A finiteness theorem for zero-cycles over \(p\)-adic fields
Let \(V\) be a smooth projective variety over a \(p\)-adic field \(k\). Let \(CH_0(V)\) and \(A_0(V)\) be the group of zero-cycles on \(V\) modulo rational equivalence and its subgroup consisting of cycles of degree zero. It is conjectured that \(A_0(V)\) is the direct sum of a finite group and a \(p'\)-divisible group. (An abelian group is called \(p'\)-divisible iff it is divisible by any integer prime to \(p\).) In the paper under review, this conjecture is proved under a weak assumption, namely that \(V\) has a regular projective flat model over the ring of integers in \(k\) on which the reduced subscheme of the special fiber has simple normal crossings. This result has not been known even when \(V\) has good reduction. This result is deduced from the following strong theorem on a cycle map, which is proved in wider context. Let \(R\) be an excellent henselian discrete valuation ring with residue field \(F\) and fraction field \(k\). (No assumption is made for the characteristic of \(k\).) Let \(X\) be a regular projective flat scheme over \(R\) on which the reduced subscheme of the special fiber has simple normal crossings. We write \(CH_1(X)\) for the Chow group of one-dimensional cycles on \(X\). Let \(n\) be a natural number prime to the characteristic of \(F\). There is a cycle class map \(\rho_X : CH_1(X)/n \to H^{2d}_{et}(X, \mathbb{Z}/n\mathbb{Z}(d))\), where \(d = \dim X - 1\). The following strong theorem is proved in this paper: If \(F\) is finite or separably closed, then \(\rho_X\) is bijective. This theorem is studied in the framework of Kato homology. We keep the same notation and assumption as the preceding paragraph. The Kato homology \(KH_a(X, \mathbb{Z}/n\mathbb{Z})\) is defined to be the homology group of a complex of the form \[ \bigoplus_{x \in X_0} H^0_{et}(x, \mathbb{Z}/n\mathbb{Z}(-1)) \leftarrow \cdots \leftarrow \bigoplus_{x \in X_a} H^a_{et}(x, \mathbb{Z}/n\mathbb{Z}(a-1)) \leftarrow \cdots \] in degree \(a\). (The leftmost term is placed in degree \(0\).) Let \(l\) be a prime number different from the characteristic of \(F\), and put \(KH_a(X, \mathbb{Q}_l/\mathbb{Z}_l) = \lim_{\to} KH_a(X, \mathbb{Z}/l^n\mathbb{Z})\). The authors propose the following conjectures: If \(F\) is separably closed, then \(KH_a(X, \mathbb{Z}/n\mathbb{Z})\) should vanish for all \(a\). If \(F\) is finite, then \(KH_a(X, \mathbb{Q}_l/\mathbb{Z}_l)\) should vanish unless \(a \not= 1\), and \(KH_1(X, \mathbb{Q}_l/\mathbb{Z}_l)\) should be isomorphic to \((\mathbb{Q}_l/\mathbb{Z}_l)^{\oplus r}\) where \(r\) is the number of the irreducible components of the special fiber of \(X\). (This is a `different weight' analogue of \textit{K. Kato}'s conjecture [J. Reine Angew. Math. 366, 142--183 (1986; Zbl 0576.12012)].) It is relatively easy to prove the conjectures for \(a \leq 1\). Roughly speaking, the conjecture for \(a=2\) is equivalent to the surjectivity of \(\rho_X\), and when \(\dim X = 3\) the conjecture for \(a=3\) implies the injectivity of \(\rho_X\). As a consequence of the theorem in the preceding paragraph, the conjectures are proved in degree \(a \leq 3\). The proof of the above results are carried out by induction on the dimension of \(X\). Among main ingredients to proceed the induction step are theorems of Bertini type and of affine Lefschetz type. The first one, due to Uwe Jannsen and the first author, affirms the existence of a `very nice' hypersurface section over \(R\). (This is a refinement of their previous result [\textit{U. Jannsen} and \textit{S. Saito}, ``Bertini theorems and Lefschetz pencils over discrete valuation rings, with applications to higher class field theory'', \url{arXiv:0911.1470}, to appear in J. Algebraic Geom.].) The second one asserts that, for the complement of such a `very nice' hypersurface, a vanishing theorem of the etale cohomology holds in a stronger form than Artin-Gabber's affine Lefschetz theorem.
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