Tag Archives: k theory

The Nonzero K-Theory of Finite Rings is Finite

Let \(R\) be a finite ring. The example we'll have in mind at the end is the ring of \(2\times 2\) matrices over a finite field, and subrings. A. Kuku proved that \(K_i(R)\) for \(i\geq 1\) are finite abelian groups. Here, \(K_i(R)\) denotes Quillen's \(i\)th \(K\)-group of the ring \(R\). In this post we will […]

A is Homotopy Equivalent to A^op via Functors

Let $ \mathcal{A}$ be a small category and $ \mathbf{B}\mathcal{A}$ its geometric realisation. It is evident that $ \mathbf{B}\mathcal{A}$ and $ \mathbf{B}\mathcal{A}^\circ$ are homotopy equivalent, and in fact homeomorphic. However, can we find functors that realise this equivalence? This post summarises some informal notes I have written on this following D. Quillen's paper Higher Algebraic […]

Preprints and Classics 1: Higher cats, squarefree, max modulus

Mostly to take a break from marking exams, I thought I'd start a new recurring series here about mathematics papers and books that I find, both new and old. The "new" will consist mainly of preprints that look interesting (to encourage me to browse the arXiv) and the "old" will consists of papers I will […]