Swan representations associated with rigid analytic curves (Q2755161)

(Review 1) NEWLINENEWLINENEWLINEThe classical construction of the Artin representation and Swan representation associated with henselian discretely valued fields of rank 1 with perfect residue field is generalized to a larger class of valued fields. This construction and a Lefschetz fixed point formula are applied to prove an Euler-Poincaré formula for the étale cohomology of rigid analytic curves. NEWLINENEWLINENEWLINE(Review 2) NEWLINENEWLINENEWLINEIn previous papers the author has introduced the concept of adic spaces, see in particular his monograph `Étale cohomology of Rigid Analytic Varieties and Adic Spaces' (Vieweg 1996; Zbl 0868.14010). To any rigid analytic variety \(X\) over a complete nonarchimedean field one can construct an adic space \(X^{ad}\); in addition to the classical points of \(X\), \(X^{ad}\) contains points whose residue field is equipped with a valuation of rank \(>1\). \newline To a finite Galois extension \(L/K\) of complete discretely valued fields one can associate the Artin and the Swan representation of \(G=\text{Gal}L/K\). In the first part of the present paper the author generalizes this construction to a certain class of fields with a valuation of higher rank which includes all valuations arising in adic spaces. As in the classical situation one constructs class functions \(a_G\) and \(sw_G\) which in the new situation turn out to be characters of virtual representations only. NEWLINENEWLINENEWLINEA central result of the paper is a Lefschetz fixed point formula for a finite morphism \(f:X \to X\) of a quasicompact smooth adic curve \(X\). Such a curve has a unique adic compactification, and to the formula contribute not only the fixed points of \(f\), but also the points in the completion (which all have residue fields with valuations of rank \(>1\)). \newline If \(F:Y \to X\) is a finite Galois covering of adic curves with Galois group \(G\), \(M \subset Y\) a locally closed constructible subset and \(L:=f^{-1}(M)\), then the Lefschetz fixed point formula can be applied to express the representations of \(G\) on the étale cohomology of \(L\) in terms of the Artin and Swan representations of \(G\). \newline In the last part of the paper \(F\) is a locally constant sheaf on the étale site of an adic curve \(X\) as above (or a slightly more general sheaf). The author proves an Euler-Poincaré formula for the restriction of \(F\) to a locally closed constructible subset \(L\) of \(X\). The main ingredient here is the Swan conductor of the restriction of \(F\) to the étale site of a point. The definition of this conductor is also generalized from discrete valuations to valuations of higher rank. Finally these results are compared to Ramero's results on meromorphically ramified sheaves [\textit{L. Ramero}, J. Alg. Geom. 7, No. 3, 405-504 (1998; Zbl 0964.14019)].