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Caractéristique d’Euler-Poincaré des faisceaux constructibles sur une surface. (French) Zbl 0533.14007
Astérisque 101-102, 193-207 (1983).
The paper contains an outline of the analogue of the formula of Grothendieck-Ogg-Shafarevich [cf. A. Grothendieck, Sémin. Géom. algébr. 1965-66, SGA 5, Lect. Notes Math. 589, Exposé No.10, 372-406 (1977; Zbl 0356.14005)] for surfaces, due to Deligne. In other words: If U is an open dense set in a connected normal projective surface X over an algebraically closed field k and $${\mathcal F}$$ a locally constant sheaf of $${\mathbb{F}}$$-vector spaces over U, $${\mathbb{F}}$$ being a finite field of characteristic different from char(k), then a formula is given for the Euler-Poincaré characteristic (with compact supports) $$\chi_ c(U,{\mathcal F})$$ in terms of the Euler-Poincaré characteristics of U and of suitable open sets of the component curves of X-U, of the rank of $${\mathcal F}$$ and some Swan conductors. The main ingredient of the proof, of which a sketch is given, is the semicontinuity theorem for the Swan conductor on a family of curves by Deligne [cf. the author, Astérisque 82-83, 173-219 (1981; Zbl 0504.14013)] applied to a Lefschetz pencil on X.
For the entire collection see [Zbl 0515.00021].
Reviewer: H.Lange

##### MSC:
 14F20 Étale and other Grothendieck topologies and (co)homologies 14F45 Topological properties in algebraic geometry 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)