Hartshorne in his book gives the 'Infinitesimal Lifting Property' as an exercise in chapter 2, section 8 and mentions this to be very important in the deformation theory of nonsingular varieties. For completeness, I record the statement below:
Let $ k $ be an algebraically closed field and $ A $ a finitely generated $ k $-algebra such that $ \operatorname{Spec} A$ is a nonsingular variety over $ k $. Let $ B $ be a $ k $-algebra and $ B' $ a square-zero extension of $ B $ by $ I $, i e., there is an exact sequence $$ 0 \rightarrow I \rightarrow B' \xrightarrow{\pi} B \rightarrow 0 $$ where $ B' $ is a $ k $-algebra and $ I $ is an ideal such that $ I^2 = 0 $. Let $ f : A \rightarrow B $ be a $ k $-algebra homomorphism. Then there is a lift $ g : A \rightarrow B' $, i.e. a $ k $-algebra homomorphism $ g $ such that $ \pi \circ g = f $.
As someone starting with deformation theory, I would like to know how/why this result is very important as well as some applications of this result. Does this result have applications while studying moduli problems/moduli spaces?
For instance, Hartshorne gives one application: For $ X $ a nonsingular variety over $ k $ and $ \mathcal{F} $ a coherent sheaf on $ X $, the set of Infinitesimal extensions of $ X $ by $ \mathcal{F} $ upto isomorphism is in one to one correspondence with $ H^1(X, \mathcal{F} \otimes \mathcal{T}_X) $ where $ \mathcal{T}_X $ is the tangent sheaf.