Cohomology of \(p\)-adic Stein spaces. (English) Zbl 1475.11110
Let \(\mathcal{O}_K\) be a discrete valuation ring of mixed characteristic \((0,p)\), with residue field \(k\) and fraction field \(K\). Let \(C\) be the complete algebraic closure of \(K\).
The authors explain how to compute the pro-étale cohomology of the analytic space \(X_C\) associated to a “semistable Stein weak formal scheme” \(X\) over \(\mathcal{O}_K\). The main theorem is that \(H^r_{\text{pro-étale}}(X_C,\mathbb{Q}_p(r))\) can be determined by the following objects:
– (some part of) the overconvergent Hyodo-Kato cohomology of the reduction \(X \otimes_{\mathcal{O}_K} k\),
– closed \(r\)-forms on \(X_C\), and
– the de Rham cohomology \(H^r_{\mathrm{DR}}(X_C)\).
Indeed, \(H^r_{\text{pro-étale}}(X_C,\mathbb{Q}_p(r))\) is the pullback of a diagram of the form \[ \left(H_{\mathrm{HK}}^{r}(X_k)\otimes \mathbb{B}_{\mathrm{st}}\right)^{\substack{N=0\\ \varphi=p^r}} \to H^r_{\mathrm{DR}}(X_C) \leftarrow \Omega^{r}(X_C)^{d=0} \]
The pro-étale cohomology and the above mentioned gadgets are related by the syntomic cohomology. Via the period morphism, the pro-étale cohomology and syntomic cohomology can be identified after a truncation. The authors then introduce a Bloch-Kato type syntomic cohomology, which is more concrete, and prove the two syntomic theories are isomorphic.
The theorem is used to calculate the étale and pro-étale cohomology of Drinfeld half-spaces.
The authors explain how to compute the pro-étale cohomology of the analytic space \(X_C\) associated to a “semistable Stein weak formal scheme” \(X\) over \(\mathcal{O}_K\). The main theorem is that \(H^r_{\text{pro-étale}}(X_C,\mathbb{Q}_p(r))\) can be determined by the following objects:
– (some part of) the overconvergent Hyodo-Kato cohomology of the reduction \(X \otimes_{\mathcal{O}_K} k\),
– closed \(r\)-forms on \(X_C\), and
– the de Rham cohomology \(H^r_{\mathrm{DR}}(X_C)\).
Indeed, \(H^r_{\text{pro-étale}}(X_C,\mathbb{Q}_p(r))\) is the pullback of a diagram of the form \[ \left(H_{\mathrm{HK}}^{r}(X_k)\otimes \mathbb{B}_{\mathrm{st}}\right)^{\substack{N=0\\ \varphi=p^r}} \to H^r_{\mathrm{DR}}(X_C) \leftarrow \Omega^{r}(X_C)^{d=0} \]
The pro-étale cohomology and the above mentioned gadgets are related by the syntomic cohomology. Via the period morphism, the pro-étale cohomology and syntomic cohomology can be identified after a truncation. The authors then introduce a Bloch-Kato type syntomic cohomology, which is more concrete, and prove the two syntomic theories are isomorphic.
The theorem is used to calculate the étale and pro-étale cohomology of Drinfeld half-spaces.
Reviewer: Dingxin Zhang (Beijing)
MSC:
| 11F85 | \(p\)-adic theory, local fields |
| 14F20 | Étale and other Grothendieck topologies and (co)homologies |
| 14F30 | \(p\)-adic cohomology, crystalline cohomology |
| 20G25 | Linear algebraic groups over local fields and their integers |
| 22Fxx | Noncompact transformation groups |
| 14G20 | Local ground fields in algebraic geometry |
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