On a Group Graded Version of BGG

Abstract

A major result in Algebraic Geometry is the theorem of Bernstein–Gelfand–Gelfand that states the existence of an equivalence of triangulated categories: gr Λ ≅ 𝒟b(Coh ℙn), where gr Λ denotes the stable category of finitely generated graded modules over the n + 1 exterior algebra and 𝒟b(Coh ℙn) is the derived category of bounded complexes of coherent sheaves on projective space ℙn. Generalizations of this result were obtained in Martínez-Villa and Saorín (2004) and from a different point of view, the theorem has been extended by Yanagawa (2004) to ℤn-graded modules over the polynomial algebra. This generalization has important applications in combinatorial commutative algebra. The aim of the article is to extend the results of Martínez-Villa and Saorín (2004) to group graded algebras in order to obtain a generalization of Yanagawa's results having in mind the application to other settings (Geigle and Lenzing, 1987).