Let J1 be the ideal J1=(x^2-y^2 , xz-yz , xz+yz) C[x,y,z] (where C is the complex numbers)
and let W1=V(J1) A^3(C) (where V means the algebraic varitey and A is affine space)
Let J2 be the ideal J2=(u^3-uv^2) C[u,v]
and let W2=V(J2) A^2(C)
I am trying to calculate the dimension of the tangent spaces to W1 and W2 at the origin and deduce that W1 and W2 are not isomorphic.
So far i have found the Jacobian matrices of J1 and J2 and evaluated them at (0,0,0) and (0,0) respectively.
This gave me a 3x3 Jacobian matrix of all zeros for J1.
And a 1x2 Jacobian matrix of all zeros for J2.
Since both matrices consist of all zeros they both have rank zero.
Then dimension of tangent space to W1 at the origin = n- rank of Jacobian evaluated at (0,0,0) = 3 - 0 = 3
Dimension of tangent space to W2 at the origin = n - rank of Jacobian evaluated at (0,0) = 2 - 0 = 2
Is this the correct way to calculate the dimensions?
How do i deduce that the W1 and W2 are not isomorphic? - is it just because the dimenstion of their tangent spaces at the same point are not equal?
Thanks for any feedback.