I am studying Qing Liu’s “Algebraic Geometry and Arithmetic Curves” and am at a loss about the section on smooth morphisms.
I am trying to understand the stability of smooth morphisms under base change, ie Proposition 4.3.38.
In my text (I think it was elaborated on in later versions) the claim is supposed to “follow immediately”.
Let me recall the definitions: an algebraic variety $X$ over a field $k$ is smooth, if $X_{\overline{k}}$ has no singular point (ie if the dimension over the residue fields of the Zariski tangent spaces matches at each point the Krull dimension of the ring of stalks).
A morphism $f: X \rightarrow Y$ of finite type with $Y$ locally Noetherian is smooth if it is flat and if for each $y \in Y$, the base change of $f$ by $\operatorname{Spec}{k(y)} \rightarrow Y$ (which is the structural morphism of the algebraic variety $X_y$ over $k(y)$) is smooth.
So my goal is to prove that a base change of a smooth morphism is smooth. Flatness is easy to check, so the question becomes the following (with the base change being an extension of algebraically closed fields, but it implies the general case):
Let $V$ be a smooth algebraic variety over an algebraically closed field $K$. Let $L$ be an algebraically closed extension of $L$. Then $V_L$ is a smooth algebraic variety over $L$.
I can’t see anything to do, because there doesn’t seem to be an “obvious” way to transfer regularity from $V$ to $V_L$, since the dimension of the stalk rings is not even preserved.
So I would be very interested in any hints or sketches of proof (hopefully that do not rely on nontrivial results not mentioned at that point in the book).
Thank you.