TAILIEUCHUNG - Đề tài " Ramification theory for varieties over a perfect field "

For an -adic sheaf on a variety of arbitrary dimension over a perfect field, we define the Swan class measuring the wild ramification as a 0-cycle class supported on the ramification locus. We prove a Lefschetz trace formula for open varieties and a generalization of the Grothendieck-Ogg-Shararevich formula using the Swan class. Let F be a perfect field and U be a separated and smooth scheme of finite type purely of dimension d over F . In this paper, we study ramification of a finite ´tale scheme V over U along the boundary, by introducing a map (). | Annals of Mathematics Ramification theory for varieties over a perfect field By Kazuya Kato and Takeshi Saito Annals of Mathematics 168 2008 33 96 Ramification theory for varieties over a perfect field By KAZuyA Kato and Takeshi Saito Abstract For an -adic sheaf on a variety of arbitrary dimension over a perfect field we define the Swan class measuring the wild ramification as a 0-cycle class supported on the ramification locus. We prove a Lefschetz trace formula for open varieties and a generalization of the Grothendieck-Ogg-Shararevich formula using the Swan class. Let F be a perfect field and U be a separated and smooth scheme of finite type purely of dimension d over F. In this paper we study ramification of a finite etale scheme V over U along the boundary by introducing a map below. We put CH0 V V lim CH0 Y V where Y runs compactifications of V and the transition maps are proper push-forwards Definition . The degree maps CH0 Y V Z induce a map deg CH0 V V Z. The fiber product V u V is smooth purely of dimension d and the diagonal Ay V V u V is an open and closed immersion. Thus the complement V u V Ay is also smooth purely of dimension d and the Chow group CHd V u V Ay is the free abelian group generated by the classes of connected components of V u V not contained in Ay. If U is connected and if V U is a Galois covering the scheme V u V is the disjoint union of the graphs rơ for ơ E G Gal V U and the group CHd V u V Ay is identified with the free abelian group generated by G 1 . The intersection of a connected component of V u V Ay with Ay is empty. However we define the intersection product with the logarithmic diagonal Ay log CHd V U V Ay --- CH0 V V Z Q using log product and alteration Theorem . The aim of this paper is to show that the map gives generalizations to an arbitrary dimension of the classical invariants of wild ramification of f V U. The image of the map is in fact supported on the wild ramification locus Proposition .

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