Category Archives: homological-algebra

## Weak dimension one rings are axiomatizable

Let $R$ be a ring. In the previous post on pure exact sequences, we called an exact sequence $0\to A\to B\to C\to 0$ of left $R$-modules pure if its image under any functor $X\otimes -$ is an exact sequence of abelian groups for any right $R$-module $X$. Here is yet another characterization of purity: Theorem. […]

## Pure exact sequences

Over the next few posts, I’ll talk more about axiomatizability of algebraic structures in first-order logic. Before I do that, we need to know about purity of exact sequences. So let’s fix a ring $R$. An exact sequence $$0\to A\to B\to C\to 0$$ in the category of left $R$ modules is called pure if for […]

## Finite-dimensional k[x]-modules: projective or not?

Let’s suppose $M$ is a nonzero projective $\Z$-module. Can it be finite? Nope. I’m sure there are plenty ways to prove it, but one way is to observe that a projective $\Z$-module is free, and hence if $M$ is nonzero it must have at least one copy of $\Z$. So, $M$ is infinite. What’s the […]

## 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 […]

## Image factorisation in abelian categories

Let $R$ be a ring and $f:B\to C$ be a morphism of $R$-modules. The image of $f$ is of course $${\rm im}(f) = \{ f(x) : x\in B \}.$$The image of $f$ is a submodule of $C$. It is pretty much self-evident that $f$ factors as $$B\xrightarrow{e} {\rm im}(f)\xrightarrow{m} C$$where $e$ is a surjective homomorphism […]

## Injective and p-injective

An $R$-module $M$ is called injective if the functor $\Hom_R(-,M)$ is exact. The well-known Baer criterion states that an $R$-module $M$ is injective if and only if for every ideal $I$ of $R$, every map $I\to M$ can actually be extended to a map $R\to M$. For example, $\Q$ is an injective $\Z$-module. If every […]

## When is a direct product of projective modules projective?

Over a field $k$, an arbitrary product of copies of $k$ is a free module. In other words, every vector space has a basis. In particular, this means that arbitrary products of projective $k$-modules are projective. Over the ring of integers, an arbitrary product of projective modules is not necessarily projective. In fact, a product […]

## First-order characterisations of free and flat…projective?

Here is an interesting question involving free, projective, and flat modules that I will leave to the readers of this blog for now. First, consider free modules. If $R$ is a ring, then every $R$-module is free if and only if $R$ is a division ring. The property of $R$ being a division ring can […]

## Dimension zero rings for three types of dimension

There are all sorts of notions of dimension that can be applied to rings. Whatever notion you use though, the ones with dimension zero are usually fairly simple compared with the rings of higher dimension. Here we’ll look at three types of dimension and state what the rings of zero dimension look like with respect […]

## Is it a projective module?

Consider a field $k$. Define an action of $k[x,y]$ on $k[x]$ by $f*g = f(x,x)g(x)$ for all $f\in k[x,y]$ and $g\in k[x]$. In other words, the action is: multiply $f$ and $g$ and then replace every occurrence of $y$ by $x$. Is $k[x]$ a projective $k[x,y]$-module? Consider first the map $k[x,y]\to k[x]$ given by \$f\mapsto […]