair two wide subcategories in Mod A. These subcategories are often coreflective, and I will address the question of which wide coreflective subcategories can be obtained in this way. When A is the Kronecker algebra, this leads us to an open problem of Henning Krause and Greg Stevenson concerning the classification of localizing subcategories in the derived category of quasi-coherent sheaves on the projective line: are there more localizing subcategories beyond the ones constructed from our understanding of the compact objects?

The talk will be based on joint work with Francesco Sentieri.