Let $k$ be a field and $X$ be a proper $k$ scheme. Futhermore let $\mathcal{H}$ be a coherent $\mathcal{O}_X$-module. Grothendieck's Finiteness Theorem says that for all $i \ge 0$ the cohomology groups $H^i(X, \mathcal{H})$ are finite vector spaces.
Let $\mathcal{F}, \mathcal{G}$ be two inverible sheaves. Consider $H^1(X, \mathcal{F}^{\vee} \otimes \mathcal{G}) = k^n$. Under some conditions for $X$ we have the identification $H^1(X, \mathcal{F}^{\vee} \otimes \mathcal{G}) = Ext^1 _X(\mathcal{O}_X, \mathcal{F}^{\vee} \otimes \mathcal{G})= Ext^1 _X (\mathcal{F},\mathcal{G})$.
$Ext^1 _X (\mathcal{F},\mathcal{G})$ describes the extension classes of $\mathcal{F}$ by $\mathcal{G}$.
By definition two extensions $\mathcal{L}, \mathcal{K}$ are equivalent if there exist a sheaf isomorphism $d: \mathcal{L} \to \mathcal{L}$ such that following diagram commutes:
$$ \require{AMScd} \begin{CD} 0 @>{} >> \mathcal{F} @>{a} >> \mathcal{L} @>{a} >> \mathcal{G} @>{} >> 0\\ @VV0V @VVidV @VVdV @VVidV @VV0V \\ 0 @>{} >> \mathcal{F} @>{b}>> \mathcal{K} @>{a} >> \mathcal{G} @>{} >> 0; \end{CD} $$
Now the question: Generally, $Ext^1 _X (\mathcal{F},\mathcal{G})$ is only a set consisting of all extension equivalence classes. But here - since it is isomorphic the vector space $k^n$ - I'm curious if this extra vector space structure tells "more" about the shape of the extension classes.
For example: How are the equivalence classes related to each other which belong in $k^n$ to the same line $l \subset k^n$. Is there a special connection between them in the commuting diagram? Or same subspace?
Or: What can we say about $d$ between two aquivalence classes lying in the same line? In $k^n$ they can be transformed to each other but simple scalar multiplication $ \cdot k$. Does this tell something about $d$?