# Tangent Space stuff and Nonsingularity

I’ve dropped the ball on making posts. Time to get back in the game.

### Three Tangent Spaces

Roughly the first definition of tangent space I came across is the following. Let be a differential manifold and . Take a smooth path with . Then . Roughly the vector space spanned by such is the tangent space.

This definition is the most intuitive to me and I like it because in the case of a hypersurface it makes it clear why the vector is normal to the surface at . Indeed let be a path, in coordinates we get functions , with . We have , so its derivative is zero, so by the chain rule

so

And evaluating at gives . That is, the dot product with a generic tangent vector is zero, so it perpendicular to the tangent space, hence normal to the surface. This definition also shows that the tangent space at is the linear subspace defined by the vanishing of

I’ve also seen the tangent space defined as the set of points derivations. Again in the case of a hypersuface and a point these are linear maps such that

.

For general we see ; here . Its clear to see that

so

So is determined by the values and any values work provided , i.e. as before. So that’s how there tangent spaces are connected.

I also see the tangent spaces defines as which I’ll write for short. To connect this to the previous tangent space, simple take more local derivations . We extend the domain of by setting

Given such a restricting it to gives an element . Conversely given , define via . These are the three tangent spaces.

### Sheaf of Differentials

Now some algebraic geometry. A lot of this is taken from page 170 of mumford’s red book. We can connect the middle definition of tangent space to the sheaf of differentials. This gives a way of defining nonsingularity for more general objects, but it involves some stuff about the tangent cone; a better discussion of the tangent cone will have to wait for another post. In any case, recall for any module a derivation is a linear map such that , where we use the module structure to make sense of the multiplication , although usually there are some constants involved i.e. where , usually is a field.

Now let’s say we have a -algebra and we have the module of differentials over k, . It has the property that

In the case of a closed point we set and . Then is exactly point derivations as in the second definition so in algebraic geometry we get another defintion

This shows in particular that the vector spaces actually bundle together to form the stalks of the sheaf of differentials .

Now about nonsingularity. In section I.5 of Hartshorne there a very down to earth definition of nonsingularity in terms of the rank of a Jacobian type matrix but when I first saw the definition I had no idea where it came from or why it worked. As I recall it is explained in Eisenbud’s book in the section about differentials, but Mumford also says something about it in his book. Basically if you know anything about tangent cones, it makes sense to define nonsingular as points where the tangent cone is equal to the tangent space (i.e. the lowest order approximation is linear at that point).

Now the tangent space always contains the tangent cone. And the tangent cone always has the same dimension as the variety (for closed points); this is proved in mumford pg 162. So and equality holds exactly when is a nonsingular point. Given the first description of tangent space as the set of points where , it follows the previous inequality turns into a reverse inequality about the rank of the Jacobian matrix, and this is roughly where that definition in I.5 comes from.

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- Published:
- December 11, 2009 / 1:09 pm

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