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”.


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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