Support theory of abelian categories-some discussion

We consider an axiomatic treatment of support theory for abelian categories. Many supports of Rosenberg and the atom support of Kanda are examples of support in the sense described here. We introduce the notion of a locally projective object, and we use it to give a generalization of the triangulated category of singularities for schemes. A result of Chen about the triangulated category of singularities is shown to hold in this generality.