The Pseudo-Hyperresolution and Applications

Abstract

Resolving objects in an abelian category by injective (projective) resolutions is a fundamental problem in mathematics, and this article aims at introducing a particular solution called “Pseudo-hyperresolutions”. This method originates in the category of unstable modules to study the minimal resolution of the reduced singular cohomology of spheres. In particular, for all integers $n \geq 0$, we can describe a large range of the minimal injective resolution of the sphere $S^n$ based on the Bockstein operation of the Steenrod algebra. Moreover, many classical constructions in algebraic topology, such as the algebraic EHP sequence or the Lambda algebra can be recovered using the Pseudo-hyperresolution method. A particular connection between spheres and the projective spaces is also established. Despite its origin, Pseudo-hyperresolutions generalize to all abelian categories. In particular, many classical construction of injective resolutions of strict polynomial functors can be reunified in view of Pseudo-hyperresolutions. As a consequence, we recover the global dimension of the category of homogeneous strict polynomial functors of finite degree as well as the Mac Lane cohomology of finite fields.