On the Hodge--Tate crystals over O_K

Abstract

We prove that a Hodge--Tate prismatic crystal on (O_K)_{\Prism} is uniquely determined by a topologically "nilpotent" operator. Using this operator, we construct a C_p-representation of G_K from a Hodge--Tate crystal in an explicit way. We then compute the cohomology of a Hodge--Tate crystal by using this operator and obtain the cohomological dimension of a crystal. In particular, we conjecture that this operator is essentially the classical Sen operator. As applications, under some mild assumption, we show the crystalline Breuil--Kisin modules admit "nilpotent connections" and give an explicit description of prismatic crystals. This "connection" is conjectured predicting the Hodge--Tate weights of associated crystalline representations.