If $R$ is a commutative unital associative ring and $A$ is an $R$-algebra of dimension $d$, which is local as a ring, then from dimension theory we know that the global dimension of $A$ must be at least $d$ (an the two conincide if $A$ is regular). Since the global dimension of $A$ is equal to the projective dimension of the residue field $\kappa$ of $A$, then we know $\operatorname{Ext}^d(\kappa,\kappa)$ must be non-trivial.
My question is what is an example of two elements in the group $\operatorname{Ext}^d(\kappa,\kappa)$, that is, what are two $d$-extensions of $\kappa$ by $\kappa$ which are not equivalent under the equivalence relation giving in $\operatorname{Ext}^d(\kappa,\kappa)$?