A Question On Indices Notation In General Relativity I am trying to make sense of this simple case in my book, but I am still baffled by the notation that is used in the indices with the commas and semicolons; I also do not understand how these are equivalent.  
$$A_{\alpha;\beta}-A_{\beta;\alpha}=A_{\alpha,\beta}-A_{\beta,\alpha}$$
 A: By definition:
\begin{align}\tag{1} A_{\mu,\nu} &\equiv \frac{\partial}{\partial x^{\nu}} \, A_{\mu} \equiv \partial_{\nu} \, A_{\mu}, \\[1ex]
\tag{2} A_{\mu;\nu} &\equiv \frac{\partial}{\partial x^{\nu}} \, A_{\mu} - \Gamma_{\nu \mu}^{\lambda} \, A_{\lambda} \equiv \nabla_{\nu} \, A_{\mu},\end{align}
If the connection is symetric: $\Gamma_{\mu \nu}^{\lambda} = \Gamma_{\nu \mu}^{\lambda}$ (Levi-Civita connection AKA Christoffel symbols, which means no torsion), then the antisymetric expression $A_{\mu;\nu} - A_{\nu;\mu}$ gives the same as $A_{\mu,\nu} - A_{\nu,\mu}$.
A: The semikolon indicates a covariant derivative, in General Relativity only covariant derivatives transform like a tensor, therefore it is so important. 
The simple comma in difference indicates a simple derivative. 
the difference between covariant derivative and simple derivative is:
$$A_{a;b} = A_{a,b} - \Gamma^c_{ba} A_c$$
where $\Gamma$ stands for the Christoffel-symbols. See a.o.: https://en.wikipedia.org/wiki/Christoffel_symbols
On the basis of this formula it can be easily shown that  
$$A_{a;b} -A_{b;a} = A_{a,b} - A_{b,a}$$
because the Christoffel-symbols are symmetric on the 2. and 3. index (if the torsion is zero).
Actually, in order to define a covariant derivative on a (pseudo)-Riemannian manifold an additional structure is required, the "connection". The Christoffel-symbols represent here the connection. 
However, for the computation of an exterior derivative of a one-form $A$
a connection is not required. Actually $ A_{a,b} - A_{b,a}$ represent the result of the exterior derivative of a one-form $A$ in components. 
The equality $A_{a;b} -A_{b;a} = A_{a,b} - A_{b,a}$ shows that an exterior derivative can be defined on a (pseudo)-Riemannian manifold without further structure introduced.
