Nonsingular and smooth of curves over noncomplete field

Question

I heard concept of smoothness and nonsingular coincidence when underlying field is complete.

My question: Could you tell me an example of curve in which smoothness and nonsingular are different?

Answer 1

I think the terminology you are looking for is not complete but perfect. Recall that a field $K$ is perfect if all of its extensions are separable. In particular, $K$ can be non-perfect only if $\mathrm{char}(K)=p>0$ in which case one can understand perfectness of $K$ as the claim that Frobenius map $K\to K,\,\,x\mapsto x^p$ is surjective. There is always a minimal perfect extension of $K$ called its perfection and denoted $K^\mathrm{perf}$. If $K$ is characteristic $0$ then $K^\mathrm{perf}=K$, and if $\mathrm{char}(K)=p>0$ then $K^\mathrm{perf}$ is equal to $\{x\in \overline{K}:x^{p^n}\in K\text{ for some }n\geqslant 0\}$.

The general fact is then the following.

Fact: Let $X$ be a variety over a field $K$. Then, the following conditions are requivalent:

1. The map $X\to\mathrm{Spec}(K)$ is smooth.
2. The scheme $X_{\overline{K}}$ is regular.
3. The scheme $X_{K^\mathrm{perf}}$ is regular.

Here I am calling regular what you are calling ‘nonsingular’. A scheme $X$ is called regular if for all points $x\in X$ the local ring $\mathcal{O}_{X,x}$ is a regular local ring.

Given this, it’s easy to find examples of a regular, but not smooth (i.e. geometrically regular) schemes.

Example: Let $K=\mathbf{F}_p(t)$ and consider $X=\mathrm{Spec}(\mathbf{F}_p(t^{\frac{1}{p}}))$. Evidently $X$ is regular—-its only local ring is $\mathbf{F}_p(t^{\frac{1}{p}})$ which is a field, and so certainly regular local. That said, it’s not geometrically regular since $X_{\overline{K}}=\mathrm{Spec}(\overline{\mathbf{F}_p(t)}[x]/(x^p))$ (why is this true?) and this scheme is not regular since it has nilpotents (why does this contradict regularity)? Alternatively, you can observe that $X\to\mathrm{Spec}(K)$ is not smooth since $\Omega^1_{X/K}\ne 0$ (why is this true?). $\blacksquare$

Can you use this example to come up with an example with a curve?

Answer 2

Question: "My question: Could you tell me an example of curve in which smoothness and nonsingular are different?"

Remark: If $f: X:=Spec(B) \rightarrow S:=Spec(A)$ where $B$ is a finitely generated $A$-algebra and $dim(X)-dim(S):=n$, it follows $f$ is formally smooth iff $\Omega^1_{B/A}$ is locally trivial of rank $n$. Look up "formally smooth" in the litterature. A noetherian local ring $A$ over a field $k$ is formally smooth over $k$ iff $A$ is geometrically regular, ie $A\otimes_k K$ is regular where $K$ is any extension of $k$.

Hence if your curve $C$ is regular but not geometrically regular over $k$, it follows $C$ is regular but not (formally) smooth over $k$.