3

Let $K$ be a field of characteristic zero and let $X$ be an irreducible variety over $K$. Let $K \subseteq L$ be a field extension. I am looking for a reference of the following facts:

-The dimension of every irreducible component of $X_L$ is $\dim X$.

-A point $x \in X$ is smooth if and only if every point of $X_L$ lying over $x$ is smooth.

I have been looking in various books, but I haven't found anything.

Hans
  • 3,539
  • 2
    Yes you are. $L[X]/PL[X]$ is not necessarily a domain. Take $\mathbb{R}\subset \mathbb{C}$ and $P=(X^{2}+1)$. @Hans: For the stuff on smoothness, I like the discussion in Görtz, Wedhorn very much. But you will find this in many introductory books – Rieux Dec 03 '15 at 20:40
  • yes, Görtz and Wedhorn is good for the smoothness, thx. and I also found a reference for the other question in the book! – Hans Dec 03 '15 at 21:35

0 Answers0