Category Archives: math

Anything mathematical.

For real? Atiyah's proof of the Riemann hypothesis

Well this is strange indeed: according to this New Scientist article published today, the famous Sir Michael Atiyah is supposed to talk this Monday at the Heidelberg Laureate Forum. The topic: a proof of the Riemann hypothesis. The Riemann hypothesis states that the Riemann Zeta function defined by the analytic continuation of $\zeta(s) = \sum_{n=1}^\infty […]

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 […]

Art vs. science in mathematical discovery

Is mathematics science or art? Mathematics resembles science. In math, data and examples are collected and hypotheses made. There is a difference in the hypotheses of math versus science: in the former they can be proved, but that could be considered a small point. But I have met mathematicians who don't consider themselves scientists, and […]

A little intro to the Jacobi symbol: Part 3

This is the final post on the Jacobi symbol. Recall that the Jacobi symbol $(m/n)$ for relatively prime integers $m$ and $n$ is defined to be the sign of the permutation $x\mapsto mx$ on the ring $\Z/n$. In the introductory post we saw this definition, some examples, and basic properties for calculation purposes. In Part […]

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 […]