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On the de Rham-Witt complex attached to a semi-stable family. (English) Zbl 0742.14015
Let $$K$$ be a complete discrete valuation field with integer ring $${\mathfrak O}_ K$$ and residue field $$k$$ ($$\hbox{ch}(K)=0$$, $$\hbox{ch}(k)=p>0$$ and $$k$$ is perfect). A proper semistable family $${\mathcal X}$$ over $${\mathfrak O}_ K$$ means that: $${\mathcal X}$$ is regular and $$X={\mathcal X}\otimes_{{\mathfrak O}_ K}K$$ is smooth over $$K$$; $$Y={\mathcal X}\otimes_{{\mathfrak O}_ K}k$$ is a reduced divisor with normal crossings in $${\mathcal X}$$.
In section 1 the author defines a “modified Hodge-Witt sheaf” $$W_ n\omega^ i_ Y$$. In section 2, one describes the structure of the “ modified de Rham-Witt complex” $$W\omega_ y^ \bullet$$ on $$Y_{et}$$, whose hypercohomology $$H^*(Y,W\omega_ Y^ \bullet)$$ is a finitely generated $$W(k)$$-module with Frobenius $$\varphi$$ and one defines an endomorphism $$N$$ on $$H^*(Y,W\omega_ Y^ \bullet)$$ such that $$N\circ\varphi=p\varphi\circ N$$ and $$N$$ is nilpotent on $$H^*(Y,W\omega_ Y^ \bullet)\otimes\mathbb{Q}$$. In section 3 one develops the Poincaré duality theory for the “modified Hodge-Witt sheaves”.

##### MSC:
 14F40 de Rham cohomology and algebraic geometry 14F30 $$p$$-adic cohomology, crystalline cohomology 14D10 Arithmetic ground fields (finite, local, global) and families or fibrations 14D07 Variation of Hodge structures (algebro-geometric aspects) 13D25 Complexes (MSC2000)
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##### References:
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