Although idempotent kernel functors 161, or equivalently, abstract localization theory with respect to hereditary torsion theories [S] or Gabriel topologies [3] were originally introduced to generalize traditional localization theory to noncommutative rings and modules, they have also been applied to sheaf theory. Consider an arbitrary ringed space (X, Q,). In [ 121 K. Suominen introduces cohomology with support in the category of sheaves of &-modules, as derived functors of certain idempotent kernel functors f,, associated to not-necessarily closed subsets Z of X, thus generalizing some work of A. Grothendieck’s [7]. On the other hand, in [2] J. P. Cahen considers localization theory in the category of quasicoherent sheaves of Q,-modules over a locally noetherian scheme X. Finally, in [14, 151 F. Van Oystaeyen and one of the authors introduce abstract localization theory for presheaves and study the behaviour of sheaves in this theory. In this note we study the relationship between these points of view and settle some problems that remained open in [ 161. In fact, it appears that the three theories are essentially equivalent, when restricted to the category of quasicoherent sheaves on a locally noetherian scheme.