A variational Tate conjecture in crystalline cohomology (Q2009217)

The article under review proves a crystalline version of the variational Tate conjecture for divisors. Here is the precisely statement. Let \(f:X\to S\) be a smooth projective morphism of smooth varieties over a perfect field of characteristic \(p>0\), let \(s \in S\) be a closed point. Let \(c\in\mathrm{H}^{2}_{\text{cris}}(X/W(k))[1/p]\) be the cycle class of an element in \(\mathrm{CH}^{1}(X_{s})\otimes \mathbb{Q}\). Then \(c\) falls in the image of the restriction map \(\mathrm{H}^{2}_{\text{cris}}(X/W(k))[1/p]\to\mathrm{H}_{\text{cris}}^{2}(X_{s}/W(k))[1/p]\) (\(c\) is ``horizontal'') if and only if there exists \(z\in\mathrm{CH}^{1}(X)\otimes\mathbb{Q}\) such that \(\mathrm{cl}(z_{X_s})|=c\). The theorem is a consequence of its local analogue (when \(S\) is the spectrum of \(A=k[\![t_1,\dots,t_m]\!]\)), discussed in Section 3. We give a rough sketch of the proof of this local theorem. Set \(Y_r = X \otimes A/(t)^{r+1}\), \(Y=Y_0\). (a) The trace map from the K-theory spectrum to the topological cyclic homology spectrum induces a homotopy pullback \[ \operatorname{holim} K(Y_i) = TC(X;p) \times_{TC(Y;p),\mathrm{tr}}^{h} K(Y). \] which induces a ladder of exact sequences of homotopy groups. This is due to [\textit{B. Dundas} and \textit{M. Morrow}, Ann. Sci. École Norm. Sup. (4) 50, 201-238 (2017; Zbl 1372.19002)]. The \(\pi_0\otimes\mathbb{Q}\) of the topological cyclic homology is the sum of \(\mathrm{H}^{i}(X,W\Omega^{i}_{X,\log})[1/p]\) (the continuous cohomology of logarithmic de Rham-Witt sheaves, see [\textit{A. Shiho}, J. Math. Sci., Tokyo 14, No. 4, 567--635 (2007; Zbl 1149.14013)]). (b) Spelling out the meaning of the exactness at \(\pi_0\), one concludes that a class \(z \in K_0(Y)\) lifts to \(\lim_{r}K_0(Y_r)[1/p]\) if and only if \(\mathrm{ch}^{\log}(z)\in\bigoplus\mathrm{H}^{i}(Y,W\Omega_{Y,\log}^{i})[1/p]\) lifts to a class in \(\bigoplus\mathrm{H}^{i}(X,W\Omega_{X,\log}^{i})[1/p]\) (c) One shows, by studying the canonical map \(\varepsilon_{\mathbb{Q}}:\mathrm{H}^{i}(X,W\Omega_{X,\log}^{i})[1/p]\to\mathrm{H}^{2i}_{\text{cris}}(X/W(k))[1/p]\), that the desired crystalline cohomology class lifts if and only if \(\bigoplus\mathrm{H}^{i}(X,W\Omega^{i}_{X,\log})[1/p]\to\bigoplus\mathrm{H}^{i}(Y,W\Omega^{i}_{Y,\log})[1/p]\) lifts. Thus we can apply (b) above to conclude the proof of the ``local theorem''. The paper also contains useful applications and remarks. For example, it is shown that the Tate conjecture for divisors follows from its surface version; remarks on the existence of Leray spectral sequences for rigid cohomology in a certain circumstance are given. The article ends with an appendix explaining and clarifying the notion of ``direct image F-isocrystal'' in the sense of \textit{A. Ogus} [Duke Math. J. 51, 765--850 (1984; Zbl 0584.14008)].