# 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 […]