I am studying an example : { (x,y) / $x^{4} = y^{2}$ } . I want to compute differential forms at (1,1) (for example).
It is an algebraic variety. It is a differential manifold...except that (0,0) is singular,so I may work on an open subset : a neighbourhood of (1,1), not containing (0,0), U. I want to see what algebraic one forms and differential forms look like, in one case using the definition , in the second, as df, where f $\in $ C $^{\infty}$(U).
I compute : an element of the maximal ideal containing (1,1), is a real polinomyal on ($x^ {2} - y) . $ So an (algebraic) form is of the form p = a ($x^ {2} - y)$ + b. In the differential manifold case, using the fact that the dimension is one , one has that a differential form is determined by a C $^{\infty}$ function on U. That is the same as a function on an open interval. Where does th second construction "see" the polynomial equation? An interval and a neighbourhood in the parabole are diffeomorphic. So algebraic information is lost?
May be it is a far fetched example, but I wanted to get a reducible polynomial, and see easily what the maximal ideal looked like.