  This article deals with the quotient category of the category of coherent
sheaves on an irreducible smooth projective variety by the full subcategory of
sheaves supported in codimension greater than c. It turns out that this
category has homological dimension c. As an application of this, we will
describe the space of stability conditions on its derived category in the case
c=1. Moreover, we describe all exact equivalences between these quotient
categories in this particular case which is closely related to classification
problems in birational geometry.
