Intuition for thinking about $R$-module of Kähler differentials, universal receptacles, derivations?

Question

Suppose $k$ is a field of characteristic zero, and $R$ is a $k$-algebra. The $R$-module of Kähler differentials $\Omega_{R/k}$ of $R$ over $k$ with generators $\{dr\}_{r \in R}$ is the module subject to the following relations:

1. $dr = 0$ if $r \in k$.
2. $d(r+s) = dr + ds$.
3. $d(rs) = r\,ds + s\,dr$.

We have a natural map $d: R \to \Omega_{R/k}$ of vector spaces which sends $r \mapsto dr$.

$\Omega_{R/k}$ is the universal receptacle for derivations of $R$ into some $R$-module $M$. If $\delta: R \to M$ satisfies the axioms above, then there exists a map of $R$-modules $\Omega_{R/k} \to M$ such that $\delta$ factors through $d$.

What is the motivation/geometric intuition of these definitions? Is there a good way of thinking about all this in context of algebraic curves? Hopefully nothing fancy is needed?

Answer 1

You want something that plays the role of cotangent bundle in nice cases, and be defined in general. It may not be clear, a priori, that the universal derivation $d:R\to \Omega_{R/k}$ will give a reasonable object but it does. If you do a few examples, you will see that it behaves quite well. If $R=k[x_1,\ldots, x_n]$ which is the ring of functions on affine space, then $\Omega_{R/k}$ is isomorphic to the module of polynomial $1$-forms $\bigoplus_i Rdx_i$, and $$df=\sum \frac{\partial f}{\partial x_i} dx_i$$ exactly as you might hope from calculus. If $R$ is a dvr, i.e. the local ring of a regular affine curve, then $\Omega_{R/k} = Rdx$, where $x$ is a uniformizer.