Commutative von Neumann Regular Rings

A ring of left global dimension zero is a ring $R$ for which every left $R$-module is projective. These are also known as semisimple rings of the Wedderburn-Artin theory fame, which says that these rings are precisely the finite direct products of full matrix rings over division rings. Note the subtle detail that “semisimple” is used here instead of “left semisimple” because left semisimple is the same thing as right semsimple.

In the commutative world, the story for Krull dimension zero is not so simple. For example, every finite commutative ring has Krull dimension zero. Indeed, if $R$ is a ring with Krull dimension greater than zero, then there would exist two distinct primes $P\subset Q$ so that $R/P$ is an integral domain that is not a field. Thus, $R$ is infinite, as every finite integral domain is a field.

The story becomes simpler if we require $R$ to have no nilpotent elements: i.e., that $R$ is reduced. In this case, a commutative ring is reduced and of Krull dimension zero if and only if every principal ideal is idempotent. Every principal ideal being idempotent means that for every $x\in R$ there is an $a\in R$ such that $xax = x$. Rings, commutative or not, satisfying this latter condition are called von Neumann regular. So:

Theorem. A commutative ring has Krull dimension zero and is reduced if and only if it is von Neumann regular.

Wild Spectral Sequences Ep. 3: Cohomological Dimension

Last time in Wild Spectral Squences 2, we saw how to prove the five lemma using a spectral sequence. Today, we’ll see a very simple application of spectral sequences to the concept of cohomological dimension in group cohomology.

We will define the well-known concept of cohomological dimension of a group $G$, and then show how the dimension of $G$ relates to the dimension of $G/N$ and $N$ for a normal subgroup $N$. We do this with a spectral sequence. Although this application will appear to be very simple, it might be a good exercise for those just learning about spectral sequences.

The Category

For the sake of conreteness, let us work in the category of $G$-modules where $G$ is a profinite group. The $G$-modules are the $\mathbb{Z}G$-modules $A$ with a continuous action of $G$ where $A$ is given the discrete topology, but we could be working with any group $G$ with suitable, minor modifications, or Lie algebras, etc.

Like in all homology theories, there is the notion of cohomological dimension: a profinite group $G$ has cohomological dimension $n\in \mathbb{N}$ if for every $r > n$ and every torsion $G$-module $A$, the group $H^r(G,A)$ is trivial. If no such $n$ exists, we say that $G$ has cohomological dimension $\infty$. The usual arithmetic rules in working with $\infty$ apply.
More »