##
**Transversal crystals of finite level.**
*(English)*
Zbl 0883.14006

In his book “\(F\)-crystals, Griffiths transversality, and the Hodge decomposition”, Astérisque 221 (1994; Zbl 0801.14004), A. Ogus introduced the notion of \(T\)-crystals and \(F\)-spans, i.e. \(p\)-isogenies in the category of crystals on some \(X/W\), where \(X\) is a smooth projective scheme over a perfect field \(k\) with Witt ring \(W\). He constructs a functor \(\alpha\) from the category of \(F\)-spans to the category of \(T\)-crystals, on any smooth logarithmic scheme in characteristic \(p\), and then he shows that this functor commutes with the formation of higher direct images. This may be interpreted as a generalization of Mazur’s fundamental theorem on the relation of Frobenius and the Hodge filtration to the case of crystalline cohomology with coefficients. As applications one finds Katz’s conjecture about Newton and Hodge polygons, and the degeneration of the Hodge spectral sequence, for the cohomology of a variety with coefficients in an \(F\)-crystal. A key result is that \(\alpha\) induces an equivalence of categories provided that one restricts to objects of width less than \(p\).

Meanwhile, P. Berthelot developed the theory of crystals with level \(m\). In the underlying paper this theory is used to extend Ogus’s theorem to objects of width less than \(p^{m+1}\). This is done via the notion of \(F\)-\(m\)-spans and \(T\)-\(m\)-crystals, and it is shown that one can identify \(T\)-\(m\)-crystals of width less than \(p^{m+1}\) with a full subcategory of \(F\)-\(m\)-spans.

After some preliminaries on transversal filtrations, \(p\)-isogenies and \(m\)-PD-structures, level \(m\) differential operators and Griffiths transversality for \({\mathcal D}^{(m)}\)-modules according to Berthelot is recalled. In the corresponding terminology and notation, a functor \(\mu\) from the category of \(p\)-torsion free Griffiths transversal \(\hat{\mathcal D}^{(m)}_{X/S}\)-modules of width less than \(p^{m+1}\) that are transversal to the \(m\)-PD-filtration \((p,\{\})\) to the category of \(F^{m+1}\)-\(p\)-isogenies of width less than \(p^{m+1}\) on \(X/S\), where \(S\) is a formal \(m\)-PD-scheme and \(X\) is a formal \(S\)-scheme, is constructed. The notion of \(p\)-\(m\)-curvature for \({\mathcal D}^{(m)}\)-modules in characteristic \(p\) is introduced and the relation between \(F^{m+1}\)-\(p\)-isogenies and Griffiths transversality is discussed. This leads to the construction of a functor \(\alpha\) from \(F^{m+1}\)-\(p\)-isogenies to filtered \(\hat{\mathcal D}^{(m)}\)-modules. Then it is shown how this proves that \(\mu\) is fully faithful. Berthelot defined the \(m\)-th crystalline site \(\text{ Cris}^{(m)}(X/S)\) of \(X/S\) and \(m\)-crystals on it. In a similar way one may define a \(T\)-\(m\)-crystal on \(\text{Cris}^{(m)}(X/S)\). An \(F\)-\(m\)-span is a \(p\)-isogeny of \(p\)-torsion free \(m\)-crystals. The main result says that a \(p\)-torsion free \(T\)-\(m\)-crystal on \(X/S\) of width less than \(p^{m+1}\) determines a unique \(F\)-\(m\)-span of width less than \(p^{m+1}\) in a functorial way, and this functor is fully faithful. Varying the level \(m\) one can improve on Ogus’s theory.

Meanwhile, P. Berthelot developed the theory of crystals with level \(m\). In the underlying paper this theory is used to extend Ogus’s theorem to objects of width less than \(p^{m+1}\). This is done via the notion of \(F\)-\(m\)-spans and \(T\)-\(m\)-crystals, and it is shown that one can identify \(T\)-\(m\)-crystals of width less than \(p^{m+1}\) with a full subcategory of \(F\)-\(m\)-spans.

After some preliminaries on transversal filtrations, \(p\)-isogenies and \(m\)-PD-structures, level \(m\) differential operators and Griffiths transversality for \({\mathcal D}^{(m)}\)-modules according to Berthelot is recalled. In the corresponding terminology and notation, a functor \(\mu\) from the category of \(p\)-torsion free Griffiths transversal \(\hat{\mathcal D}^{(m)}_{X/S}\)-modules of width less than \(p^{m+1}\) that are transversal to the \(m\)-PD-filtration \((p,\{\})\) to the category of \(F^{m+1}\)-\(p\)-isogenies of width less than \(p^{m+1}\) on \(X/S\), where \(S\) is a formal \(m\)-PD-scheme and \(X\) is a formal \(S\)-scheme, is constructed. The notion of \(p\)-\(m\)-curvature for \({\mathcal D}^{(m)}\)-modules in characteristic \(p\) is introduced and the relation between \(F^{m+1}\)-\(p\)-isogenies and Griffiths transversality is discussed. This leads to the construction of a functor \(\alpha\) from \(F^{m+1}\)-\(p\)-isogenies to filtered \(\hat{\mathcal D}^{(m)}\)-modules. Then it is shown how this proves that \(\mu\) is fully faithful. Berthelot defined the \(m\)-th crystalline site \(\text{ Cris}^{(m)}(X/S)\) of \(X/S\) and \(m\)-crystals on it. In a similar way one may define a \(T\)-\(m\)-crystal on \(\text{Cris}^{(m)}(X/S)\). An \(F\)-\(m\)-span is a \(p\)-isogeny of \(p\)-torsion free \(m\)-crystals. The main result says that a \(p\)-torsion free \(T\)-\(m\)-crystal on \(X/S\) of width less than \(p^{m+1}\) determines a unique \(F\)-\(m\)-span of width less than \(p^{m+1}\) in a functorial way, and this functor is fully faithful. Varying the level \(m\) one can improve on Ogus’s theory.

Reviewer: W.W.J.Hulsbergen (Haarlem)

### MSC:

14F30 | \(p\)-adic cohomology, crystalline cohomology |

14G15 | Finite ground fields in algebraic geometry |

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

### Keywords:

Griffiths transversality; isogeny; filtered differential operator; Frobenius structure; crystals; span### Citations:

Zbl 0801.14004
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\textit{B. Le Stum} and \textit{A. Quirós}, Ann. Inst. Fourier 47, No. 1, 69--100 (1997; Zbl 0883.14006)

### References:

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[7] | [K] , Nilpotent connexions and the monodromy theorem : applications of a result of Turritin, Publ. Math. I.H.E.S, 39 (1970), 355-412. · Zbl 0221.14007 |

[8] | [O1] , F-Crystals and Griffiths Transversality, Proc. Int. Symp. on Algebraic Geometry, Kyoto 1977 (1977), 15-44. · Zbl 0427.14007 |

[9] | [2] , , Devil’s staircase route to chaos in a forced relaxation oscillator, Ann. Inst. Fourier, 44-1 (1994 · Zbl 0801.14004 |

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