Category Archives: ring-theory

Krull dimension of Laurent series rings

Let $R$ be a Noetherian commutative ring and let $\dim(R)$ denote the Krull dimension of $R$. For the polynomial ring $R[x]$, we have $\dim(R[x]) = 1 + \dim(R)$. In fact, the same is true if we replace the polynomial ring by the power series ring: again $\dim(R[[x]]) = 1 + \dim(R)$. The situation is a […]

The nonvanishing K_2(Z/4)

We saw previously that $K_2(F) = 0$ for a finite field $F$, where $K_2$ is the second $K$-group of $F$. It may be helpful to refer to that post for the definitions of this functor. I thought that it might be disappointing because we did all that work to compute the second $K$-group of a […]

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 […]

The multiplicative group of a finite field is cyclic

Today, we are going to prove that if $F$ is a finite field, the the multiplicative group $F^\times$ of $F$ is a cyclic group. Let’s start with an example of this phenomenon. Take $\F_5 = \{0,1,2,3,4\}$. Now, if we take powers of $2$ modulo $5$ we get $$2,2^2=4,2^3 = 3, 2^4 = 1.$$ Therefore, we […]

Abelian categories: examples and nonexamples

I’ve been talking a little about abelian categories these days. That’s because I’ve been going over Weibel’s An Introduction to Homological Algebra. It’s a book I read before, and I still feel pretty confident about the material. This time, though, I think I’m going to explore a few different paths that I haven’t really given […]