What the correct notion of “Kähler differentials” on a sufficiently nice adic spaces (rigid space, perhaps) ? Given, a smooth variety $X$ over a perfect field $k$ of some positive characteristic $p$ and a formal lift $mathcal{X} to X$ from $k$ to $W(k)$, equipped with a lift of Frobenius $sigma_{mathcal{X}}$, then thanks to Bhatt, Lurie, and Matthew, one knows that the de Rham-Witt complex $WOmega^*_{X/k}$ becomes the usualy de Rham complex of $X/k$ after reduction modulo $p$, i.e.:

$$WOmega^*_X/p cong_{qis} Omega^*_{X/k}$$

and furthermore, that:

$$WOmega^*_{mathcal{X}/W(k)} cong_{qis} Omega^*_{mathcal{X}/W(k)}$$

which means that:

$$Omega^*_{mathcal{X}/W(k)}/p cong_{qis} Omega^*_{X/k}$$

My initial guess was that one can obtain the de Rham complex $Omega^*_{mathcal{X}_{eta}/W(k)}$ of the generic fibre $eta$ of $mathcal{X}/W(k)$ (where the pullback is taken in the category of adic spaces) by simply performing extension scalars to $W(k)(1/p)$ on $Omega^*_{mathcal{X}/W(k)}$, but then I got stuck when I tried figure out whether or not there is a non-discrete topology on $H^i(Omega^*_{mathcal{X}_{eta}/W(k)(1/p)})$. Even if there is such a topology, I also have no idea how to prove the exactness of $Omega^*_{mathcal{X}_{eta}/W(k)(1/p)}$.

By extension, how does one define connections on vector bundles on a given adic space ? If $mathcal{E}$ is a vector bundle on $mathcal{X}_{eta}$, then will a connection simply be a homomorphism of $mathcal{O}_{mathcal{X}_{eta}}$-modules:

$$nabla: mathcal{E} otimes_{mathcal{O}_{mathcal{X}_{eta}}} mathcal{O}_{mathcal{X}_{eta}} to mathcal{E} otimes_{mathcal{O}_{mathcal{X}_{eta}}} Omega^1_{mathcal{X}_{eta}/W(k)(1/p)}$$

or is there a subtlety I’m overlooking ?