The Graded Preprojective Algebra of a Quiver

The paper introduces a new grading on the preprojective algebraof an arbitrary locally finite quiver. Viewing the algebra asa left module over the path algebra, the author uses the gradingto give an explicit geometric construction of a canonical collectionof exact sequences of its submodules. If a vertex of the quiveris a source, the above submodules behave nicely with respectto the corresponding reflection functor. It follows that whenthe quiver is finite and without oriented cycles, the canonicalexact sequences are the almost split sequences with preprojectiveterms, and the indecomposable direct summands of the submodulesare the non-isomorphic indecomposable preprojective modules.The proof extends that given by Gelfand and Ponomarev in thecase when the finite quiver is a tree. 2000 Mathematics SubjectClassification 16G10, 16G70.