On the structure of étale motivic cohomology (Q515879)

The author proves structure results for the integral etale motivic cohomology groups \(H^i_{\mathrm{\'et}}(X,\mathbb{Z}(n))\) for \(X\) smooth projective over a algebraically closed field, a finite field, a \(p\)-adic local field or over \(C \setminus S\) for \(C\) the spectrum of the ring of integers of a number field or a smooth proper curve over a finite field and \(S \subseteq C\) a finite non-empty set of closed points. Many examples are given. The main results are as follows: Theorem 1.1. Let \(k\) be an algebraically closed field of characteristic \(p \geq 0\), \(i \neq 2n\) and \(d = \mathrm{dim}\,X\). Then there is an isomorphism \(H^i_{\mathrm{\'et}}(X,\mathbb{Z}(n)) \cong D^i(n) \oplus C^i(n) \oplus P^i(n)\) where 1. \(D^i(n)\) is uniquely divisible and \(= 0\) for \(i > \min\{2n,n+d\}\). 2. The prime-to-\(p\) torsion part \(C^i(n)\) is isomorphic to \(\bigoplus_{\ell \neq p}\mathbb{Q}_\ell/\mathbb{Z}_\ell^{r_\ell} \oplus F^i(n)\), invariant under extensions of algebraically closed fields, and vanishes unless \(1 \leq i \leq 2d+1\). The group \(F^i(n) = \prod_{\ell \neq p}\mathrm{Tor}\,H^i_{\mathrm{\'et}}(X,\mathbb{Z}_\ell)\) is a finite group independent of \(n\), and \(r_\ell \in \mathbb{N}_0\) is independent of \(n\) and \(\ell\) if \(n \neq \frac{i-1}{2}\). 3. The \(p\)-primary torsion part \(P^i(n)\) is isomorphic to \(\mathbb{Q}_p/\mathbb{Z}_p^s \oplus P^i_c(n)\), with \(P^i_c(n)\) an extension of a finite group \(F^i_p(n)\) by a finitely generated torsion \(W(k)\)-module \(U^i(n)\), and \(P^i(n) = 0\) unless \(n+1 \leq i \leq n+d+1\) and \(0 \leq n \leq d\). The proof proceeds by showing \(H^i_{\mathrm{\'et}}(X,\mathbb{Z}(n)) \otimes \mathbb{Q}/\mathbb{Z} = 0\), studying \(H^i_{\mathrm{\'et}}(X,\mathbb{Z}_\ell(n))\) for \(\ell \neq p\) (they are finitely generated \(\mathbb{Z}_\ell\)-modules, torsion-free for almost all \(\ell\) by \textit{O. Gabber} [C. R. Acad. Sci., Paris, Sér. I 297, 179--182 (1983; Zbl 0574.14019)]) and \(H^i_{\mathrm{\'et}}(X,\mathbb{Z}_p(n))\). Proposition 1.2. In Theorem 1.1, 1. \(F^i(n)\) and \(F^i_p(n)\) are Pontrjagin dual to \(F^{2d+1-i}(d-n)\) and \(F^{2d+1-i}_p(d-n)\), respectively. 2. If \(n,u \geq 0\), \(i \neq 2u+1\), and \(2d+2-i \neq 2n+1\), then the Tate module of \(C^{2d+2-i}(n)\) is Pontrjagin dual to \(C^i(u)/F^i(u)\). 3. \(U^i(n) \cong \mathrm{Hom}_{W(k)}(U^{2d+2-i}(d-n), CW(k))\) for \(CW(k)\) the co-Witt vectors. The proof uses Poincaré duality for \(\ell \neq p\) and \textit{J. S. Milne}'s duality [Am. J. Math. 108, 297--360 (1986; Zbl 0611.14020); addendum ibid. 137, No. 6, 1703--1712 (2015), Theorem 11.1] for the \(p\)-part. Theorem 1.3. Let \(k\) be a \(p\)-adic field of residue characteristic \(p\). Then \(H^i_{\mathrm{\'et}}(X,\mathbb{Z}(n))\) is the direct sum of a finite group and a group which is uniquely \(\ell\)-divisible for \(\ell \neq p\), if either \(X\) has good reduction and \(i \not\in \{2n-1,2n,2n+1,2n+2\}\), or if \(i \not\in \{n,\ldots,n+d+2\}\). Theorem 1.4. If \(X\) is smooth projective over \(C \setminus S\) for \(C\) the spectrum of the ring of integers of a number field or a smooth proper curve over a finite field and \(S \subseteq C\) a finite non-empty set of closed points, then \(\mathrm{III}^{3,1}(X) := \mathrm{ker}(H^i_{\mathrm{\'et}}(X,\mathbb{Z}(n)) \to \bigoplus_{v \in S}H^i(X \times K_v,\mathbb{Z}(n))\) is finite if and only if the Tate-Shafarevich group \(\mathrm{III}(\mathrm{Pic}^{0,\mathrm{red}}_{X_\eta/\eta})\) of the Picard variety of the generic fibre is finite.