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A cohomological construction of Swan representation over the Witt ring. I, II. (English) Zbl 0699.14026

From the introduction: Let \(K\) be a complete discrete valuation field with residue field \(k\). We assume \(k\) is a perfect field of characteristic \(p>0\). For a finite Galois extension \(M/K\) with Galois group \(G\), the Swan character \(\text{Sw}_ G: G\to\mathbb Z\) is defined as follows: \(\text{Sw}_ G(\sigma)=(1-v_ M(\sigma (\pi_ M)-\pi_ M))\cdot f\) for \(1\neq \sigma \in I\), \(\text{Sw}_ G(\sigma)=0\) for \(\sigma\) \(\not\in I\), \(\text{Sw}_ G(1)=-\sum_{1\neq \sigma \in G}\text{Sw}_ G(\sigma) \).
Here \(I\) denotes the inertia group, \(\pi_ M\) a prime element of \(M\), \(v_ M\) the normalized valuation of \(M\) and \(f\) the degree of the residue field extension. Then it is a classical result that \(\text{Sw}_ G\) is a character of a linear representation of \(G\) and that it can be defined over the \(\ell\)-adic field \(\mathbb Q_{\ell}\) \((\ell \neq p)\) (respectively the fraction field of the Witt ring \(W(k)\)). We call it the Swan representation of \(G\) and denote it by \(\text{Sw}_{G,\ell}\) (resp. \(\text{Sw}_{G,p}).\)
In this note we construct \(\text{Sw}_{G,p}\) cohomologically (or geometrically) when \(K\) is of equal characteristic \(p\). We use a new theory of de Rham-Witt complex with logarithmic poles, which supplies us nice \(p\)-adic cohomology for open varieties.
The content of this note is as follows. In part I (= §1-2) we introduce the de Rham-Witt complex with logarithmic poles, and construct \(\text{Sw}_{G,p}\) in part II (= §3).

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
12G05 Galois cohomology
13F35 Witt vectors and related rings
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