Tag Archives: k theory

The K-theory of finite fields: a synopsis

In my previous post, I proved that if $F$ is a finite field, then multiplicative group $F^\times$ is a cyclic group. This fact will play a small part in our description today of the $K$-theory of $F$. We will start by describing the classical $K$-theory of $F$ and then briefly talk about Quillen’s computation of […]

Land and Tamme’s new result on the K-theory of pullbacks

Or: just what is a Milnor square? A Milnor square is a certain pullback in the category of associative rings. It happens when we have a ring $R$, a two-sided ideal $I\subseteq R$ and a ring homomorphism $f:R\to S$ such that the restriction $f:I\to f(I)$ is an isomorphism of rings without identity. Since $I$ is […]

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 […]