Generic Hodge and special Hodge--Witt polygons

Abstract

We prove that the Hodge polygon of the generic fiber of a proper smooth $p$-adic formal scheme over $\mathcal O_C$ lies on or above the generalized Hodge--Witt polygon of its special fiber. The argument combines prismatic cohomology with Ekedahl's diagonal theory of coherent Raynaud complexes. For each integral cutoff and each finite iteration length, a finite iterated Nygaard construction produces a perfect $A_{\mathrm{inf}}$-complex. Its individual-degree lengths along the generic divisor branches are weighted Hodge sums. A maximal-minor specialization argument bounds their sum by the corresponding crystalline length. On the special fiber, an exact derived modification of coherent Raynaud complexes has length growth governed by weighted Hodge--Witt numbers, with bounded error. Passing to the limit in these numerical lengths gives the comparison, allowing negative Hodge--Witt numbers and crystalline torsion. No projectivity, algebraization, or descent to a discretely valued field is assumed.