Example Extended
If V is a subvariety of an n-dimensional vector space, defined by an ideal I, then R = Fn/I, where Fn is the ring of smooth/analytic/holomorphic functions on this vector space. The Zariski tangent space at x is
- mn / ( I+mn2 ),
where mn is the maximal ideal consisting of those functions in Fn vanishing at x.
In the planar example above, I = <F>, and I+m2 =
Read more about this topic: Zariski Tangent Space
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