Galois Descent, Explained through Graphs

In this post, I'll describe a theory which classically originates in and is usually applied in algebraic geometry, but which doesn't really depend on it. The intent is that if you don't know AG, you can skip any bits in which varieties are discussed and still come away with a pretty complete understanding of the … Continue reading Galois Descent, Explained through Graphs

Geometrically Interpreting Normalization

At some point I've googled things like "X intuition" as X has ranged across pretty much every basic term in algebraic geometry. Most I've done this several times for. Algebraic geometry is a language, and like any language you need to be exposed to its quirks and underlying ideas repeatedly to fold them into your … Continue reading Geometrically Interpreting Normalization

A Worked Example of the Interpolation Algorithm

I've been trying to make it a habit to actually calculate things lately (at some point I really should get around to learning about Groebner bases...), so I thought I'd try actually, with a pen and paper, extending a regular function with the interpolation algorithm of my last post. As it turns out, it works … Continue reading A Worked Example of the Interpolation Algorithm

The Remainder Theorem, Lagrange Interpolation, and closed subschemes

Recently Bill Shillito posted on Twitter about a correspondence between Lagrange interpolation and the Chinese remainder theorem (which I'll call the remainder theorem) that I found pretty interesting. I'll expand on this connection a little by describing how both Lagrange interpolation and the remainder theorem can be derived from a geometric fact accessible to anyone who … Continue reading The Remainder Theorem, Lagrange Interpolation, and closed subschemes

Connectedness and Compactness: First Interpretations

I wanted to write something much more beginner friendly than what I usually post. When you've had some exposure to topological ideas - usually from a real analysis course - but aren't soaked in the concepts, you tend to take the long way around when thinking about connectedness and compactness. It doesn't help when the … Continue reading Connectedness and Compactness: First Interpretations

Four Interpretations of the Yoneda Lemma

This post won't require any more background than the definitions of categories, functors, natural transformations, and limits, with some inessential discussion of sheaves being in there as well. A bit of notation: I'll name categories with fraktur symbols. Given a locally small category $latex \mathfrak C$, I'll write $latex \mathfrak C(x,y)$ to indicate the set … Continue reading Four Interpretations of the Yoneda Lemma

In defense of the Zariski topology

As a beginner in algebraic geometry, few things are more annoying than trying to wrap one's head around putting the Zariski topology on your affine schemes. It's not that there's anything technically demanding in the definition, but... why? The feeling is that in constructing $latex \mathbb A^n_R$, we've taken $latex R^n$, a perfectly nice $latex … Continue reading In defense of the Zariski topology

Adjunctions are Imperfect Conjugations

Having built up 'here and there' knowledge of category theory over time makes it easy to learn the definition of an adjunction, but motivation for it often seems to consist of just throwing examples at the reader. I'll assume you've seen plenty of these and are looking for a broad, imprecise, but meaningful description of … Continue reading Adjunctions are Imperfect Conjugations