On the image of \(p\)-adic regulators. (English) Zbl 0928.14014
Introduction: Let \(V\) denote a complete discrete valuation ring with a fraction field \(K\) of characteristic 0 and a perfect residue field \(k\) of positive characteristic \(p\), let \(V_0=W(k)\) denote the ring of Witt vectors with coefficients in \(k\), and \(K_0\) its fraction field. Set \(G_K=\text{Gal}(\overline K/K)\). Let \(X\) be a smooth and projective scheme over \(V\).
For \(i\geq 0\), \(2i-n-1\geq 0\), consider the Galois representation \(L=H^n_{\text{ét}} (X_{ \overline K}, \mathbb{Q}_p(i))\). In the work of S. Bloch and K. Kato on the special values of \(L\)-functions of motives [in: The Grothendieck Festschrift, Vol. I, Prog. Math. 86, 333-400 (1990; Zbl 0768.14001)], the question of describing the images of Soulé’s \(p\)-adic Chern character \(r_p^{\text{ét}}: K_{2i-n-1}(X) \otimes\mathbb{Q}\to H^1(G_K,L)\), for \(2i-n-1\geq 1\), and of the étale cycle mass map \(\text{cl}_X: (CH^i(X) \otimes \mathbb{Q})_{ \text{hom}\sim 0}\to H^1(G_K,L)\) arises. We will show here that, as Bloch and Kato predicted, they are contained in the subgroups \(H^1_f(G_K,L)\) of crystalline extensions. The factorization through \(H_f^1\) of the cycle class map was shown earlier by I. Nekovář [in: Semin. Théorie Nombres Paris 1990-91, Prog. Math. 108, 127-202 (1993; Zbl 0859.11038)] via the de Rham conjecture. The problem can be reduced to a question of existence and compatibility between various constructions in the theory of \(p\)-adic periods. Briefly, our argument goes as follows. There is a Chern character and a cycle class map into a version of syntomic cohomology \(H^*_f(X,{\mathcal K}(*))\) (introduced by W. Niziol in a preprint: “Cohomology of crystalline smooth sheaves”) based on convergent crystalline cohomology. The relevant theory of Chern classes in the convergent crystalline cohomology is described in the appendix to the paper.
The map \(h:H_f^n(X, {\mathcal K} (i))\to H^n_{\text{ét}} (X_K, \mathbb{Q}_p(i))\), which is constructed in the preprint cited above, commutes with the \(K\)-theory characteristic classes, hence with the regulators and cycle classes. This is shown by mapping both the syntomic and the étale cohomology to the cohomology of the topos of ‘rigid smooth sheaves’ introduced by G. Faltings [in: Algebraic analysis, geometry and number theory, Proc. JAMI Inaugur. Conf., Baltimore 1988, 25-88 (1989; Zbl 0805.14008)]. We show that this last cohomology also carries \(K\)-theory characteristic classes. Now, the standard methods allow to reduce the question to the comparison of Chern classes of line bundles on projective spaces, which can be done explicitly. Finally, since we have proved in the preprint cited above that the map \(h\) is compatible with the spectral sequences \(H^i_f(G_K,H^j_{\text{ét}} (X_{\overline K}, \mathbb{Q}_p(n))) \Rightarrow H_f^{i+j} (X, {\mathcal K}(n))\), \(H^i(G_K, H^j_{\text{ét}} (X_{\overline K}, \mathbb{Q}_p(n))) \Rightarrow H_{\text{ét}}^{i+j} (X_K, \mathbb{Q}_p(n))\), we are done.
For \(i\geq 0\), \(2i-n-1\geq 0\), consider the Galois representation \(L=H^n_{\text{ét}} (X_{ \overline K}, \mathbb{Q}_p(i))\). In the work of S. Bloch and K. Kato on the special values of \(L\)-functions of motives [in: The Grothendieck Festschrift, Vol. I, Prog. Math. 86, 333-400 (1990; Zbl 0768.14001)], the question of describing the images of Soulé’s \(p\)-adic Chern character \(r_p^{\text{ét}}: K_{2i-n-1}(X) \otimes\mathbb{Q}\to H^1(G_K,L)\), for \(2i-n-1\geq 1\), and of the étale cycle mass map \(\text{cl}_X: (CH^i(X) \otimes \mathbb{Q})_{ \text{hom}\sim 0}\to H^1(G_K,L)\) arises. We will show here that, as Bloch and Kato predicted, they are contained in the subgroups \(H^1_f(G_K,L)\) of crystalline extensions. The factorization through \(H_f^1\) of the cycle class map was shown earlier by I. Nekovář [in: Semin. Théorie Nombres Paris 1990-91, Prog. Math. 108, 127-202 (1993; Zbl 0859.11038)] via the de Rham conjecture. The problem can be reduced to a question of existence and compatibility between various constructions in the theory of \(p\)-adic periods. Briefly, our argument goes as follows. There is a Chern character and a cycle class map into a version of syntomic cohomology \(H^*_f(X,{\mathcal K}(*))\) (introduced by W. Niziol in a preprint: “Cohomology of crystalline smooth sheaves”) based on convergent crystalline cohomology. The relevant theory of Chern classes in the convergent crystalline cohomology is described in the appendix to the paper.
The map \(h:H_f^n(X, {\mathcal K} (i))\to H^n_{\text{ét}} (X_K, \mathbb{Q}_p(i))\), which is constructed in the preprint cited above, commutes with the \(K\)-theory characteristic classes, hence with the regulators and cycle classes. This is shown by mapping both the syntomic and the étale cohomology to the cohomology of the topos of ‘rigid smooth sheaves’ introduced by G. Faltings [in: Algebraic analysis, geometry and number theory, Proc. JAMI Inaugur. Conf., Baltimore 1988, 25-88 (1989; Zbl 0805.14008)]. We show that this last cohomology also carries \(K\)-theory characteristic classes. Now, the standard methods allow to reduce the question to the comparison of Chern classes of line bundles on projective spaces, which can be done explicitly. Finally, since we have proved in the preprint cited above that the map \(h\) is compatible with the spectral sequences \(H^i_f(G_K,H^j_{\text{ét}} (X_{\overline K}, \mathbb{Q}_p(n))) \Rightarrow H_f^{i+j} (X, {\mathcal K}(n))\), \(H^i(G_K, H^j_{\text{ét}} (X_{\overline K}, \mathbb{Q}_p(n))) \Rightarrow H_{\text{ét}}^{i+j} (X_K, \mathbb{Q}_p(n))\), we are done.
MSC:
| 14F30 | \(p\)-adic cohomology, crystalline cohomology |
| 14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |
| 14G20 | Local ground fields in algebraic geometry |
| 19F27 | Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) |
