The way I see this is the following, which I learnt from Sernesi's book "Deformations of algebraic schemes".

Assume that you have an *infinitesimal deformation* $\xi$ of a nonsingular scheme $X$ over $\textrm{Spec}(A)$, i.e. a flat map $\mathcal{X} \to \textrm{Spec}(A)$ together with an identification of $X$ with the fibre over $\textrm{Spec}(k)$.

Then, whenever one has a surjection $B \to A$ of local Artinian $k$-algebras, it is natural to ask whether one can lift $\xi$ to a deformation $\xi'$ over $\textrm{Spec}(B)$, i.e. whether one can find $\mathcal{X}' \to \textrm{Spec}(B)$ which extends $\mathcal{X} \to \textrm{Spec}(A)$.

In the case of small extensions, the answer is very simple: in fact, given $\xi$ and any small extension $e$ of $A$, there is associated an element $o_{\xi}(e) \in H^2(X, T_X)$, called "obstruction", which is zero if and only if a lifting $\xi'$ of $\xi$ to $\textrm{Spec}(B)$ exists.

Moreover, if $o_{\xi}(e)=0$ then there is a natural transitive action of $H^1(X, T_X)$ on the set of isomorphism classes of liftings $\xi'$.

Finally, the correspondence $e \to o_{\xi}(e)$ defines a $k$-linear map

$o_{\xi} \colon \textrm{Ex}_k(A,k) \to H^2(X, T_X).$

Summing up, whenever one has a small extension $0 \to I \to B \to A \to 0$ and a smooth scheme $X$, it is possible to lift over $\textrm{Spec}(B)$ exactly those infinitesimal deformations $\xi$ of $X$ over $\textrm{Spec}(A)$ such that the corresponding obstruction $o_{\xi}(e)$ vanishes; moreover, the isomorphism classes of liftings form a homogeneous space under the natural action of the group $H^1(X, T_X)$ of first-order deformations of $X$.

For instance, given the small extension

$0 \to \frac{(t^{n-1})}{(t^n)} \to \frac{k[t]}{(t^n)} \to \frac{k[t]}{(t^{n-1})} \to 0$,

the obstruction map is non-zero exactly for those "directions" in the $n-1$-th infinitesimal neighborhood of $X$ which cannot be "fattened" into the $n$-th infinitesimal neighborhood.

In particular, if $H^2(X, T_X)=0$ then *every* infinitesimal deformation of $X$ over $\textrm{Spec}(A)$ can be lifted to an infinitesimal deformation over $\textrm{Spec}(B)$.

smalland it adds a "direction"... – Qfwfq Jan 27 '11 at 22:57