Weight filtrations on log crystalline cohomologies of families of open smooth varieties
The main goal of this book is to construct a theory of weights for the log crystalline cohomology of families of open smooth varieties in characteristic \(p>0\), analogous to Deligne's theory of mixed Hodge structures assigned to families of open smooth varieties over the complex number field. As their basic results the authors list: \(p\)-adic purity, functoriality, the weight-filtered base change theorem, the weight filtered Künneth formula, convergence of the weight filtration, weight filtered Poincaré duality and the \(E_2\)-degeneration of \(p\)-adic weight spectral sequences. Let \(U\) be a smooth variety over the complex number field \({\mathbb C}\). Let \(X\) be a smooth variety over \({\mathbb C}\) with a simple normal crossings divisor \(D\) such that \(U=X-D\). Let \(j:U\to X\) be the natural open immersion. Set \(D^{(0)}=X\) and, for a positive integer \(k\), let \(D^{(k)}\) be the disjoint union of all \(k\)-fold intersections of the different irreducible components of \(D\). Consider the sheaf \(\Omega_{X/ {\mathbb C}}^i(\log(D))\) of differential \(i\)-forms on \(X\) with logarithmic poles along \(D\). Counting the number of logarithmic poles of local sections of \(\Omega_{X/ {\mathbb C}}^i(\log(D))\) defines the weight filtration \(P=\{P_k\}_{k\in{\mathbb Z}}\) on \(\Omega_{X/ {\mathbb C}}^i(\log(D))\). Let \(a^{(k)}:D^{(k)}\to X\) for \(k\in{\mathbb Z}_{\geq0}\) be the natural morphism of schemes over \({\mathbb C}\). Then we have the well known Poincaré isomorphism \[ \text{Res}:\text{gr}_k^P\Omega_{X/ {\mathbb C}}^{\bullet}(\log(D))\cong a_*^{(k)}(\Omega_{D^{(k)}/ {\mathbb C}}^{\bullet}\otimes \omega(D^{(k)}/{\mathbb C})(-k)) \] where \(\omega(D^{(k)}/{\mathbb C})\) is the orientation sheaf of \(D^{(k)}/ {\mathbb C}\). It induces the weight spectral sequence \[ E_1^{-k,h+k}=H^{h-k}(D^{(k)},\Omega_{D^{(k)}/ {\mathbb C}}^{\bullet}\otimes \omega(D^{(k)}/{\mathbb C}))(-k)\Rightarrow H^h(X,\Omega_{X/ {\mathbb C}}^{\bullet}(\log(D))). \] Moreover, in the filtered derived category of bounded below filtered complexes of \({\mathbb C}_{X^{\text{an}}}\)-modules we have an isomorphism \[ (\Omega_{X/ {\mathbb C}}^{\bullet}(\log(D)),P)\cong (Rj^{\text{an}}_*({\mathbb Z}_{U^{\text{an}}})\otimes{\mathbb C},\tau) \] where \(\tau\) denotes the canonical filtration, and we have the spectral sequence \[ E_1^{-k,h+k}=H^{h-k}((D^{\text{an}})^{(k)},\omega((D^{\text{an}})^{(k)}/{\mathbb C}))(-k)\Rightarrow H^h(U^{\text{an}},{\mathbb Z}). \] If \(X\) is proper it yields the weight filtration on \(H^h(U^{\text{an}},{\mathbb Z})\) and, tensored with \({\mathbb C}\), it coincides with the previous spectral sequence (by GAGA and the Poincaré lemma). In fact, Deligne has generalized this story to families as follows. Let \(f:X\to S\) be a proper smooth morphism of schemes of finite type over \({\mathbb C}\). Let \(D\) be a relative simple normal crossings divisor on \(X\) over \(S\). Set \(U=X-D\) and let \(f\) also denote the structural morphism \(U\to S\). Then \(R^hf^{\text{an}}_*({\mathbb Q}_{U^{\text{an}}})\) is a local system, and there exists a filtration \(P\) on \(R^hf^{\text{an}}_*({\mathbb Q}_{U^{\text{an}}})\) by sub local systems such that the induced filtration \(P_s\) on the stalk \(R^hf^{\text{an}}_*({\mathbb Q}_{U^{\text{an}}})_s=H^h((U^{\text{an}})_s,{\mathbb Q})\) for \(s\in S^{\text{an}}\) is obtained from the second spectral sequence above. The purpose of this book is to imitated these constructions in the context of log crystalline cohomology. Thus, the authors consider a PD-scheme \((S,{\mathcal I},\gamma)\) with a quasicoherent ideal sheaf \({\mathcal I}\) on which \(p\) is nilpotent, and a smooth morphism \(f:X\to S_0=\underline{\text{Spec}}_S({\mathcal O}_S/{\mathcal I})\) with a relative simple normal crossings divisor \(D\) on \(X\) over \(S_0\). One of the main results is then the construction of a functorial spectral sequence \[ E_1^{-k,h+k}=R^{h-k}f_{D^{(k)}/S,*}({\mathcal O}_{D^{(k)}/S}\otimes\omega_{\text{crys}}(D^{(k)}/S))(-k)\Rightarrow R^hf_{(X,D)/S,*}({\mathcal O}_{(X,D)/S}) \] where \(\omega_{\text{crys}}(D^{(k)}/S)\) is the crystalline orientation sheaf of \(D^{(k)}/S\). This spectral sequence is compatible with Frobenius actions. There is also a \(p\)-adic version (if \(p\) is no longer nilpotent on \(S\)). The construction of this spectral sequence relies on a profound investigation of various filtered crystalline complexes on Zariski and crystalline sites, and their interplay. As usual in crystalline cohomology, and in particular in log crystalline cohomology, since \(f\) lifts only locally, one must cut \(X\) and \(S\) into small pieces which in a favourable way admit nice embeddings into characteristic zero situations (in this book: admissible embedddings). On these pieces one may perform the desired geometric constructions as suggested by the paradigm over the complex numbers as recalled above, and then glue together. As is to be expected, this requires a substantial amount of technical work --- which the authors carry out with great energy. The book begins with a 40 pages chapter on filtered derived categories and topoi, the correct abstract setting for the matter of the present book. It should be useful beyond the particular purpose of this book. The second chapter is the technical core of the book. Besides the topics already mentioned it contains: the log linearization functor, vanishing cycle sheaves, log crystalline cohomology with compact support, filtered log de Rham-Witt complex, filtered convergent \(F\)-isocrystals, the filtered log Berthelot-Ogus-isomorphism, \(\ell\)-adic weight spectral sequences. The third and last chapter gives an outline of the construction of the weight filtration and the calculation of the slope filtration on the rigid cohomology of a separated scheme of finite type over a perfect field of characteristic \(p>0\). They are constructed and calculated by using de Jong's alteration theorem, Tsuzuki's proper cohomological descent for rigid cohomology, the comparison theorem bewteen log crystalline cohomology and rigid cohomology, and the results from the second chapter of this book.
scientific article; zbMATH DE number 5317211