Given a scheme $X$ over a field $k$ and a closed point $x$ with residue field $k(x)$ and inclusion $i:{x}\rightarrow X$ one can consider the following abelian groups
(1)$Ext^1_{\mathcal O_X}(i_*k(x),i_*k(x))$
and
(2)$Ext^1_{\mathcal O_{X,x}}(k(x),k(x))$.
The second one is just seen as Ext-group in the sense of modules over a ring.
Are they isomorphic?
I would define a map from (1) to (2) by just taking stalk in $x$, but I dont really see how one would come from (2) to (1).
Addition:
And is there a structure of a $k-$ vector space on (2) such that the iso is also one of $k-$spaces? (1) surely has a $k-$vector space structure as the scheme is defined over $k$.