2013/08/02 by Peter Mannisto, Mannisto, Peter
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #14F20 #14G10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Alkaloids: synthesis and pharmacology #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1308.0472
openalex publication_date 2013/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define 1-motives of a variety X over a perfect field of positive characteristic which realize the etale cohomology groups of X in dimension and codimension one. This is the analogue in positive characteristic of previous results of Barbieri-Viale and Srinivas, except that we only consider the etale realization but also consider compactly supported cohomology. The dimension-1 case (called the Picard 1-motives) can be done by standard techniques, and indeed this case is probably well known. But the codimension-one case (Albanese 1-motive) requires stronger tools, namely a strong version of de Jong's alterations theorem and some cycle class theory on smooth Deligne-Mumford stacks which may be of independent interest. Unfortunately, we only succeed in defining the Albanese 1-motive for a variety X over an algebraically closed base field, and only up to isogeny. As a corollary to our definition of these 1-motives we deduce some independence of l results when X is a variety over a finite field.