I now want to talk about some of the material in Hartshorne, II.8.  First, we need some preliminaries from commutative algebra.

Let ${A}$ be a commutative ring, ${B}$ an ${A}$-algebra, and ${M}$ a ${B}$-module. Then an ${A}$-derivation of ${B}$ in ${M}$ is a linear map ${D: B \rightarrow M}$ satisfying ${D(a)=0}$ for ${a \in A}$ and

$\displaystyle D(bb') = (Db) b' + b (Db').$

The set of all such derivations forms a ${B}$-module ${\mathrm{Der}_A(B,M)}$. If we regard this as a set, clearly, we have a contravariant functor

$\displaystyle \mathrm{Der}_A(B, -): B-\mathrm{mod} \rightarrow \mathrm{Set}$

because if ${B \rightarrow B'}$ is a homomorphism of ${A}$-algebras, we can pull back a derivation.

Before proceeding, I should say something about the canonical example. Let ${M}$ be a smooth manifold and ${O_x}$ the local ring (of germs of smooth functions) at ${x \in M}$. Then ${\mathbb{R}}$ becomes an ${O_x}$-module if the germ ${f}$ acts by multiplication by ${f(x)}$. More precisely, we have an exact sequence

$\displaystyle 0 \rightarrow m_x \rightarrow O_x \rightarrow \mathbb{R} \rightarrow 0$

for ${m_x \subset O_x}$ the maximal ideal of functions vanishing at ${x}$, and this is the way ${\mathbb{R}}$ is an ${O_x}$-module.

Anyway, an ${\mathbb{R}}$-derivation ${O_x \rightarrow \mathbb{R}}$ is just a tangent vector at ${x}$.

Now back to the algebraic theory. It turns out that the functor ${\mathrm{Der}_A}$ is representable. In other words, for each ${A}$-algebra ${B}$, there is a ${B}$-module ${\Omega_{B/A}}$ such that

$\displaystyle \hom_B( \Omega_{B/A}, M) \simeq \mathrm{Der}_A(B,M) ,$

the isomorphism being functorial. In addition, there must be a “universal” derivation ${d: B \rightarrow \Omega_{B/A}}$ (corresponding to the identity ${\Omega_{B/A} \rightarrow \Omega_{B/A}}$ in the above functorial isomorphism), that any derivation factors through.

The construction of ${\Omega_{B/A}}$ is straightforward. We define it as the ${B}$-module generated by symbols ${db, b\in B}$, modulo the relations ${da = 0}$ for ${a \in A}$, ${d(b+b') = db + db'}$, and ${d(bb') = b' db + b db'}$. It is now clear that we have a functorial isomorphism as above. Now, ${\Omega_{B/A}}$ is called the module of Kahler differentials of ${B}$ over ${A}$. (more…)

Finally, it’s time to get to the definition of a Riemann surface. A Riemann surface is a connected one-dimensional complex manifold. In other words, it’s a Hausdorff space ${M}$ which is locally homeomorphic to ${\mathbb{C}}$ via charts (i.e., homeomorphisms) ${\phi_i:U_i \rightarrow V_i}$ for ${U_i \subset M, V_i \subset \mathbb{C}}$ open and such that ${\phi_j \circ \phi_i^{-1}: V_i \cap V_j \rightarrow V_i \cap V_j}$ is holomorphic.

Examples

Evidently an open subset of a Riemann surface is a Riemann surface. In particular, an open subset of ${\mathbb{C}}$ is a Riemann surface in a natural manner.

The Riemann sphere ${P^1(\mathbb{C}) := \mathbb{C} \cup \{ \infty \}}$ or ${S^2}$ is a Riemann sphere with the open sets ${U_1 = \mathbb{C}, U_2 = \mathbb{C} - \{0\} \cup \{\infty\}}$ and the charts

$\displaystyle \phi_1 =z, \ \phi_2 = \frac{1}{z}.$

The transition map is ${\frac{1}{z}}$ and thus holomorphic on ${U_1 \cap U_2 = \mathbb{C}^*}$.

An important example comes from analytic continuation, which I will briefly sketch below. A function element is a pair ${(f,V)}$ where ${f: V \rightarrow \mathbb{C}}$ is holomorphic and ${V \subset \mathbb{C}}$ is an open disk. Two function elements ${(f,V), (g,W)}$ are said to be direct analytic continuations of each other if ${V \cap W \neq \emptyset}$ and ${f \equiv g }$ on ${V \cap W}$. By piecing together direct analytic continuations on a curve, we can talk about the analytic continuation of a function element along a curve (which may or may not exist, but if it does, it is unique).

Starting with a given function element ${\gamma = (f,V)}$, we can consider the totality ${X}$ of all equivalence classes of function elements that can be obtained by continuing ${\gamma}$ along curves in ${\mathbb{C}}$. Then ${X}$ is actually a Riemann surface. (more…)