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\(p\)-adic weight spectral sequences of log varieties. (English) Zbl 1108.14015

The paper under review deals with various cohomological properties of log-schemes in the sense of Fontaine-Illusie-Kato [K. Kato, in: Algebraic analysis, geometry and number theory, Proc. JAMI Inaugur. Conf. Baltimore/MD (USA) 1988, 191–224 (1989; Zbl 0776.14004)]. More specifically, let \(k\) be a perfect field of characteristic \(p> 0\) and \(s= (\text{Spec}(k),\mathbb{N}\oplus k^*)\) a log-point. Denote by \(W\) the Witt ring of the ground field \(k\). For a proper simple normal crossing log-variety (SNCL) of pure dimension over the log-point \(s\), say \(X/s\), one has the so-called log-crystalline cohomology groups \(H^i_{\text{log-crys}}(X/W)\) of \(X/s\) over \(W\), \(i\in\mathbb{N}\), together with the so-called \(p\)-adic weight spectral sequence of \(X/s\) satisfying
\[ E^{-k,i+k}_1\Rightarrow H^i_{\text{log-crys}}(X/W). \]
This spectral sequence was originally established by A. Mokrane [Duke Math. J. 72, No. 2, 301–337 (1993; Zbl 0834.14010)] more than 10 years ago, along with a conjecture predicting its degeneration at \(E_2\) modulo torsion.
Part I of the present paper not only provides a complete proof of Mokrane’s conjecture, but also generalizes a number of Mokrane’s results for finite fields to general perfect fields of characteristic \(p> 0\), culminating in a theorem that ensures the \(E_2\)-degeneration of the \(p\)-adic weight spectral sequence of an open smooth variety which is the complement of a simple normal crossing divisor in a proper smooth variety over the perfect ground field \(k\). The latter result delivers an extension of an analogous result for finite ground fields, which had been obtained by B. Chiarellotto and B. Le Stum [J. Reine Angew. Math. 546, 159–176 (2002; Zbl 0993.14006)] slightly earlier. The author’s generalization is based on a recent comparison theorem by A. Shiho [J. Math. Sci. Univ. Tokyo, 9, 1–163 (2002; Zbl 1057.14025)], thereby replacing the method of rigid cohomologies used by Chiarellotto and Le Stum.
Part II of the paper under review enhances the theory of (idealized) log de Rham-Witt complexes by elaborating rigorous proofs for some unproven facts, folklore statements, or other technical tools occasionally used in the current literature on the subject. In this vein, the author helps clarify the delicate picture of the highly involved theory of log de Rham-Witt complexes, also by correcting some mistakes in published papers and by constructing important counterexamples.
After developing a perfect account of the theory of formal de Rham-Witt complexes (a la Cartier-Dieudonné-Raynaud), the author completes a number of fundamental results of O. Hyodo and K. Kato on semi-stable reduction and crystalline chohomology [in: Periodes \(p\)-adiques, Séminaire de Bures (1988), Astérisque 223, 221–268 (1994; Zbl 0852.14404)] by his refined approach. In the remaining four sections of the paper, various results from A. Mokrane’s work (cited above) are scrutinized, refined and complemented. This includes some further properties of the \(p\)-adic weight spectral sequence, a closer investigation of \(p\)-adic monodromy operators, and a complete construction of the so-called \(p\)-adic Steenbrink complex (à la A. Mokrane).
As the author points out, the construction of the \(p\)-adic weight spectral sequence for a family of open SNCL-varieties, by a different method, will be the subject of a forthcoming paper.

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
14G20 Local ground fields in algebraic geometry
11G25 Varieties over finite and local fields
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
18G40 Spectral sequences, hypercohomology
18G35 Chain complexes (category-theoretic aspects), dg categories
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