2018/03/07 by Vladimir Guletskiĭ, Guletskii, Vladimir · 1 citation
Mathematics · #14A20 #14C25 #14D23 #14J29 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1803.02907
openalex publication_date 2018/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let S be a Noetherian scheme, and let X be a scheme over S. Under mild assumptions, one can construct the connected infinite symmetric power \rm Sym∞ (X/S), whose group completion \rm Sym∞ (X/S)+ is an abelian group object in the category of set valued sheaves on the Nisnevich site over S. Viewing this completion as the space of relative 0-cycles on X/S, we construct the sheaf of Kähler differentials Ω1_\rm Sym∞ (X/S)+, and the tangent sheaf T_\rm Sym∞ (X/S)+. We prove that the category of étale neighbourhoods at a point P on the space of 0-cycles is cofiltered. Applying the stalk functor, we also obtain the stalk of the tangent sheaf at P, whose tensor product with the residue field is the needed tangent space to the space of 0-cycles at P.