Download or read book Tate Duality in Positive Dimension and Applications written by Zev Setchen Rosengarten and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In Part I, we generalize classical Tate duality (local duality, nine-term exact sequence, etc.) for finite discrete Galois modules (i.e., finite etale commutative group schemes) over global fields to all affine commutative group schemes of finite type (the "positive-dimensional" case), building upon recent work of Cesnavicius generalizing Tate duality to all finite commutative group schemes (the "zero-dimensional" case). We concentrate mainly on the more difficult function field setting, giving some remarks about the easier number field case along the way. In Part II, we give applications of this extension of Tate duality to the study of Picard groups, Tate-Shafarevich sets, and Tamagawa numbers of linear algebraic groups over global function fields.