Algebraic K-theory of projective spaces associated to Zⁿ-graded rings

Abstract

It is known that the K-theory of the projective line P¹, over an arbitrary commutative ring splits into two copies of the K-theory of the ground ring. In this thesis, first we generalised this result to the case of an arbitrary strongly Z-graded ring R. The projective line associated with R is indirectly defined by specifying the corresponding category of quasi-coherent sheaves. The process, perhaps surprisingly, works very much like in the "classical" case with notions from algebraic geometry like sheaf cohomology and twisting sheaves being transferred to the new setting. However the aforementioned family of twisting sheaves from algebraic geometry now depends on a two-parameter construction instead of just one. Loosely following the pattern of the proof by Quillen, the K-theoretical splitting for the projective line is established.

We then show how this template can be expanded upon by looking at the projective plane and strongly Z²-graded rings. After establishing the K-theoretical splitting result here we move to fully generalising the K-theoretical splitting for the projective space.

Date of Award	Dec 2024
Original language	English
Awarding Institution	Queen's University Belfast
Sponsors	Northern Ireland Department for the Economy
Supervisor	Thomas Huettemann (Supervisor)