Syntomification of rigid-analytic varieties over $\mathbb{Q}_p$

I will describe a stacky approach to syntomic cohomology of rigid spaces over $\mathbb{Q}_p$, which is an analytic analogue of the syntomification of a $p$-adic formal scheme of Drinfeld and Bhatt--Lurie. This yields a notion of syntomic cohomology of rigid spaces with coefficients which satisfies Poincaré duality, affords a theory of Chern classes and compares both to Hyodo--Kato and proétale cohomology.

    In the first part of the talk, I will explain what syntomic cohomology is, why one should care about a stacky approach and what the syntomification looks like geometrically. In the second part, I will show how one can use this geometric point of view to obtain cohomological comparison theorems.