Category Archives: math

Anything mathematical.

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. […]

A quick intro to Galois descent for schemes

This is a very quick introduction to Galois descent for schemes defined over fields. It is a very special case of faithfully flat descent and other topos-descent theorems, which I won't go into at all. Typically, if you look up descent in an algebraic geometry text you will quickly run into all sorts of diagrams […]

Dividing a square into triangles of equal area

Take a square and divide it down a diagonal, dividing the square into two triangles. Drawing the opposite diagonal now divides it into four triangles. In these two examples, we divided a square into an even number of triangles, all with equal area. Can we divide a square into an odd number of nonoverlapping triangles, […]

Yet another group that is not Hopfian

A few weeks ago I gave an example of a non-Hopfian finitely-presented group. Recall that a group $G$ is said to be Hopfian if every surjective group homomorphism $G\to G$ is actually an isomorphism. All finitely-generated, residually finite groups are Hopfian. So for example, the group of the integers $\Z$ is Hopfian. Another example of […]

A zero-dimensional ring that is not von Neumann regular

An associative ring $R$ is called von Neumann regular if for each $x\in R$ there exists a $y\in R$ such that $x = xyx$. Now let $R$ be a commutative ring. Its dimension is the supremum over lengths of chains of prime ideals in $R$. So for example, fields are zero dimensional because the only […]

A finitely generated flat module that is not projective

Let's see an example of a finitely-generated flat module that is not projective! What does this provide a counterexample to? If $R$ is a ring that is either right Noetherian or a local ring (that is, has a unique maximal right ideal or equivalently, a unique maximal left ideal), then every finitely-generated flat right $R$-module […]