2010/07/22 by Ahmed Abbes, Takeshi Saito, Abbes, Ahmed +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT
paper · pdf · doi:10.48550/arxiv.1007.3873
openalex publication_date 2010/07/22 · arxiv created 2011/10/23 · arxiv updated 2011/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This article is devoted to studying the ramification of Galois torsors and of ℓ-adic sheaves in characteristic p>0 (with ℓ\not=p). Let k be a perfect field of characteristic p>0, X be a smooth, separated and quasi-compact k-scheme, D be a simple normal crossing divisor on X, U=X-D, Λ be a finite local \mathbb Z_ℓ-algebra, F be a locally constant constructible sheaf of Λ-modules on U. We introduce a boundedness condition on the ramification of F along D, and study its main properties, in particular, some specialization properties that lead to the fundamental notion of cleanliness and to the definition of the characteristic cycle of F. The cleanliness condition extends the one introduced by Kato for rank one sheaves. Roughly speaking, it means that the ramification of F along D is controlled by its ramification at the generic points of D. Under this condition, we propose a conjectural Riemann-Roch type formula for F. Some cases of this formula have been previously proved by Kato and by the second author (T.S.).