# ag.algebraic geometry – Differential forms on rigid analytic/adic spaces

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 ?