arXiv subjects
Publications and source records attributed to Nguyen The Cuong.
The present note presents some results about the mod-2 cohomology, modulo nilpotent elements elements of the fiber E of a decomposable map $ψ$ : K(Z, 2) $\rightarrow$ K(Z/2, p). This is more an announcement and a brief description of the tools that are used: Lannes' T functor and the Eilenberg-Moore spectral sequence.
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.
A relation between Schur algebras and Steenrod algebra is shown in [Hai10] where to each strict polynomial functor the author associates an unstable module. We show that the restriction of Hai's functor to the subcategory of strict polynomial functors of a given degree is fully faithfull.
We discuss two extensions of results conjectured by Nick Kuhn about the non-realization of unstable algebras as the mod $p$ singular cohomology of a space, for $p$ a prime. The first extends and refines earlier work of the second and fourth authors, using Lannes' mapping space theorem. The second (for the prime $2$) is based on an analysis of the $-1$ and $-2$ columns of the Eilenberg-Moore spectral sequence, and of the associated extension. In both cases, the statements and proofs use the relationship between the categories of unstable modules and functors between $\Fp$-vector spaces. The second result in particular exhibits the power of the functorial approach.
This note investigate some finiteness properties of the category U of unstable modules. One shows finiteness properties for the injective resolution of finitely generated unstable modules. One also shows a stabilization result under Frobenius twist for Ext-groups.