The Pseudo-hyperresolution and Applications

Abstract

Homological algebra techniques can be found in almost all modern areas of mathematics. Many interesting problems in mathematics can be formulated, computed, or can find their equivalence in terms of Ext-groups. For instance, important (co)homology theories, such as the Mac Lane cohomology for rings or the Hochschild and cyclic homology of commutative algebras can be defined as Ext-groups in suitable functor categories; homotopical invariants can also gain information from homological data with the help of the unstable Adams spectral sequence, whose input takes the form of Ext-groups in the category of unstable modules over the Steenrod algebra. Therefore, the constructions of explicit injective (projective) resolutions in an abelian category is of great importance. In this article, we introduce a new method, called Pseudo-hyperresolution, to study such constructions. This method originates in the category of unstable modules, and aims at building explicit resolutions for 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 infinite complex projective space is also established. Despite its origin, Pseudo-hyperresolution generalizes to all abelian categories. In particular, many explicit resolutions of classical strict polynomial functors can be reunified in view of Pseudo-hyperresolution. 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.