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 ζ a primitive p-th root of unity.We introduce a new notion of calculus over W , which 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 ∆-connection on a finite free module is equivalent to a prismatic vector bundle on W [ζ ].As a corollary of a theorem of Bhatt and Scholze, we finally obtain that a ∆-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.Introduction 1301 1. Twisted powers 1306 2. Absolute calculus 1314 3. Higher ∆-derivatives 1326 4. The p-adic situation 1333 5. Absolute hypercalculus 1338 6. Absolute differential operators 1350 7. Application to prismatic crystals 1359 8. De Rham prismatic crystals 1364 9. Hodge-Tate prismatic crystals 1373 10.Prismatic F-crystals 1379