Perfect Category-graded Algebras

In a perfect category every object has a minimal projective resolution. We give a sufficient condition for the category of modules over a category-graded algebra to be perfect. AMS Subject Classification (2000): 18E15, 16W50. In [6] the second author explored homological properties of algebras graded over a small category. Our interest in these algebras arose from our research on the homological properties of Schur algebras, but we believe that they play an important organizational role in representation theory in general. Recall that an abelian category C is called perfect if every object of C has a projective cover (see Section 1). The existence of projective covers for every object guarantees the existence of minimal projective resolutions for every object in the category. The category C is called semi-perfect if every finitely generated object has a projective cover. We say that a category-graded algebra A is (semi)-perfect if the category of A-modules is (semi)-perfect. In [6] it was given a criterion for category-graded algebras to be semi-perfect. This criterion is sufficient to ensure that all category-graded algebras which appear in [5] are semi-perfect. But this is not enough to prove the existence of a minimal projective resolution for some of them, as the kernel of a projective cover may not be finitely generated. In this article we fill this gap by giving a criterion for a category-graded algebra to be perfect. Now we introduce the notions related with category-graded algebras that will be needed and explain the main result in more detail. Recall that, given a small category C, a C-graded algebra (see [6]) is a collection of vector spaces Aα parametrised by the arrows α of C, with preferred elements es ∈ A1s for every object s of C and a collection of maps μα,β : Aα ⊗Aβ → Aαβ for every composable pair of morphisms α, β of C. For a ∈ Aα and b ∈ Aβ we shall write ab for μα,β(a ⊗ b). For every composable triple α, β, and γ of arrows Received January 29, 2010. The second named author was supported by FCT grant SFRH/BPD/31788/2006. Financial support by CMUC is gratefully acknowledged by both authors.