February 15, 2010

Kahler differentials

Posted by Akhil Mathew under algebra, category theory, commutative algebra | Tags: , , |
[2] Comments 

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…)

 

December 4, 2009

Riemann surfaces

Posted by Akhil Mathew under analysis, complex analysis | Tags: , , , |
[6] Comments 

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…)