## 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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### References:

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