Separability of $\mathbb{C}[x]$ over its $\mathbb{C}$-subalgebras

Question

For commutative rings $R \subseteq S$, recall that $S$ is separable over $R$, if $S$ is a projective $S \otimes_R S$-module, via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$.

Question 1: Is $\mathbb{C}[x]$ separable over $\mathbb{C}[x^2,x^3]$?

More generally,

Question 2: Is it possible to characterize all $\mathbb{C}$-subalgebras $\mathbb{C} \subset R \subset \mathbb{C}[x]$ such that $\mathbb{C}[x]$ is separable over $R$?

According to wikipedia: "Moreover, an algebra $S$ is separable if and only if it is flat when considered as a right module of $S \otimes_R S$ in the usual way". Here $\mathbb{C}[x^2,x^3] \subset \mathbb{C}[x]$ is not flat; I am not sure if there is a connection between flatness or non-flatness of $\mathbb{C}[x^2,x^3] \subset \mathbb{C}[x]$ and $\mathbb{C}[x] \otimes_{\mathbb{C}[x^2,x^3]} \mathbb{C}[x] \subset \mathbb{C}[x]$.

If, for example, $\mathbb{C}[x^2,x^3] \subset \mathbb{C}[x] \otimes_{\mathbb{C}[x^2,x^3]} \mathbb{C}[x]$ is flat (I do not know if this is true or false), then flatness of $\mathbb{C}[x] \otimes_{\mathbb{C}[x^2,x^3]} \mathbb{C}[x] \subset \mathbb{C}[x]$ would imply flatness of $\mathbb{C}[x^2,x^3] \subset \mathbb{C}[x]$, which is false.

(This question may be relevant. Also asked in MSE).

Edit: After receiving a comment that "it's unlikely that you can characterise all $R$ for which $R \subseteq \mathbb{C}[x]$ is separable, I would like to change Question 2 to the following question:

Question 3: Is it possible to characterize all $h \in \mathbb{C}[x]$, such that $\mathbb{C}[x]$ is separable over:

(i) $A=\mathbb{C}[h]$.

(ii) $B=\mathbb{C}+(h)$, where $(h)$ denotes the ideal of $\mathbb{C}[x]$ generated by $h$.

Example: If $h=x^2$, then $B=\mathbb{C}+(x^2)=\mathbb{C}[x^2,x^3] \subseteq \mathbb{C}[x]$ is not separable (first comment below).

Remark: Denote $h(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0$.

According to Corollary 8 with $h(Z)=c_nZ^n+c_{n-1}Z^{n-1}+\cdots+c_1Z+c_0-h$, we obtain an answer to Question 3(i): $\mathbb{C}[x]$ is separable over $\mathbb{C}[h]$ iff $\deg(h)=1$ (namely, $B=\mathbb{C}[x]$).

Still, I am not sure what can be said about Question 3(ii); could it be that for any $h$ of degree $\geq 2$, $B= \mathbb{C}+(h) \subseteq \mathbb{C}[x]$ is inseparable? If not, could one present a counterexample of minimal $\deg(h)$?

Notice that by this question we have: $B=\mathbb{C}+(h)=\mathbb{C}[h,xh,\ldots,x^{n-1}h]$.

Lemma 4.1 seems relevant; however, here $B$ is integrally closed in its field of fractions $\mathbb{C}(x)$ iff $B=\mathbb{C}[x]$, so we are not able to apply Lemma 4.1.

Any help is welcome! Thank you very much!

Answer 1

As I wrote in my comment, these extensions are known in algebraic geometry as unramified, which can be tested computationally by the vanishing of $\Omega_{S/R}$. (In differential geometry, this means $Y \to X$ is an immersion, i.e. injective on tangent spaces.)

Lemma. Let $h \in \mathbf C[x]$ be a polynomial of degree $n \geq 2$. Then $\mathbf C+(h) \subseteq \mathbf C[x]$ is unramified is and only if $h$ is squarefree.

Proof. The subalgebra $\mathbf C + (h)$ is generated as $\mathbf C$-algebra by $h, xh, x^2h, \ldots, x^{n-1}h$, since $x^n h$ is a linear combination of $h^2$ and $x^ih$ for $i < n$. So the question is whether the map \begin{align*} \mathbf A^1 &\to \mathbf A^n\\ x &\mapsto \big(h(x),xh(x),\cdots,x^{n-1}h(x)\big) \end{align*} is unramified. The Jacobian is $$\Big(\begin{matrix}h'(x) & xh'(x)+h(x) & x^2h'(x)+2xh(x) & \cdots & x^{n-1}h'(x) + (n-1)x^{n-2}h(x)\end{matrix}\Big),$$ and the ideal $I$ generated by the entries is just $(h,h')$. So $\Omega_{\mathbf A^1/\mathbf A^n} = k[x]/I$ is trivial if and only if $h$ and $h'$ have no factors in common, i.e. if and only if $h$ is squarefree. $\square$

A similar argument easily computes that the map \begin{align*} \mathbf A^1 &\to \mathbf A^1\\ x &\mapsto h(x) \end{align*} is never unramified if $n \geq 2$, since $h'(x)$ has at least one root. Thus, $\mathbf C[h] \subseteq \mathbf C[x]$ is never unramified if $\deg h \geq 2$.