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\(p\)-adic Hodge theory. (English) Zbl 0764.14012
Let \(V\) be a complete discrete valuation ring of characteristic 0 with perfect residue ring of characteristic \(p>0\), \(K\) be the field of fractions of the ring \(V\). In the paper a proof is presented for the hypothesis of the existence of a Hodge-Tate decomposition, i.e. the existence of a natural \(\text{Gal} \overline K/K\)-invariant isomorphism \[ H^ n_{et}(X_ K\otimes_ K\overline K,\mathbb{Z}_ p)\otimes_{\mathbb{Z}_ p}\hat{\overline K}\simeq\bigoplus_{a+b=n}H^ a(X_ K,\Omega^ b_{X_ K/K})\otimes_ K\hat{\overline K}(- b).\tag{*} \] Here, \(\hat{\overline K}\) is the \(p\)-adic completion of the algebraic closure \(\overline K\) of the field \(K\), “\((-b)\)” is the Tate twist, \(X_ K\) is a smooth proper scheme over the field \(K\). An analogue of this decomposition is also proved for open subvarieties in \(X_ K\), which are complements of a divisor with normal crossings. One of the steps of the proof is the case where \(X_ K=X\otimes_ VK\), \(X\) is a smooth proper scheme over the ring \(V\) (i.e., \(X_ K\) has good reduction). In this case, there exists an assertion exacter than \((*)\) the proof of which is based on the existence of a certain intermediate cohomology theory \({\mathcal H}^*(X)\). This theory is constructed with the help of a topology of the scheme \(X\) which is coarser than the étale topology of the general fiber of \(X_ K\) and consists of neighborhoods realized in the spectra of rings obtained by adjunction of the \(p^ n\)- th roots of invertible sections of the structure sheaf \({\mathcal O}_ X\). The closeness of étale cohomologies of the scheme \(X_{\overline K}=X\otimes\overline K\) and de Rham cohomologies of the scheme \(X_ K\) to the cohomologies \({\mathcal H}^*(X)\) follows from the property of the neighborhoods of étale topology of the general fiber of \(X_ K\) over neighborhoods of the new topology to be “almost étale”.

14F30 \(p\)-adic cohomology, crystalline cohomology
14G20 Local ground fields in algebraic geometry
14F20 Étale and other Grothendieck topologies and (co)homologies
14F40 de Rham cohomology and algebraic geometry
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