Posted by Jason Polak on 03. May 2014 · Write a comment · Categories: modules · Tags:

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 look at an example, slightly less simple than \(K_1\) of finite fields, showing that these groups can be arbitrarily large. Before we do this, let us briefly go over why this is true

But even before this, can you think of an example showing why this is false for \(i=0\)?
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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 K-Theory: I", so grab the notes or read the summary below:

Given any functor $ f:\mathcal{A}\to\mathcal{B}$, and an object $ B\in\mathcal{B}$, we can consider the category $ f^{-1}(B)$ consisting of objects $ A\in \mathcal{A}$ such that $ f(A) = B$. The morphisms of $ f^{-1}(B)$ are defined to be all the morphisms that map to $ 1_B$ under $ f$. Let us apply this to the following situation:

Given any small (or skeletally small) category $ \mathcal{A}$, we can construct another category $ S(\mathcal{A})$ as follows: the objects of $ S(\mathcal{A})$ are the arrows $ X\to Y$ of $ \mathcal{A}$, and a morphism $ (X\to Y)\to (X'\to Y')$ is a pair of morphisms $ X'\to X$ and $ Y\to Y'$ in $ \mathcal{A}$ making the obvious square commute. Now, we can consider the functor $ s:S(\mathcal{A})\to \mathcal{A}$ sending the object $ X\to Y\in S(\mathcal{A})$ to the object $ X\in\mathcal{A}$.
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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 likely read (to encourage me to read, or at least skim, more papers).

New Preprints

  1. Clark Barwick, "On the algebraic K-theory of higher categories": For a while I've wanted to learn a bit about higher category theory, but I've not yet found any application or motivation for it in something I'm already really interested in (including the stuff here). Perhaps this paper by Barwick will change my mind: algebraic K-theory may be best viewed as a functor with a universal property that generalises the well-known universal properties known for the classical K groups.
  2. Booker, Hiary, and Keating, Detecting squarefree numbers": Under the Generalised Riemann Hypothesis the authors propose an algorithm to test whether an integer is squarefree, without needing the number's factorisation.
  3. Shalit, "A sneaky proof of the maximum modulus principle": This is a proof of the maximum modulus principle in complex analysis, now with even less complex analysis! This paper also appeared in the American Mathematical Monthly and the author wrote a blog post about it as well.

View the whole post to reveal the hidden classic:
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