Let $X$ be an $S$-scheme. Under what conditions, if any, is the cotangent sheaf $\Omega_{X/S}$ isomorphic to the tangent sheaf $\Theta_{X/S}$ as $\mathcal{O}_X$- modules? For example, given a nonsingular variety $V$ over a field $k$, what conditions could $V$ satisfy to ensure that such an isomorphism would exist?

@PraphullaKoushik Here are my thoughts: If $X$ is a smooth variety of dimension $n$ over a field $k$, then we know that the $\Omega_{X/k}$ is locally free of rank $n$, and so is $\Theta_{X/k}$. At each point, we could find various isomorphisms between the stalks of $\Omega_{X/k}$ and $\Theta_{X/k}$ (this just follows because they are free modules over the respective local rings). However, this does not admit globally, i.e. there does not neccesarily exist a global section admitting the desired isomorphism. If we were to remove a particular subset of points from the variety, then we might be able to find such an isomorphism. However, I was curious if there were particular characterizations of the variety for which this is known!

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