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…)

I learned the material in this post from the book by Humphreys on Lie algebras and representation theory.

Recall that if {A} is any algebra (not necessarily associative), then the derivations of {A} form a Lie algebra {Der(A)}, and that if {A} is actually a Lie algebra, then there is a homomorphism {\mathrm{ad}: A \rightarrow Der(A)}. In this case, the image of {\mathrm{ad}} is said to consist of inner derivations.

Theorem 1 Any derivation of a semisimple Lie algebra {\mathfrak{g}} is inner.


To see this, consider {\mathrm{ad}: \mathfrak{g} \rightarrow D :=Der(\mathfrak{g})}; by semisimplicity this is an injection. Let the image be {D_i}, the inner derivations. Next, I claim that {[D, D_i] \subset D_i}. Indeed, if {\delta \in D} and {\mathrm{ad} x \in D_i}, we have

\displaystyle [\delta, \mathrm{ad} x] y = \delta( [x,y]) - [x, \delta(y)] = [\delta(x),y] = (\mathrm{ad}(\delta(x)))y.

In other words, {[\delta, \mathrm{ad} x] = \mathrm{ad}(\delta(x))}. This proves the claim.

Consider the Killing form {B_D} on {D} and the Killing form {B_{D_i}} on {D_i}. The above claim and the definition as a trace shows that {B_D|_{D_i \times D_i} = B_{D_i}}. (more…)