3

I'm working on an example I made up to try to understand how etale maps work. Let $X = Spec(k[x,y]/(y-x^2))$ be the parabola over an algebraically closed field. Consider the projection map to $\mathbb{A}^1 = Spec(k[t])$ down to the $x$-coordinate, i.e. $\pi: X \to \mathbb{A}^1$ is the map induced by $t \mapsto x$.

For this map to be etale it has to be a local diffeomorphism, i.e. it induces an isomorphism on tangent spaces. But I'm having a little trouble working out the computation explicitly. Here's what I have on my own.

We know $T_{X,p}$ for $p = (\alpha, \alpha^2)$ (i.e. corresponding to the ideal $(x-\alpha, y-\alpha^2)$ is equal to the kernel of the Jacobian map (writing $f$ for $y-x^2$)

$( \partial{f}/\partial x(\alpha) \text{ | } \partial{f}/\partial y(\alpha)) = (-2\alpha \text{ | } 1)$.

This is the line given by $y = 2\alpha x$. Now how do I explicitly compute $d\pi: \{(x,2\alpha x)\} \to T_{\mathbb{A}^1, \pi(p)}$? As $\pi(p) = (t-\alpha)$ corresponds to just the point $(\alpha, 0)$, my guess is the induced map is also projection to the first coordinate (hence an isomorphism), but perhaps I'm missing a good enough definition to see why it should be that (I'm trying to do this by piecing together a few things I've heard about Zariski tangent spaces).

If there's a good source to do these types of explicit computations, especially anything related to explicit etale maps I'd appreciate seeing it. This example is far too simple to be the end of the story, so I'd like to try a harder one after I get this down.

Edit: In particular, a source with a general formula/proof for how to explicitly compute differential maps on tangent spaces would be helpful.

TangentSpace23
  • 31
  • 1
    Why not just use the fact that $\pi$ is an isomorphism, by writing the inverse induced by $x\mapsto t, y\mapsto t^2$? – Mohan Oct 01 '16 at 23:47
  • 1
    I suppose that works in this case, but it certainly doesn't generalize well to what I'm looking for. – TangentSpace23 Oct 02 '16 at 05:04

1 Answers1

0

For a general discussion of how to compute the map on Zariski tangent spaces from a morphism of affine varieties, see the excellent discussion in this answer. Following that setup, here $\pi$ is the restriction of the polynomial map $\mathbb{A}^2\to\mathbb{A}^1$, $(x,y)\mapsto x$ (projection onto the first coordinate; we'll also call this $\pi$). The map on tangent spaces at $P=(\alpha,\alpha^2)\in X$ is the restriction to $T_{X,P}$ of the Jacobian matrix of this polynomial map: $$ \operatorname{Jac}_{\pi,P}= \begin{pmatrix} \left.\frac{\partial\pi}{\partial x}\right|_P & \left.\frac{\partial\pi}{\partial y}\right|_P \end{pmatrix} = \begin{pmatrix} 1 & 0 \end{pmatrix} $$ So as you predicted, the induced map on tangent spaces $d_P\pi:T_{X,P}\to T_{\mathbb{A}^1,\pi(P)}$ is just projection onto the first coordinate.

mbert
  • 177