Differentials on the arc space after de Fernex and Docampo

We will present the main results  in [dFD] and the ideas behind them. In particular, given an arbitrary scheme $ X $ over a base scheme $ S $ we will explain how to describe the sheaf of relative differentials (with respect to $ S $) of its arc and jets spaces (respectively $ \Omega_{(X/S)_{\infty}/S} $ and $ \Omega_{(X/S)_n/S} $) from the sheaf of relative differentials of $ X $ ($ \Omega_{X/S} $), and how to deduce an interesting decomposition of the fiber of $ \Omega_{(X/S)_n/S} $ over a jet.

[dFD] De Fernex, T. and Docampo, R. Differentials on the arc space. https://arxiv.org/pdf/1703.07505.pdf, preprint 2017.