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:

 [1] G. Faltings: F-isocrystals on open varieties. Results and conjectures (1988) (preprint). · Zbl 0736.14004 [2] J.-M. Fontaine: Groupes de ramification et representations d’Artin. Ann. Sci. Ecole Norm Sup., 4, 337-392 (1971). · Zbl 0232.12006 [3] H. Gillet and W. Messing: Cycle classes and Riemann-Roch for crystalline cohomology. Duke Math. J., 55, 501-538 (1987). · Zbl 0651.14014 [4] A. Grothendieck (redige par I. Bucur) : Formule d’Euler-Poincare en cohomologie etale. SGA 5, Springer LNM n^\circ 589, pp. 372-406 (1977). · Zbl 0356.14005 [5] L. Illusie et M. Raynaud: Les suites spectrales associees au complexe de de Rham-Witt. Publ. Math. IHES, 57, 73-212 (1983). · Zbl 0538.14012 [6] K. Kato: The limit Hodge structures in the mixed characteristic case. Manuscript (1988). [7] N. M. Katz: Local-to-global extensions of representations of fundamental groups. Ann. Inst. Fourier, 36, 69-106 (1986). · Zbl 0564.14013 [8] J.-P. Serre: Sur la rationalite des representations d’Artin. Ann. of Math., 72, 406-420. · Zbl 0202.32803
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