How to show that the cuspidal cubic is not smooth using the formal smoothness criterion? Let $X/k$ be the cuspidal cubic over an algebraically closed field? How do I show that this is not formally smooth?
The definition I am using for formally smooth is the one at the stacks project:
https://stacks.math.columbia.edu/tag/02GZ
What I really want to know is the following : a smooth morphism should roughly correspond to being a submersion, yes? If so, why is the cuspidal cubic not smooth? I guess we can be change to a scheme over which the map on tangents spaces is not smooth? 
 A: You can guess such a choice of $T$ and $T'$ from the "picture". Consider the cuspidal cubic $y^2=x^3$. The line $x=0$ intersects the cusp at order 2, but it appears to leave the curve afterwards. 
In equations, consider the ring map $k[x,y]/(x^3-y^2)\to k[\epsilon]/(\epsilon^2)$ given by $x\to 0$ and $y\to \epsilon$. This is well-defined since $(0)^3-(\epsilon^2)=0$ if we are modding out by $\epsilon^2$.
Now, let's try to lift this map to $k[\epsilon]/(\epsilon^3)$. Suppose we have a map $k[x,y]/(x^3-y^2)\to k[\epsilon]/(\epsilon^3)$, where $x$ maps to $a\epsilon^2$ and $y$ maps to $\epsilon+b\epsilon^2$. Then, substituting in the equation $x^3=y^2$, we get $0=(a\epsilon^2)^3=(\epsilon+b\epsilon^2)^2=\epsilon^2$, which is a contradiction. 
The way I think about this is that the cuspidal curve is a 1-dimensional thing, but it's tangent space at the origin is 2-dimensional. Therefore, there are some "phantom directions" that are in the tangent space but can't be extended to actual "jets" along the curve. 
