Mathematics – K-Theory and Homology
Scientific paper
2011-01-15
Mathematics
K-Theory and Homology
PhD thesis, University of Leicester, March 2010
Scientific paper
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.
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