The cohomology of lambda-rings and Psi-rings

TL;DR

This thesis develops a cohomology theory for diagrams of algebras, including lambda- and Psi-rings, and applies it to K-theory, revealing connections to homotopy groups of spheres and classical results.

Contribution

It introduces a spectral sequence linking diagram cohomology to individual algebra cohomology, specifically for lambda- and Psi-rings, with applications to K-theory and topology.

Findings

Spectral sequence connecting diagram cohomology to algebra cohomology.

Cohomology of K-theory of spheres relates to homotopy groups.

Provides a new proof of Adams' classical result.

Abstract

In this thesis we develop the cohomology of diagrams of algebras and then apply this to the cases of the $\lambda$-rings and the $\Psi$-rings. A diagram of algebras is a functor from a small category to some category of algebras. For an appropriate category of algebras we get a diagram of groups, a diagram of Lie algebras, a diagram of commutative rings, etc. We define the cohomology of diagrams of algebras using comonads. The cohomology of diagrams of algebras classifies extensions in the category of functors. Our main result is that there is a spectral sequence connecting the cohomology of the diagram of algebras to the cohomology of the members of the diagram. $\Psi$-rings can be thought of as functors from the category with one object associated to the multiplicative monoid of the natural numbers to the category of commutative rings. So we can apply the theory we developed for the diagrams of algebras to the case of $\Psi$-rings. Our main result tells us that there is a spectral sequence connecting the cohomology of the $\Psi$-ring to the Andr\'{e}-Quillen cohomology of the underlying commutative ring. The main example of a $\lambda$-ring or a $\Psi$-ring is the $K$-theory of a topological space. We look at the example of the $K$-theory of spheres and use its cohomology to give a proof of the classical result of Adams. We show that there are natural transformations connecting the cohomology of the $K$-theory of spheres to the homotopy groups of spheres. There is a very close connection between the cohomology of the $K$-theory of the $4n$-dimensional spheres and the homotopy groups of the $(4n-1)$-dimensional spheres.