Representation of Associative Algebras and Coherent Sheaves

It is proved that a triangulated category generated by a strong exceptional collection is equivalent to the derived category of modules over an algebra of homomorphisms of this collection. For the category of coherent sheaves on a Fano variety, the functor of tightening to a canonical class is described by means of mutations of an exceptional collection generating the category. The connection between mutability of strong exceptional collections and the Koszul property is studied. It is proved that in the geometric situation mutations of exceptional sheaves consist of present sheaves and the corresponding algebra of homomorphisms is Koszul and selfconsistent.