Koszul resolutions

Abstract

Resolutions, which generalize the classical Koszul resolutions, are constructed for a large class of augmented algebras including the Steenrod algebra and the universal enveloping algebras. For each such algebra A A , an explicit differential algebra K ¯ ∗ ( A ) \bar K\ast (A) is described such that (1) K ¯ ∗ ( A ) \bar K^ \ast (A) is a small quotient algebra of the cobar complex and (2) the homology of K ¯ ∗ ( A ) \bar K^ \ast (A) is the cohomology algebra H ∗ ( A ) H ^ \ast (A) . The resolution of May for restricted Lie algebras in characteristic 2 is retrieved and a simple derivation of the resolution of Kan et al. of the Steenrod algebra is given.