Zariski Tangent Space - Example Extended

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 = +m2.

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