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 M be a differential manifold and p \in M.  Take a smooth path \gamma \colon (-1,1) \to M with \gamma(o) = p.  Then \gamma'(0) \in T_p M.  Roughly the vector space spanned by such \gamma'(o) is the tangent space.

This definition is the most intuitive to me and I like it because in the case of a hypersurface Z(f(x_1,...,x_n)) \subset \mathbb{A}^n it makes it clear why the vector (f_{x_1}|_P, ..., f_{x_n}|_P) is normal to the surface at P.  Indeed let \gamma(t) be a path, in coordinates we get functions x_1(t),..., x_n(t), with (x_1(0), ..., x_n(0)) = P.  We have f(x_1(t),..., x_n(t)) = 0, so its derivative is zero, so by the chain rule

0 = df(x_i(t)) = \sum_i \frac{\partial f}{\partial x_i} \frac{dx_i}{dt}dt

so

\frac{d}{dt}f(x_i(t)) = \frac{df(x_i(t))}{dt} = \sum_i \frac{\partial f}{\partial x_i} \frac{dx_i}{dt}

And evaluating at t=0 gives \sum_i f_{x_i}|_P \cdot x_i'(o) = (f_{x_1}|_P,..., f_{x_n}|_P)\cdot (x_1'(0),..., x_n'(0)) = 0.  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 P is the linear subspace defined by the vanishing of

\sum_i \alpha_i x_i

\alpha_i = \frac{\partial f}{\partial x_i}|_P

I’ve also seen the tangent space defined as the set of points derivations.  Again in the case of a hypersuface and a point P these are k linear maps D \colon k[x_1,...,x_n]/(f) \to k such that

D(hg) = h(P)D(g) + D(h)g(P).

For general h = \sum_\alpha c_\alpha x^\alpha we see D(h) = \sum_\alpha c_\alpha D(x^\alpha); here x^\alpha = x_1^{\alpha_1}\cdots x_n^{\alpha_n}.  Its clear to see that

D(x^\alpha) = \sum_i \frac{\partial x^\alpha}{\partial x_i}|_P D(x_i)

so

D(h) = \sum_i \frac{\partial h}{\partial x_i}|_P D(x_i)

So D is determined by the values D(x_i) = \lambda_i and any values work provided D(f) = 0, i.e. (\lambda_1,..., \lambda_n) \cdot (f_{x_1}, ..., f_{x_n}) = 0 as before.  So that’s how there tangent spaces are connected.

I also see the tangent spaces defines as \hom(m_P/m_P^2, k) which I’ll write (m/m^2)^* for short.  To connect this to the previous tangent space, simple take more local derivations D \colon \bigl(k[x_1,...,x_n]/(f)\bigr)_{m_p} \to k.  We extend the domain of D by setting

D(h/g) = D(hg^{-1}) = h(P)D(g^{-1}) + D(h)/g(P)

= -h(P)D(g)/g(P)^2 + D(h)/g(P)

Given such a D restricting it to m/m^2 gives an element \phi \in (m/m^2)^*.  Conversely given \phi, define D via D(g) = \phi(g - g(P)).  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 R module M a derivation D \colon R \to M  is a linear map such that D(rs) = rD(s) + D(r)s, where we use the R module structure to make sense of the multiplication rD(s), although usually there are some constants involved i.e. B \to R where D(b) = 0,\ b \in B, usually B is a field.

Now let’s say we have a k-algebra R and we have the module of differentials over k, \Omega_{R/k}.  It has the property that

Der(R, M) \cong \hom_k(\Omega_{R/k}, M)

In the case of a closed point p \in Z(f) = X we set R = \bigl(k[x_1,...,x_n]/(f)\bigr)_{m_p} and M = R/m_P = k.  Then Der(R,M) is exactly point derivations as in the second definition so in algebraic geometry we get another defintion

T_pX = Der(R,M) = \hom_k(\Omega_{R/k}, k) \cong \hom_{k(P)} (\bigl(\Omega_{X/k}\bigr)_P, k(P))

This shows in particular that the vector spaces m_P/m_P^2 actually bundle together to form the stalks of the sheaf of differentials \Omega_{X/k}.

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 \dim T_P X \ge \dim TC_P X = \dim X and equality holds exactly when P is a nonsingular point.  Given the first description of tangent space as the set of points a_i where \sum_i f_{x_i}|_P a_i = 0, 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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