Diagonal morphism of regular variety is a regular embedding Let $X$ be a regular $k-$variety (i.e. all of its local rings are regular) of pure dimension $d$. Then I would like to show that the diagonal morphism $X\rightarrow X\times_k X$ is a regular embedding of codimension $d$, to be able to prove exercise $12.2.M$ in Ravi Vakil's "Foundations of Algebraic Geometry".

The preceding exercise shows that any closed embedding $Y \rightarrow Z$ with $Y$ regular and $Z$ regular at all points in the image of $Y$ is a regular embedding. I'd like to use this (particularly because Vakil lists $12.2.M$ as a consequence of the preceding exercise), only I'm having trouble proving that $X \times_k X$ is regular at points on the diagonal. Thus what I would really like to know is:


  Let $X$ be a regular $k-$variety. Then $X \times_k X$ is regular at points on the diagonal.


Since $X$ is finite type, I can explicitly describe the local rings of $X$ as quotients of local rings of affine space, and use the kernels involved to explicitly describe the local rings of $X \times_k X$ as quotients of the local rings of $X$, but I can't then do very much with this, and I'm not sure that this is even the right approach.

Edit: Some of the answers suggest that regularity is not enough to show that the diagonal is a regular embedding. One provides an example of a regular scheme $W$ where $W \times W$ isn't regular at the diagonal, so the approach taken in this question is certainly flawed.
Without some hypothesis on $k$ I don't think it's true that the diagonal embedding $X \to X \times_k X$ is regular. Like take everybody's favorite regular but non-smooth scheme over the non-perfect field $k:= \Bbb{F}_p(t)$. Namely, the scheme $\operatorname{Spec} \Bbb{F}_p(t)[x]/(x^p - t)$. We first compute the tensor product 

$$\begin{eqnarray*} \Bbb{F}_p(t)[x]/(x^p - t) \otimes_{\Bbb{F}_p(t)} \Bbb{F}_p(t)[y]/(y^p - t) &amp;\cong&amp; \Bbb{F}_p(t)[x,y]/(x^p - t, y^p - t) \\ 
&amp;\cong&amp; L[y]/(y-\alpha)^p \end{eqnarray*}$$
where $L := \Bbb{F}_p(t)[x]/(x^p-t)$ and $\alpha$ is the image of $x$ in $L$.

We see that the tensor product is a local non-reduced  of dimension $0$. Hence the diagonal $X \to X \times_k X$ is a map from a point to a fat point and so is not regular.

Edit: Ravi told me via email that there is indeed an error in the notes with regards to this question.
Sadly I cannot comment. In any case:

(1) make clear what $k$ is (any field?) It depends greatly on the properties of $k$, as was already mentioned by A.G.

(2) unlike what some people think, there is no standard definition of "variety". E.g. everyone assumes *of finite type" (I think), but some assume irreducible, geometrically integral, whatever. For that reason you should always state what you mean by it.

(3) Over any field "smoothness" is equivalent to geometric regularity (see EGA IV), and thus "smoothness" is equivalent to regularity over a perfect field.

(4) The notion of smoothness is superior to regularity, exactly because it behaves very well with respect to base change. Over nonperfect fields I think regularity is less interesting.