  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.
