ag.algebraic geometry - Modules and Square Zero Extensions

If $X$ is a scheme and $mathcal M$ is a quasi-coherent sheaf on $X$ then we can
form a sheaf of rings $mathcal A := mathcal O_X oplus mathcal M$, on which multiplication
of sections is given just by the same formula as for $R oplus M$.

The pair $(X,mathcal A)$ is then a scheme which is an infinitesimal thickening of
$X$, and this is precisely how you pass from a quasi-coherent sheaf to the corresponding
thickening; it is just a sheafified version of the construction in your posting.

(Regarding cohomology, in your question you seemed most interested in the case when
$X =$ Spec $R$ is affine, in which case quasi-coherent sheaves have vanishing higher cohomology, so I'm not sure there is much to say about this.)

Added in response to comment below: To see how these come up geometrically,
consider for example a $k$-scheme $X$ embedded diagonally into $X times X$.
(Here $k$ is a field, and everything is happening over Spec $k$.)

Let $mathcal I_X$ be the ideal sheaf on $X times X$ cutting out the diagonal,
and consider the square-zero thickening
$mathcal O_{Xtimes X}/mathcal I_X^2$ of $X$.

This sits in the short exact sequence
$$0 to Omega^1_X = mathcal I_X/mathcal I_X^2
to mathcal O_{Xtimes X}/mathcal I_X^2 to mathcal O_{Xtimes X}/I_X = mathcal O_X
to 0.$$ The projection $p_1:Xtimes X to X$ gives a spliting of this short exact
sequence, and so we find that $mathcal O_{Xtimes X}/mathcal I_X^2 = mathcal O_X oplus
Omega^1_X$.

Recapitulating, we see that in the special case $mathcal M = Omega^1_X$, then
$(X, mathcal O_X oplus Omega^1_X)$ is equal to the first order infinitesimal neighbourhood of $X$ in $Xtimes X$.

Suppose for example that $X$ is a smooth curve, so that $Omega^1_X$ is a line-bundle.
Then $(X,mathcal O_X oplus Omega^1_X)$ is locally like the dual numbers (as you observe
in your comment) but is globally twisted (unless $X$ is an elliptic curve, i.e. the genus is 1, which is the one case when $Omega^1_X$ is actually trivial).

This should give you some sense of how these kinds of objects arise geometrically (and
why one would consider other examples rather than just the dual numbers).