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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”.

##### MSC:
 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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