Logarithmic Hodge–Witt Forms and Hyodo–Kato Cohomology☆

The aim of this paper is to extend some results of Illusie and Raynaud [13] to the Hyodo–Kato cohomology [9 17 19 23 24]. Let S be the spectrum of a perfect field k of characteristic p > 0 endowed with a fine log structure and let X be a proper, log smooth, and of Cartiertype fine log scheme over S. Recall that the Hyodo–Kato cohomology groups H X/W S of X are defined as the limit over n of the crystalline cohomology groups H X/Wn S of X over the Teichmüller lifting Wn S . These are finitely generated W -modules, on which the Frobenius endomorphism of X induces a σ-linear isogeny φ, where W = W k and σ is the Frobenius automorphism of W . We study the slopes of the corresponding crystals, and especially their integral slopes, using the alternative description of the Hyodo–Kato cohomology groups as H X W X/S , where W X/S is the de Rham–Witt complex of X/S [9]. First of all, generalizing the Illusie–Raynaud finiteness theorem [13, (II, 2.2)] we prove that R X W X/S , as an object of D R , where R is the Raynaud ring W F V +W F V d, has bounded cohomology, consisting of coherent complexes of R-modules (Theorem 3.1). This implies the degeneration modulo torsion of the slope spectral sequence as well as the Mazur–Ogus-type results [2, Sect. 8; 22, Sect. 7] concerning the Newton polygon of H X/W S and the Hodge polygon given by the Hm−q X q X/S . Next, we study the integral slopes. To do this we