A noncommutative version of Beilinson's theorem

Abstract

We prove that the category of representations of the N-Kronecker quiver is derived equivalent to the category qgr ( k 〈 X 1 , … , X N 〉 / ( ∑ i = 1 N X i 2 ) ) which is considered as the category of coherent sheaves on the noncommutative projective scheme associated to k 〈 X 1 , … , X N 〉 / ( ∑ i = 1 N X i 2 ) . This theorem is easily proved by applying Orlov's theorem. On the other hand, in our proof, a method of noncommutative projective geometry is used, and the quadratic relation ∑ i = 1 N X i 2 naturally arises from Auslander–Reiten theory.