Logarithmic Hodge–Witt Forms and Hyodo–Kato Cohomology☆
Logarithmic Hodge–Witt Forms and Hyodo–Kato Cohomology☆
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对数 Hodge-Witt 形式和 Hyodo-Kato 上同调☆
DOI：
10.1006/jabr.2001.8802
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发表时间：
2002
影响因子：
0.9
通讯作者：
Pierre Lorenzon
中科院分区：
文献类型：
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作者：
Pierre Lorenzon
本文的目的是将Illusie和Raynaud [13]的一些结果推广到Hyodo-Kato上同调[9 17 19 23 24].设S是特征p > 0的理想域k的谱，且具有细对数结构，X是S上的真对数光滑的Cartier型细对数概型.回想一下，X的Hyodo-Kato上同调群HX/WS被定义为X的晶体上同调群HX/Wn S在Teichmüller提升Wn S上的n个极限。这些是X生成的W -模，在其上X的Frobenius自同态诱导出一个σ-线性同构φ，其中W = Wk，σ是W的Frobenius自同构。我们研究了相应晶体的斜率，特别是它们的积分斜率，使用Hyodo-Kato上同调群的替代描述为H X W X/S，其中W X/S是X/S的de Rham-Witt复形[9]。首先，推广了Illusie-Raynaud有限性定理[13，（II，2.2）]，证明了R X W X/S作为DR的一个对象，其中R是Raynaud环W FV +W FV d，具有由R-模的凝聚复形构成的有界上同调（定理3.1）。这意味着斜率谱序列的退化模挠率以及Mazur-Ogus型结果[2，Sect. 8; 22，Sect. 7]关于H X/W S的Newton多边形和由Hm−q X q X/S给出的Hodge多边形。接下来，我们研究积分斜率。为此我们
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