I'm learning some group scheme stuff and there's the following result: If $A$ is Hopf $k$- algebra, then $\Omega_{A/k} \simeq A \otimes_k I/I^2$, where $I$ is the augmentation ideal.
I know the formal proof of this and i'm fine with that. But i was trying to figure out a geometric interpretation of this, that makes it an intuitive and in some sense expected(or at least hoped by analgy with something known) result. More precisely i'd like to see it as the formal counterpart of some geometric result. Here is my guess, and i wonder if it is appropiate:
I'd like to see it as the analogue on some result on Lie group. So i tought that an analogous of $\Omega_{A/k}$ is the space of global section of the cotangent bundle. Indeed once you write down as a k-algebra with some presentation by coordinates, then you have expression of the form $\sum_{i=1}^{n}a_i(x_1,...,x_n)dx_i$ with the relations condition given by the constraints, that is by some generators of the ideal of the variety. So now is known that the tangent bundle on a Lie group is trivial, since once you define a basis for the tangent plane in a point you can move smoothly the n vectors around the variety using the regular group operation and so you get n independent global section, that gives you that this bundle is trivial. So the cotangent bundle is trivial, so the space of global section is $C^{\infty}(M) \otimes {T_{e_G}}^*$. This result read formally becomes $A \otimes_k I/I^2$ being $A$ the ring of coordinates and $I/I^2$ the cotangent plane at the origin.
Do you find this an appropriate analogy or is there some big misunderstanding behind it?