2023/08/03 by Bertrand Toën, Toen, Bertrand
Computer Science · Mathematics · Pharmacology, Toxicology and Pharmaceutics · #Advanced Algebra and Logic #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Alkaloids: synthesis and pharmacology #Data Management and Algorithms #FOS: Mathematics #Graph theory and applications #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2308.01670
openalex publication_date 2023/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In [CS01, Page 109] Grothendieck sketches the construction of a complex J_*(X) or commutative pro-algebraic groups, associated to a smooth variety X, and for which each Ji(X) is a product of local factors called the local generalized jacobians. The purpose of this note is to recast this construction in the setting of higher algebraic group stacks for the fppf topology. For this, we introduce a notion of algebraic homology associated to a scheme which is a universal object computing fppf cohomology with coefficients in group schemes. We endow this algebraic homology with a filtration by dimension of supports, and prove that, when X is smooth, J_*(X) appears as the E1-page of the corresponding spectral sequence. In a final part we partially extends our constructions and results over arbitrary bases.