TITLE: Diagonal morphism of regular variety is a regular embedding
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: This is an answer to a question in Hoot's comments (I'm sorry not been able to post it as a comment by lack of reputation): "I wonder if demanding that they be geometrically reduced patches things up?"
No, $X\times_kX$ is regular (if and) only if $X$ is smooth over $k$. More generally, if $A$ is a noetherian ring, $B$ a local flat $A$-algebra such that $B\otimes_AB$ is regular, then $B$ is an essentially of finite type smooth $A$-algebra. The finiteness condition follows from Corollaire (3.6) in Ferrand, Monomorphismes et morphismes absolument plats. An then, for the smoothness, if $B\otimes_AB$ is regular, then $B$ is regular since it is a retract, and therefore the surjective map $B\otimes_AB \to B$ is a closed regular embedding. We apply now Bourbaki, Algèbre Commutative chap. 10, § 7 n. 10 Théorème 4, (i) ⟺ (v).