I’ll now say a few words on formal smoothness. This happens to be closely related to the theory of the cotangent complex (namely, the cotangent complex provides a clean criterion for when a morphism is formally smooth). Ultimately, I would like to aim first for the result that a formally smooth morphism of finite presentation is flat, and thus to characterize such morphisms via the geometric idea of “smoothness” (even though the algebraic version of formally smooth is pure commutative algebra).

1. What is formal smoothness?

The idea of a smooth morphism in algebraic geometry is one that is surjective on the tangent space, at least if one is working with smooth varieties over an algebraically closed field. So this means that one should be able to lift tangent vectors, which are given by maps from the ring into ${k[\epsilon]/\epsilon^2}$.

This makes the following definition seem more plausible:

Definition 1 Let ${B}$ be an ${A}$-algebra. Then ${B}$ is formally smooth if given any ${A}$-algebra ${D}$ and ideal ${I \subset D }$ of square zero, the map $\displaystyle \hom_A(B, D) \rightarrow \hom_A(B, D/I)$

is a surjection.

So this means that in any diagram there exists a dotted arrow making the diagram commute. (more…)