Let $X$ a finite type scheme over an algebraically closed field $k$. Let $x\in X$, such that there exists a neighborhood $U$ of $x$, such that the sheaf of differentials $\Omega^{1}_{U}$ decomposes into:

$\Omega_{U}=E\oplus T$, where $E$ is a locally free sheaf of rank $d$ and $T$ just coherent.

Can we prove that locally for étale topology around $x$, $U$ is isomorphic to $\mathbb{A}^{d}\times Y$, for a certain scheme $Y$?

Let $X$ be an integral finite type scheme over $\mathbb{C}$. Let $x\in X$, such that there exists a neighborhood $U$ of $x$, such that the sheaf of differentials $\Omega^{1}_{U}$ decomposes into:

$\Omega_{U}=E\oplus T$, where $E$ is a locally free sheaf of rank $d$ and $T$ just coherent.

Can we prove that locally for étale topology around $x$, $U$ is isomorphic to $\mathbb{A}^{d}\times Y$, for a certain scheme $Y$?