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

Absolute calculus and prismatic crystals on cyclotomic rings