Non-singularity and differentials

From a technical point of view, the big drawback about the tangent cone is that it is non-linear. It is always easier to handle essentially linear objects. The most natural way around this is to study the linear hull of the tangent cone, which we will call the tangent space. Let X be a scheme of finite type over At, and let x X be a closed point. 

When dealing with linear subspaces of An, and, more generally, with any affine spaces , i.e., schemes isomorphic to An, it is possible to get confused 

Notice that V equals the set of closed points of Vsch. In fact, by the Nullstel-lensatz, every maximal ideal of Ry is the kernel of a homomorphism 

Bearing in mind, then, that linear subspaces of An are essentially the same as subvector spaces of kn, I want to show how easy it is to compute the tangent space to an affine scheme Spec (k [Xi. Xn] /A) in An at any closed point. First look at the origin. Note the lemma  

If we translate X so that (ai.a ) is shifted to the origin, it is then defined by the equations 

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