Absolute calculus and prismatic crystals on cyclotomic rings

Let $p$ be a prime, $W$ the ring of Witt vectors of a perfect field $k$ of characteristic $p$ and $\zeta$ a primitive $p$th root of unity. We introduce a new notion of calculus over $W$ that we call absolute calculus. It may be seen as a singular version of the $q$-calculus used in previous work, in the sense that the role of the coordinate is now played by $q$ itself. We show that what we call a weakly nilpotent $\mathbb\Delta$-connection on a finite free module is equivalent to a prismatic vector bundle on $W[\zeta]$. As a corollary of a theorem of Bhatt and Scholze, we finally obtain that a $\mathbb\Delta$-connection with a frobenius structure on a finite free module is equivalent to a lattice in a crystalline representation. We also consider the case of de Rham prismatic crystals as well as Hodge-Tate prismatic crystals.