Prismatization over $\mathbf{Z}$

The aim of this article is to given an extension of the prismatization functor for $p$-adic formal schemes (whose construction was first sketched by Drinfeld and then given by Bhatt-Lurie) to all schemes over $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathbf{Z})$. We then prove some basic properties of this extension (algebraicity, flatness for syntomic morphisms, perfectness of cohomology) and show that for smooth schemes over $\mathbf{Q}$ this construction recovers (a version of) the filtered de Rham stack.