The Pseudo-hyperresolution and Applications

TL;DR

This paper introduces the Pseudo-hyperresolution method, a new homological algebra technique for constructing explicit resolutions in abelian categories, with applications in algebraic topology, functor categories, and cohomology theories.

Contribution

The paper presents the Pseudo-hyperresolution method, enabling explicit resolutions in unstable modules and other categories, unifying classical constructions and computing cohomological invariants.

Findings

Describes minimal injective resolutions of spheres using Steenrod algebra

Recovers algebraic EHP sequence and Lambda algebra via Pseudo-hyperresolution

Determines global dimension of strict polynomial functors and Mac Lane cohomology

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.