ABSTRACT: This is a report on joint work with R. Martinez Villa. We show that  (noncommutative) sheaf cohomology of Koszul modules over a Koszul quiver algebra is naturally isomorphic to Vogel cohomology of the Koszul-dual modules over the Koszul-dual algebra. In the case the original algebra is of finite global dimension, each module of type FP_infinity can be canonically approximated by a shift of a Koszul submodule with the cokernel of the approximation being of finite length. Consequently, over such algebras the above isomorphism extends to all modules of type FP_infinity. Thus over noetherian Koszul quiver algebras of finite global dimension the isomorphism holds for arbitrary finitely generated modules. This includes commutative polynomial algebras over a field.
 
 

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