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MODULES WITH PURE RESOLUTIONS OVER KOSZUL ALGEBRAS
Preview AbstractLet A be a Koszul algebra and M a finitely generated graded A-module. Suppose that M is generated in degree 0 and has a pure resolution. We prove that, if rℰ(M) ≠ 0 then M is Koszul; and if in addition M is not projective, then the converse is true as well, where r denotes the graded Jacobson radical of the Yoneda algebra
of A, and 
denotes the Ext module of M.Modules with pure resolutions over Koszul algebras revisited
Preview AbstractI construct a Koszul algebra A and a finitely generated graded A-module M that together form a counterexample to a recently published claim. M is generated in degree 0 and has a pure resolution, and the graded Jacobson radical of the Yoneda algebra of A does not annihilate the Ext module of M, but nonetheless M is not a Koszul module.